Representation Theory: Selected Papers

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES

Managing Editor: Professor I.M. James,Mathematical Institute, 24-29 St Giles,Oxford

I. General cohomology theory and K-theory, P.HILTON4. Algebraic topology, J.F.ADAMS5. Commutative algebra, J.T.KNIGHT8. Integration and harmonic analysis on compact groups, R.E.EDWARDS9. Elliptic functions and elliptic curves, P.DU VAL10. Numerical ranges II, F.F.BONSALL & J.DUNCANII. New developments in topology, G.SEGAL (ed.)12. Symposium on complex analysis, Canterbury, 1973, J.CLUNIE

& W.K.HAYMAN (eds.)13. Combinatorics: Proceedings of the British Combinatorial Conference

1973, T.P.McDONOUGH & V.C.MAVRON (eds.)15. An introduction to topological groups, P.J.HIGGINS16. Topics in finite groups, T.M.GAGEN17. Differential germs and catastrophes, Th.BROCKER & L.LANDER18. A geometric approach to homology theory, S.BUONCRISTIANO, C.P. BOURKE

& B.J.SANDERSON20. Sheaf theory, B.R.TENNISON21. Automatic continuity of linear operators, A.M.SINCLAIR23. Parallelisms of complete designs, P.J.CAMERON24. The topology of Stiefel manifolds, I.M.JAMES25. Lie groups and compact groups, J.F.PRICE26. Transformation groups: Proceedings of the conference in the University

of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI27. Skew field constructions, P.M.COHN28. Brownian motion, Hardy spaces and bounded mean oscillations,

K.E.PETERSEN29. Pontryagin duality and the structure of locally compact Abelian

groups, S.A.MORRIS30. Interaction models, N.L.BIGGS31. Continuous crossed products and type III von Neumann algebras,

A.VAN DAELE32. Uniform algebras and Jensen measures, T.W.GAMELIN33. Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE34. Representation theory of Lie groups, M.F. ATIYAH et al.35. Trace ideals and their applications, B.SIMON36. Homological group theory, C.T.C.WALL (ed.)37. Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL38. Surveys in combinatorics, B.BOLLOBAS (ed.)39. Affine sets and affine groups, D.G.NORTHCOTT40. Introduction to Hp spaces, P.J.KOOSIS41. Theory and applications of Hopf bifurcation, B.D.HASSARD,

N.D.KAZARINOFF & Y-H.WAN42. Topics in the theory of group presentations, D.L.JOHNSON43. Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT44. Z/2-homotopy theory, M.C.CRABB45. Recursion theory: its generalisations and applications, F.R.DRAKE

& S.S.WAINER (eds.)46. p-adic analysis: a short course on recent work, N.KOBLITZ47. Coding the Universe, A.BELLER, R.JENSEN & P.WELCH48. Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds.)

49. Finite geometries and designs, P.CAMERON, J.W.P.HIRSCHFELD& D.R.HUGHES (eds.)

50. Commutator calculus and groups of homotopy classes, H.J.BAUES51. Synthetic differential geometry, A. KOCK52. Combinatorics, H.N.V.TEMPERLEY (ed.)53. Singularity theory, V.I.ARNOLD54. Markov processes and related problems of analysis, E.B.DYNKIN55. Ordered permutation groups, A.M.W.GLASS56. Journees arithmetiques 1980, J.V.ARMITAGE (ed.)57. Techniques of geometric topology, R.A.FENN58. Singularities of smooth functions and maps, J.MARTINET59. Applicable differential geometry, F.A.E.PIRANI & M.CRAMPIN60. Integrable systems, S.P.NOVIKOV et al.61. The core model, A.DODD62. Economics for mathematicians, J.W.S.CASSELS63. Continuous semigroups in Banach algebras, A.M.SINCLAIR64. Basic concepts of enriched category theory, G.M.KELLY65. Several complex variables and complex manifolds I, M.J.FIELD66. Several complex variables and complex manifolds II, M.J.FIELD67. Classification problems in ergodic theory, W.PARRY & S.TUNCEL68. Complex algebraic surfaces, A.BEAUVILLE69. Representation theory, I.M.GELFAND et. al.70. Stochastic differential equations on manifolds, K.D.ELWORTHY

London Mathematical Society Lecture Note Series : 69

Representation Theory

Selected Papers

I.M.GELFAND M.I.GRAEVI.N.BERNSTEIN V.A.PONOMAREVS.I.GELFAND A.M.VERSHIK

CAMBRIDGE UNIVERSITY PRESS

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www.cambridge.orgInformation on this title: www.cambridge.org/9780521289818

With the exception of Two Papers on Representation Theory, Introduction to"Schubert Cells and Cohomology of the Spaces G/P" and Four Papers onProblems in Linear Algebra, the original Russian versions of these papers(which were first published in Uspektin Matematicheskikh Nauk.) are © VAAPand the English translations are © The British Library

This collection © Cambridge University Press 1982

First published 1982

Re-issued in this digitally printed version 2008

A catalogue record for this publication is available from the British Library

Library of Congress Catalogue Card Number: 82-4440

ISBN 978-0-521-28981-8 paperback

CONTENTS

Two papers on representation theoryG.SEGAL 1

Representations of the group SL(2,R), where R is aring of functions (RMS 28:5 (1973) 87-132)A.M.VERSHIK, I.M.GELFAND & M.I.GRAEV 15

Representations of the group of diffeomorphisms(RMS 30:6 (1975) 1-50)A.M.VERSHIK, I.M.GELFAND & M.I.GRAEV 61

An introduction to the paper "Schubert cells andcohomology of the spaces G/P"G.SEGAL 111

Schubert cells and cohomology of the spaces G/P(RMS 28:3 (1973) 1-26)I.N.BERNSTEIN, I.M.GELFAND & S.I.GELFAND 115

Four papers on problems in linear algebraC-M.RINGEL 141

Coxeter functions and Gabriel's theorem(RMS 28:2 (1973) 17-32)I.N.BERNSTEIN, I.M.GELFAND & V.A.PONOMAREV 157

Free modular lattices and their representations(RMS 29:6 (1974) 1-56)I.M.GELFAND & V.A.PONOMAREV 173

Lattices, representations and algebras connected withthem I (RMS 31:5 (1976) 67-85)I.M.GELFAND & V.A.PONOMAREV 229

Lattices, representations and algebras connected withthem II (RMS 32:1 (1977) 91-114)I.M.GELFAND & V.A.PONOMAREV

TWO PAPERS ON REPRESENTATIONTHEORY

Graeme Segal

These two papers are devoted to the representation theory of two infinitedimensional Lie groups, the group SL2(R)* of continuous maps from a spaceX into SL2(R), and the group Diff(X) of diffeomorphisms (with compactsupport) of a smooth manifold X. Almost nothing of a systematic kind isknown about the representations of infinite dimensional groups, and the mathe-matical interest of studying these very natural examples hardly needs pointingout.

Nevertheless the stimulus to the work came from physics, and I shall try toindicate briefly how the representations arise there. Physicists encountered notthe groups but their Lie algebras, the algebra $x of maps from X to the Liealgebra <j of SL2 (R), and the algebra Vect(JT) of vector fields on X. The spaceX is physical space R3. Choosing a basis in g, to represent $x is to associatelinearly to each real-valued function/on R3 three operators /,-(/) (/ = 1, 2, 3),such that

Vt(f),J,(g)] = 2 cUkJk(f8)>k

where c^k are the structural constants of $ . In quantum field theory one

writes Jt(f) as f(x)ji(x)dx, where /,- is an operator-valued distribution.R3

Then the relations to be satisfied are

where 5 is the Dirac delta-function.Similarly, to represent Vect(R3) is to associate operators P(f) to vector-

valued functions/: R3 -• R3 so that [P(f),P(g)] =P(h), where

h= -g.K

Graeme Segal

Writing P(f) = 2 ] fi{x)Pi(x)dx this becomes

lPi(x)9 Pjiy)} = dfix -y)Pj(y) - 8f(x -y)Pi(x), (**)

where dk = 35 /dxk.Operators with the properties of jf(x) and pt(x) arise commonly in quantum

field theory in the guise of "current algebras". For example, if one has a com-plex scalar field given by operators \p(x) (for JC GR3) which satisfy either com-mutation or anticommutation relations of the form [\IJ*(X)9 ]p(y)]± = 8(x —y),then the "current-like" operators pfa) defined by

OXj

satisfy (**). Similarly if one has an TV-component field \p satisfying± = 8a!38(x ~y)> an(* ol9 . . .,an BIQNX N matrices representing

* (the generators of a Lie algebra a then the operators jt{x) -satisfy (*). (These examples are taken from [3].)

In connection with the quantization of gauge fields it is also worth mention-ing that, as we shall see below, the most natural representation of the group ofall smooth automorphisms of a fibre bundle is its action on L2 (E), where E isthe space of connections ("gauge fields") in the bundle, endowed with aGaussian measure.

Representations of the group SL(2, R ) * .

This paper is concerned with the construction of a single irreducible unitaryrepresentation of the group Gx of continuous maps from a space X equippedwith a measure into the group G = SL2(R). (In this introduction I shall always

think of G as SUX 1? i.e. as the complex matrices I ~ - I such that

\a\2-\b\2 = \.)An obvious way of obtaining an irreducible representation of Gx is to

choose some point x of X and some irreducible representation of G byoperators {Ug}g(EG on a Hilbert space//, and to make Gx act onHthroughthe evaluation-map at x, i.e. to make/E Gx act on H by U^xy Thisrepresentation can be regarded as analogous to a "delta-function" at x. Moregenerally, for any finite set of points JCJ , . . ., xn in X and correspondingirreducible representations g •-> U^ of G on Hilbert spaces Hx , . . . , / / „ onecan make G act irreducibly on the tensor product H1 ® • • • (8) Hn by assigningt o / G Gx the operator lA)\ . 0 •.. ® £/(«) The object of the paper is to

generalize this construction and produce a representation on a "continuoustensor product" of a family of Hilbert spaces {Hx } indexed by the points of X(and weighted by the measure on X). There is a simple criterion for deciding

Two papers on representation theory 3

whether a representation is an acceptable solution of the problem, in view ofthe following remark. For any representation U of Gx and any continuous map0: X -+ X there is a twisted representation <j>*U given by (0*C/)/ = U^. Therepresentation to be constructed ought to have the property that 0*£/isequivalent to U whenever 0 is a measure-preserving homeomorphism of X, i.e.for each such 0 there should be a unitary operator T such that ty0 = r 0 UfT^1.

The paper describes six different constructions of the representation, butonly three are essentially different. Of these, one, described in §4 of the paper,is extremely simple, but not very illuminating because it is a construction aposteriori. I shall deal with it first. For any group F and any cyclic unitaryrepresentation of F on a Hilbert space H with cyclic vector £ E H ("cyclic"means that the vectors Uy £, for all y E F, span a dense subspace of H) one canreconstruct the Hilbert space and the representation from the complex-valuedfunction y •-> ^(7) = < £, Uy% > on F. To see this, consider the abstract vectorspace Ho whose basis is a collection of formal symbols Uy% indexed by 7 E F.An inner product can be introduced in Ho by prescribing it on the basiselements:

The group F has an obvious natural action on Ho, preserving the inner pro-duct. Then H is simply the Hilbert space completion of Ho. The function îs called the spherical function of the representation corresponding to £ E/7.

In our case the group F = Gx has an abelian subgroup Kx, where K - SO2

is the maximal compact subgroup of G, and it turns out that the desiredrepresentation H contains (up to a scalar multiple) a unique unit vector £invariant under Kx. The corresponding spherical function is easy to describe.The orbit of £ can be identified with Gx jKx, i.e. with the maps of X intoG/K, which is the Lobachevskii plane. (I shall always think of G/K as the openunit disk in C with the Poincare metric.) Given two maps/j, f2 : X -• G/Kthe corresponding inner product is

exp j logsechpCflxx), f(x2))dx,

where p is the G-invariant Lobachevskii or Poincare metric on G/K. This meansthat the spherical function ^ is given by

where, if g =( - - I E G, then \p(g) = | a I"1. To see that this construction does\b a/

define a representation of GA the only thing needing to be checked is that theinner product is positive. That is done in §4.2. But of course it is not clear

4 Graeme Segal

from this point of view that the representation is irreducible.A more illuminating construction of the representation is to realize the con-

tinuous tensor product as a limit of finite tensor products. To do this weactually represent the group of L1 maps from X to G, i.e. the group obtainedby completing the group of continuous maps in the L1 metric (cf. §3.4). TheL1 maps contain as a dense subgroup the group of step-functions X -> G, andit is on the subgroup of step-functions that the representation is concretelydefined.

If one is to form a limit from the tensor products of increasing numbers ofvector spaces then the vector spaces must in some sense get "smaller". It hap-pens that the group SL2(R) has the comparatively unusual property (cf. below)of possessing a family (called the "supplementary series") of irreduciblerepresentations//^ (where 0 < X < 1) which do in a certain sense "tend to"the trivial one-dimensional representation as X -* 0. Furthermore there is an iso-metric embeddingHx+tl ->Hx<8>Hp whenever X + n < 1. Now for any partitionv of X into parts Xx, . . ., Xn of measures Xj, . . ., Xn one can consider the groupGv of those step-functions X -> G which are constant on the steps Xt. Thegroup G acts on :W = Hx 0 • • • ® Hx ; and when a partition v' is a refinement

i n

of v then #?v is naturally contained in :Jfv<. Accordingly, the group U Gv of allV

step-functions acts on UJfy, and the desired representation is the completionV

of this.The construction just outlined is carried out in §2 of the paper. A variant

is described in §3, where the representations {Hx} of the supplementary seriesare replaced by another family {Lx} with analogous properties — the so-called"canonical" representations. These are cyclic but not irreducible, and Lx con-tains Hx as a summand. In terms of their spherical functions Lx tends to Hx asX -• 0. The spherical function \px of Lx is very simple, given by yjx(g)= \a \~x

when g = I — — \. In other words, Lx is spanned by vectors £M indexed by u in

the unit disk G/K, and < £w, £M > > = sechxp(w, u'). Notice that the size of thegenerating G-orbit {£M} in Lx tends to 0 as X -• 0.

The remaining constructions exploit a quite different idea, which is useful inother situations too, as we shall see. I shall explain it in general terms.

Gaussian measures on affine spaces

Suppose that a group T has an affine action on a real vector space E with aninner product; i.e. to each y G T there corresponds a transformation of E of theform v »-> T(y)v + £(7), where T(y): E -* E is linear and orthogonal, andP(y) E E. Then there is an induced unitary action of F on the space L2 (E) offunctions on E which are square-summable with respect to the standardGaussian measure e ""v" * dv. Because this measure is not translation-invariant

Two papers on representation theory

we have to define Uy: L2 (E) -» L2 (E) bv

(Ky-t/)

where the factor

3> (l/) = ei i g Jis to achieve unitarity.

The importance of this construction is that the representation of F on L2 (E)may be irreducible even when the underlying linear action on E by 7 »-> 7X7) ishighly reducible. If the linear action Tis given then the affine action isevidently described by the map |8: T-+E. This is a "cocycle", i.e.0(77') = 18(7) + 7X7)0(7'), and it is easy to see that the affine space is preciselydescribed up to isomorphism by the cohomology class of 0 in H1 (F; E). Onesometimes speaks of "twisting" the action of F on L2(E) by means of j3.

Apart from the description just given there are two other useful ways oflooking at L2 (E). The first of these is as a "Fock space". For the Gaussianmeasure on E the polynomial functions are square-summable, and are dense inL2(E). So L2(E) can be identified with the Hilbert space completion of thesymmetric algebra S(E) of E. (A little care is necessary here: to make thenatural inner product in S(E) correspond to the Gaussian inner product inL2(E) one must identify Sn (E) not with the homogeneous polynomials on Eof degree n, but with the "generalized Hermite polynomials" of degree n.)

The other way of approaching L2 (E) is to observe that it contains (and isspanned by) elements e° for each yG£, with the property that

<ev, ev') = e{v>v>). . . . (f)

This means that L2(E) can be obtained from the abstract free vector spacewhose basis is a set of symbols {ev} V(=E by completing it using the inner pro-duct defined by (f). Better still, one can start with symbols e and define

(e e .) = e - | l l w - w ' l l 3 •

this makes it plain that the construction uses only the affine structure of E.(Of course ev = e~\ l | u | l V.)

The group SUX t j acts on the circle Sx, and has a very natural affine actionon the space EK of smooth measures on S1 with integral X. Ex is a coset of thevector space Eo of smooth measures with integral 0. The invariant norm in Eo

is given by

nn>0

2

when a = V aneinedd.

Then L2{EX) is the "canonical representation" LX2 mentioned above. (This isstated, not quite precisely, as Theorem (7.1) of the paper.) For if o^ is theDirichlet measure on S1 corresponding to u in the unit disk G/K (i.e. 0^ is the

5 Graeme Segal

transform of dd by any element of G which takes 0 to u) then Xe E Ex and

1|| Xc -X<vll2 = X2 log coshp(u, u').

Returning to the group we are studying, Gx has an affine action (pointwise)on Ex, the space of maps X -» Ex. (Notice that the linear action of Gx on Ex

is highly reducible.) The space L2(EX) has the appropriate multiplicative pro-perty with respect to X: for if X is a disjoint union X = X1 U X2 then£xx u i 1 = Exx x £x2 a n d L2 ^x, vx2) = L2Êx1)(^L2 ^£X2 } I n t h e p a p e r

the equivalence of the representations on L2(EX) and on the continuous tensor

product of § § 2 and 3 is proved by calculating the spherical functions, but it isquite easy to give an explicit embedding of the continuous tensor product inL2(Ex). For if Y is a part of X with measure X then the map E\ -*E^X given

by /»-• X~* J / i s compatible with the Gaussian measures, so that L2(E^jx)Y

is a subspace of L 2 (Ej), and for any par t i t ion X = Xx U . . . U I n wi thm(Xt) = Xf we have

CL2(E?1) ®- - -® L2(E?n)

Indeed it is pointed out in [10] that for any affine space E the space L2(EX)can always be interpreted as a continuous tensor product of copies of L2{E)indexed by the points of X.

The irreducibility of the representation can be seen very easily in the Fockversion. For the cocycle 0 vanishes on the abelian subgroup Kx, and so underKx the representation breaks up into its components Sn (Ex), on which Kx

acts just by multiplication operators. The characters of Kx which arise are alldistinct, so the irreducibility of the representation follows from the fact thatthe vacuum vector is cyclic, which is easily proved (cf. §5.2).

The three approaches to L2 (Ex) are described in §§5,6 and 7 of the paper.In connection with § 6 notice that to give an affine action of F on a vectorspace E is the same thing as to give a linear action on a vector space H togetherwith an invariant linear form /: H-• R such that I'1 (0) = E: the affine space isthen I'1 (1). (There is no point, in §6, in considering functions/: X -*H otherthan those satisfying l(f(x)) = 1, and the formulae become less cumbersomeunder that assumption.)

§5 describes the Fock space version, but not quite in the standard form. Thespace E1 of measures on the circle can be identified (by Fourier series) with aspace of maps Z -• C. Accordingly Ex is a space of maps I X Z -> C, and thesymmetric power Sk (Ex) is a space of maps


I x . . . x I x Z x . . . x Z->C« - k - « - * - * .

which are symmetric in the obvious sense. The effect of this point of view is toidentify L2 (Ex) with a space of functions on the free abelian group generatedby the space X, i.e. on the space whose points are "virtual finite subsets"XrtfXj of X, with ni G Z. This is intriguing, but whether it is more than acuriosity it is hard to say.

That concludes my account of the contents of the paper itself; but I shallmention some related matters. The most obvious question to ask is what classof groups G the method applies to. As it stands it evidently does not work forgroups for which the trivial representation is isolated in the space of allirreducible representations. This excludes all compact groups, as for them theirreducible representations form a discrete set. The isolatedness of the trivialrepresentation has been cleverly investigated by Kazhdan [5], who proved inparticular that among semisimple groups the trivial representation is isolatedif the group contains SL3(R) as a subgroup. The only simple groups notexcluded by Kazhdan's criteria are SOW1 and SUW1 - notice thatPSL2(R) = SO2)1 and PSL2(C) = SO3j . For these the method works just asfor SL2(R). (For example SOW1 is the group of all conformal transformationsof Sn ~l, and the affine space Ex used above can be replaced by the space ofmeasures on Sn ~x with integral X.)

A class of groups for which the trivial representation is not isolated consistsof the semidirect products GxV, where G is a compact group with anorthogonal action on a real vector space V. Indeed if 12 is a G-orbit in V anelement gGG acts naturally on L2 (£2), and u G F can be made to act bymultiplication by the function el{ViU} \ When the compact orbit £2 is close tothe origin in V the representation Z,2(£2) is close to the trivial representation(in the sense of its spherical function). Furthermore GxV has an obviousaffine action on V: the induced action on L2 (V) is the direct integral of theirreducible representations L2(£l) for all orbits fiCF.

Thus the methods of the paper apply to all groups of the form GxV. Theimportance of this is that it provides a way of constructing a representationof the group (Gx )sm of smooth maps from a manifold X to a compact groupG. For a smooth map / : X -> G induces a map of tangent bundles Tf: TX ->• TG,and this can be regarded as a map which to each point x EX assigns a"l-jet"/(x) <EJX G = GH (T*X ® 0) where $ is the Lie algebra of G. Asthe groups Jx G is of the form Gx V the method of the paper provides arepresentation of the group F of bundle maps TX^TG. (The fact that Jx Gdepends on x, giving rise to a bundle of groups on X, is not important.) Fcontains (Gx)sm as a subgroup, and it turns out that the representationconstructed remains irreducible when restricted to (Gx ) s m , at least whendim(X) > 4. That is proved in the papers [ 1 ] and [2].

It is interesting to notice that the group of bundle maps F is just the semi-

8 Graeme Segal

direct product (Gx )sm H £2* (X; g), where SI1 (X; g ) is the space of 1-formson X with values in g ; and the associated affine space E is the space of con-nections in the trivial G-bundle on X. The fact that the space of therepresentation of (Gx)sm is L2 (E) is, of course, suggestive from the pointof view of gauge theories in physics.

Representations of the group of diffeomorphisms

This paper is devoted to the representation theory of the group Diff(X)of diffeomorphisms with compact support of a smooth manifold X. (Adiffeomorphism has compact support if it is the identity outside a compactregion.)

The most obvious unitary representation of Diff(X) is its natural action onH= L2 (X), the space of square-summable ^-densities on X. (By choosing asmooth measure m onX one can identify L2 (X) with the usual space offunctions/on X which are square-summable with respect to m. Then the actionof a diffeomorphism \p o n / will be/*-*/, where

f{x)= J){x)* fW~lx)

and J^ (x) = dm(\p ~l x)/dm(x). But it is worth noticing that L2 (X) is canoni-cally associated to X, and does not involve m.)

From H a whole class of irreducible representations of Diff(X) can beobtained by the well-known method introduced by Weyl to construct therepresentations of the general linear groups. For any integer n the symmetricgroup Sn acts on the «-fold tensor product R®n = H ® . • • ® H by permut-ing the factors, and the action commutes with that of Diff(X). It turns out thatunder Diff(X) x Sn the tensor product decomposes

p

Vp ® Wp,

where {Wp} is the family of all irreducible representations of Sn, and Vp is acertain irreducible representation of Diff(X). More explicitly, Vp is the spaceof L2 functions X x . . . x X -+ Wp which are equivariant with respect to Sn :

«— n —*•

thus it makes sense even when p is not irreducible, and Vp ®p' = Vp © Vp'.The class of representations { Vp}, which were first studied by Kirillov, isclosed under the tensor product: if p and a are representations of Sn and Sm

then Vp ® Va = Vp'a, where p • o is the representation ofSn+m inducedfrom p (g) o. All of this is explained in § 1 of the paper.

It is then natural to ask, especially when X is not compact, whether newrepresentations of Diff(Z) can be constructed by forming some kind of infinitetensor product H®°° and decomposing it under the infinite symmetric groupS°° of all permutations of {1, 2, 3 , . . .}. This question is the main subject of thepaper, and it is considered in the following way.

The L2 functions Xn -• Wp are the same as those Xn -+ Wp where


Xn C Xn is the space of ^-triples of distinct points. The symmetric group Sn

acts on Xn, and the quotient space is B^n\ the space of «-point subsets of X.Diff(X) acts transitively on B^n\ and there is a unique class of quasi-invariantmeasures on it. The representation Vp can be regarded as the space of sectionsof a vector bundle on B^ whose fibre is Wp. An appropriate infinite analogueof B^ is the space Tx of infinite "configurations" in X, i.e. the space ofcountable subsets y of X such that y n K is finite for every compact subsetK of X. This space, and the probability measures on it, play an important rolein both statistical mechanics and probability theory. One can imagine thepoints of a configuration as molecules of a gas filling X, or as faulty telephones.

Diff(X) does not act transitively on Tx : two configurations are in the sameorbit only if they coincide outside a compact region. Nevertheless one candefine (in many ways) measures on Fx which are quasi-invariant and ergodicunder Diff(X). For each such measure fj. there is an irreducible representation1/^ of Diff(X) on L2 (Tx ; JU). More generally, for each representation p of afinite symmetric group Sn there is an irreducible representation Up : it is thespace of sections of the infinite dimensional vector bundle on Tx whose fibreis the representation Hp of S°° induced from the representation p 0 1 ofSn x S™. (S™ denotes the subgroup of permutations in S°° which leave1, 2, . . ., n fixed.) More explicitly, one can consider a covering space Tx n

of Yx defined by

Fx n is locally homeomorphic to Tx, and therefore a measure JJL on Fx definesa measure ju, the "Campbell measure", on Tx n . The space of therepresentation Up is the space of maps Fx n -• Wp which are Sn -equivariant andsquare-summable for ju.

The simplest and most important measures on Tx are the Poisson measuresHx (parametrized by X > 0), for which the measure of the set

/ > \ n{y E Tx:card(7 n K) = n } is ( —j- I e ~Km, where m is the measure of K. More

can be said about the representations £/£ = Up in the Poisson case:(i) They form a closed family under the tensor product, and have the follow-

ing simple behaviour(a) U£=UK® Vp, and(b) Ux®Uy = Ux + x,.

(ii) Ux is what is called in statistical mechanics an "N/V limit". In otherwords, if X is the union of an expanding sequence X1 C X2 C X3 C . . . ofopen relatively compact submanifolds such that XN has volume X"1 TV thenL2 (Tx; pix) is the limit as TV -> °o of the spaces L2

ym ((XN )^) of symmetric L2

functions of TV points in XN. (This is explained in [4], [7], [8].)(hi) Ux has a more concrete realization as L2(EX), where Ex is an affine

space with a Gaussian measure (and an affine action of Diff(X)). Ex is the space

10 Graeme Segal

of |-densities/on X which are close to the standard Lebesgue \-densityfx — (X dx)7 as x -* °°, in the sense that f~fx belongs to H = L2 (X). This is anaffine space associated to the vector space H and the cocycle j3: Diff(X) -> Hgiven by

where J (x) = dm(ty ~x x)/dm(x) as before. As we have seen when discussingthe representations of Gx, L2(Ex) can also be regarded as a Fock spaceS(H) = ® L2 (Xn), but with the natural action of Diff(Z) twisted by the

n>0

cocycle j3. Because /3 vanishes on the subgroup Differ, m) of measure-preservingdiffeomorphisms we see that in the Poisson case the representations associatedto infinite configurations break up and give us nothing new when restricted toDiffGT, m).

In the paper the equivalence of L2 (Ex) and L2 (Fx) is proved by consideringthe spherical functions, but it can also be described explicitly as a sequence ofmaps L2

sym (Xn )-*L(Tx). In fact L2 (X) -* L(TX) takes X*/to the function

[ fix)dx9

while Is2

ym (X x X) -> L2 {Yx) takes X/ to

f(x,y)dxdy,

and so on.The fact that there is a Gaussian realization of the representation is closely

connected with the property of the Poisson measure /xx called "infinitedivisibility". The latter means that if X is the disjoint union of two pieces Xx

and X2, so that Fx = Fx x Fx up to sets of measure zero, then

Hx = JJLX^ x Mx2\ where JJL^ is the projection of JJLK on Fx.. This implies that

when the representation Ux of Differ) is restricted to the subgroupDiffer) x Diff(.r2) it becomes U[xi) <8> U[x*}, a property which mustcertainly be possessed by a construction of the type of L2(EX).

The reader may at first be confused by the fact that the affine action onL2 (Ex) used in this paper is the Fourier transform of the natural one used inthe paper on Gx. Perhaps it is worth pointing out explicitly that if a group Gacts orthogonally on a real vector space H with an inner product, andj3: G -> H is a cocycle, and L2 (H) is formed using the standard Gaussianmeasure, then the following two unitary actions of G on L2{H) are unitarilyequivalent

(a) gÂg, where (Ag<t>)(h) = e^h^ ~^h~Hm* <t>(g-l(h~l


(b) g »Bgi where (Bg<p)(h) = ei{P^>h) (j>(g~l h).

The automorphism of L2(H) relating them is characterized by

e{a,h)-l\\a\\i ^êi{a,h)

for all a^H. The important thing to notice about it is that it takes polynomialsto polynomials.

I shall conclude this account by drawing attention to the matters treatedrather sketchily in Appendix 2, as I think they are interesting and deserve tobe investigated further. The representations we obtained from Tx were con-structed from a particularly simple family {Hp } of irreducible unitary repres-entations of the uncountable discrete group S°°. But the group which seemsmore obviously relevant — because a diffeomorphism with compact supportcan move only finitely many points of a configuration — is the countablegroup S^ of the permutations of the natural numbers which leave almost allfixed. The representations/^ restrict to irreducible representations of S^ ;but most representations of S^ , notably the one-dimensional sign repres-entation, do not extend to S°°. (There is a natural compact convex set ofprimary representations of S^ which has been elegantly described by Thoma[9]. It is the family of all primary representations which admit a finite trace.It contains the trivial representation, the sign representation, and the regularrepresentation. All members are of type II except for the two one-dimensional representations.) Menikoff [8] has constructed a representation ofDiff(X) corresponding to the sign representation of S^ as an N/V limit of thefermionic space £^.ew ((SN )N) of antisymmetric functions of N particles in XN.

Can one construct a representation of Diff(X) corresponding to any unitaryrepresentation HofSÎK possible method is described in Appendix 2. Let uschoose an arbitrary rule for ordering thej>oints of each configuration 7 E F^ •This gives us a map s: Fx -* X°° (where X°° is the space of orderedconfigurations), which clearly cannot be continuous. We require of the order-ing only "correctness": if 7 and 7' differ only in a compact region then thesequences s(y) and 5(7') are required to coincide after finitely many terms.Consider the subspace As = S^ ms(Tx) of X°°. It is invariant underDiff(X) x S^, and AJS^ = Tx. It was proved in §2.3 of the paper that forany quasi-invariant ergodic measure /LI on Fx there is a quasi-invariant ergodicmeasure jl o n A r Then the space of 5^-invariant maps As -> H which aresquare-summable with respect to £ affords a unitary representation of Diff(X)associated to (n, H, s). The extent of its dependence on the arbitrary andinexplicit choice of s is rather unclear, as is its relation to the N/V limit of thephysicists. But the method does seem to produce, at least, a large supply oftype II representations of Diff(X).

Vershik and Kerov [11] have proved that Thoma's family of representationscan be obtained as limits of finite-dimensional representations of the finitesymmetric groups. (One associates representations ofSn+1 to representations

12 Graeme Segal

of Sn by induction.) I imagine that this description should permit one both toconstruct the corresponding representations of Diff(X) as N/V limits, and todescribe them in terms of a Gaussian measure.

Note. The definition of the topology of Tx given in the paper does not seemquite correct. One method of obtaining it is as follows.

The topology on the space Bx of finite configuration in X is obvious anduncontroversial. For a connected open manifold X the connected componentsof Bx are the B^ for n- 0, 1, 2, . . . If Y is an open relatively compact sub-manifold of X let us topologize Ty as a quotient space of Bx. Then if Y isconnected so is Ff. Now define Tx as lim Ty, where Y runs through all open

relatively compact submanifolds of X. This means that a configuration movescontinuously precisely when it appears to move continuously to every observerwith a bounded field of vision. ^

An alternative definition is: F^ has the coarsest topology such that / : F^ -* Ris continuous for every continuous function/: X-> R with compact support,where f(y)=

It is easy to see (cf. [6]) that F^ is metrizable, separable, and complete. Onthe other hand if X°° is given the product topology then the map X°° -> Tx isnot continuous, and I do not see how to obtain F^ as a quotient space frornâsensible topology on X°°. (It does not seem, however, that the topology on X°°plays a significant role in the paper.)

In conclusion, notice that the fundamental group of Tx is S°°; but of courseFjf is not locally simply connected. The map TXn -* F^, for example, is a localhomeomorphism, but not a locally trivial fibration.

References

[1] I. M. Gelfand, M. I. Graev and A. M. Versik, Representations of the group of smoothmappings of a manifold X into a compact Lie group. Compositio Math., 35 (1977),299-334.

[2] I. M. Gelfand, M. I. Graev and A. M. Versik, Representations of the group of functionstaking values in a compact Lie group. Compositio Math., 42 (1981), 217-243.

[3] G. Goldin, Non-relativistic current algebras as unitary representations of groups. J.Math. Phys., 12 (1971), 462-488.

[4] G. Goldin, K. J. Grodnik, R. Powers and D. Sharp, Non-relativistic current algebras inthe # / F limit. J. Math. Phys., 15 (1974), 88-100.

[5] D. A. Kazhdan, The connection of the dual space of a group with the structure of itsclosed subgroups. Functsional. Anal, i Prilozhen 1 (1967), 71—74.= Functional Anal. Appl. 1 (1967), 63-66.

[6] K. Matthes, J. Kerstan and J. Mecke, Infinitely divisible point processes. John Wiley,1978.

[7] R. Menikoff, The hamiltonian and generating functional for a non-relativistic localcurrent algebra. J. Math. Phys., 15 (1974), 1138-1152.


[8] R. Menikoff, Generating functionals determining representations of a non-relativisticlocal current algebra in the N/V limit. J. Math. Phys., 15 (1974), 1394-1408.

[9] E. Thoma, Die unzerlegboren, positiv-definiten Klassenfunktionen der abzahlbarunendlichen symmetrischen Gruppe. Math. Z., 85 (1964), 40—61.

[10] A. M. Vershik, I. M. Gelfand and M. I. Graev, Irreducible representations of the groupGx and cohomology. Functsional. Anal, i Prilozhen., 8 (1974), 67—69.= Functional Anal. Appl., 8 (1974), 151-153.

[11] A. M. Vershik and S. V. Kerov, Characters and factor-representations of the infinitesymmetric group. Doklady AN SSSR 257 (1981), 1037-1040.

REPRESENTATIONS OF THE GROUP SL (2, R) ,WHERE R IS A RING OF FUNCTIONS

Dedicated toAndrei Nikolaevich Kolmogorov

A. M. Vershik, I. M. Gel'fand, and M. I. Graev

We obtain a construction of the irreducible unitary representations of the group ofcontinuous transformations X -*• G, where X is a compact space with a measure m andG == PSL(2, R), that commute with transformations in X preserving m.

This construction is the starting point for a non-commutative theory of generalizedfunctions (distributions). On the other hand, this approach makes it possible to treat therepresentations of the group of currents investigated by Streater, Araki, Parthasarathy, andSchmidt from a single point of view.

Contents

Introduction 15§ 1. Some information on the representations of the group of

real 2 X 2 matrices , 1 7§2. A construction of the multiplicative integral of representations

of G = PSL(2, R) 27§3. Another construction of the multiplicative integral of

representations of G = PSL(2, R) 32§4. A representation of Gx associated with the Lobachevskii plane 39§5. A representation of Gx associated with a maximal compact

group K C G 42§6. Another method of constructing a representation of G 1 . . 5 2§7. Construction with a Gaussian measure , 57

References , 60

Introduction

One stimulus to the present work was the desire to extend the theoryof generalized functions to the non-commutative case. Let us explain whatwe have in mind.

Let R be the real line, X a compact manifold, and fix) an infinitelydifferentiable function on X with values in R, that is, a mapping X -• R.A group structure arises naturally on the set of functions f(x), which wedenote by R^. Irreducible unitary representations of this group are defined

15

16 A.M. Vershik, I. M. GeVfand, and M. I. Graev

by the formula f(x) i—* eilV\ where / is a linear functional in the space of"test" functions /(x). Thus, to each generalized function (distribution)there corresponds an irreducible representation of Rx. If we replace R byany other Lie group G, then it is natural to ask for the construction ofirreducible unitary representations of the group Gx, regarded as a naturalnon-commutative analogue to the theory of distributions. Such an attemptwas made in [ 1 ] , § 3.

However, our progress was only partial. We succeeded in defining distri-butions with support at a single point or at a finite number of points (forthe group SU{2)) — analogues to the delta function and its derivatives; wewere also able to introduce the concept of a derivative and show that 5'is the derivative of 5. The work came to a halt because we did not succeedin introducing the concept of an integral, without which the theory ofgeneralized functions cannot go on.

The problem of constructing an integral for Gx can be stated as follows.Suppose that an X measure m is given. We have to find irreducibleunitary representations of Gx that go over into equivalent ones undertransformations of X preserving m. Reducible representations of this kindcan be constructed without special difficulty. However, even the case G = Rindicates that for our purposes reducible representations are unsuitable.

For a long time it was not clear to the authors whether such irreduciblerepresentations exist for semisimple groups G. Finally we succeeded inconstructing such representations for a number of semisimple groups,namely, groups in which the identity representation is not isolated in theset of all irreducible unitary representations.

In this paper we analyze in detail only the case of the group SL(2, R).The fact is, as experience with representation theory shows, that an under-standing of any new situation is impossible without a preliminary study ofthe group SL2 from all points of view.

We have performed the construction of the integral several times, eachtime from a somewhat different standpoint. The order in which we havewritten down the various constructions corresponds more or less to theorder in which we thought them out. The first construction proceeds froma very simple idea: to obtain the multiplicative integral as the limit of atensor product of representations, each member of the product being acloser approximation to the identity representation than the last, moreprecisely, to the point of the representation space to which the identityrepresentation is attached.

From the last few sections it is clear that this representation can also beinterpreted in terms of the cocycles of Streater, Araki, Parthasarathy andSchmidt (more precisely, it is not the 1-cocycles that play the fundamentalrole, but rather the reducible representations from Ext1). The proof of theirreducibility of these representations is a new feature in our constructions.

Representations of the group SL(2,R) 17

At the end of §6 we construct two other projective unitary represen-tations of the group (PSL(2, R))x.

The construction of the integral for all other groups Gx for which Gsatisfies the condition that the identity representation is not isolated inthe set of all irreducible representations will be presented elsewhere.

The integral constructed in this paper provides us with a constructiverepresentation of the group Gx, which in the terminology of mathematicalphysics is the group of currents of G.

Thus, this paper can also be regarded as a survey, from a somewhatdifferent standpoint, of work on the representations of the group ofcurrents.

Representations of the group of currents have been widely studied by anumber of authors (Streater, Araki, Parthasarathy and Schmidt). See [4],[7] — [11], [13], [14], [15], and the further literature cited in the surveypapers [6] and [12].

§ 1 . Some information on the representationsof the group of real 2 X 2 matrices

1. Representations of the supplementary series. We consider here the

group G = PSL(2, R) of real matrices g = f 1 with determinant 1 in

which g and — g are identified. This group is known to be isomorphic to

the group of complex matrices of the form U - ,where \a\2 - \j5\2 = 1\P a/

and g and — g are again identified. In what follows we use either the firstor the second definition of G, as convenient.

Let G be given in the second form.We introduce the space K of continuous infinitely differentiable functions

on the unit circle | f | = 1 in the complex plane. With each real number Xin the interval 0 < X < 1 we associate a representation 7\ of G in K.

if i Z i*<-2

In K there is a positive definite Hermitian form (f1, f2\ that is invariantunder the operators Tx'-^

(2) (/,,/) r f/,,/a)x= , f -3u ,( [4 V JI r ^ —-— j o b

We deno te by Hx the comple t ion of K in the no rm \\f\\\ = (f, f)\.

' The numerical factor is chosen so that (1,1)\= 1, where 1 is the function identically equal to unity.

18 A.M. Vershik, I. M. GeVfand, and M. I. Graev

It is evident that the Tx can be extended to unitary operators in Hx.DEFINITION. A unitary representation in the Hilbert space Hx, as

defined by (1), is called a representation 7\ of the supplementary seriesof G.

It is known that all representations Tx, 0 < X < 1, of the supplementaryseries are irreducible and pairwise inequivalent.1^

It is sometimes convenient to specify the representations of the sup-plementary series in another manner. Let G be given in the first form. Foreach X, 0 < X < 1, we introduce the space 3)^ of continuous real func-tions such that f{x) = O(\x \x~2) as x -> ± °°. In 3)%, we introduce thepositive definite Hermitian form (/i, f2)\:

(3) (/i, /2)x= j

A representation of the supplementary series acts in the Hilbert spaceobtained by completing 3)k in the norm \\f\\\ — (/, f)\. The representationoperators have the following form:

2. Canonical representations of G. Some unitary representations, whichwe call canonical, of the group G of 2 X 2 real matrices play an importantrole in our work. These very pretty representations of the matrix groupare interesting for their own sake. We present two methods of specifyingthe canonical representations.

2a. THE FIRST METHOD. We specify G in the second form.Further, welet K C G be the maximal compact subgroup in G consisting of the

matrices of the formV 0 e-"} •

Of fundamental importance for the first specification of the canonicalrepresentations is the function

THEOREM 1.1. For any X > 0 the function \px(g) is positive definiteon G and constant on the double cosets of K.

PROOF. The fact that i//x(#) is constant on the double cosets of K isobvious. That it is positive definite is a consequence of the following twolemmas.

*' (1) defines a unitary representation 7^ also in the interval 1 < \ < 2. The scalar product (/x , / a ) \in the space of this representation is defined as the analytic continuation of the function of \ defined inthe domain Re \ < 1 by the convergent integral (2).

The representations 7^ and T2_x, 0 < \ < 2, are known to be equivalent; hence we can always restrictour attention to the interval 0 < \ < 1.


We denote by $\(g) the zonal spherical function of the representation\ of the suppletnentary series with respect to K, normalized so that(e) = 1.LEMMA 1.1. The function 0\(g) is continuous and differentiable in X at

X = 0. Also lim <px(g) = 1 and

The proof follows easily from the explicit form of

where

is the Legendre function.LEMMA 1.2. The function

positive definite for n > 0.

PROOF. By Lemma 1.1, <ftp?i (g) it is evident that

,(g) — l is conditionally positive definite, that is, 2 ——V &1./ > 0X i, j

under the condition that 2 5» = 0. It then follows that exp (\i (p? ^ )

is positive definite for /u > 0. Since the positive definite functions form a

weakly closed set, the limit exp (a *\« 1 n) = limexp (u ^ ^ p " )

is positive definite.DEFINITION. The unitary representation of G defined by the positive

definite function \px(g), X > 0, is called canonical. A cyclic vector £x inthe space of a canonical representation for which (T(g)Z\, £ ) = \px(g) iscalled canonical

Let us construct a canonical representation. We consider the spaceY — K\G, which is a Lobachevskii plane. Let y0 be the point of Y thatcorresponds to the coset of the identity element. We define the kernel

2), where X > 0, on the Lobachevskii plane by the formula

where yx = yogi, y2 = ^0^2- By Theorem 1.1 this kernel is positive

definite.We consider the space of all finite continuous functions on Y. We

denote by Lx the completion of this space in the norm:

I I / IP =

20 A. M. Vershik, I. M. Gel'fand, andM. I. Graev

where dy is an invariant measure on Y.A canonical representation of G is defined by operators in Lx of the

form( 7 W ) ( y ) = fiyg).

(That the operators T(g) are unitary and form a representation of G isobvious.)

THEOREM 1.2. / / X > 1, then a canonical representation in Lx splitsinto a direct integral over the representations of the principal continuousseries of G. If 0 < X < 1, then

where HK is the space of the representation 7\ of the supplementary series,and Lx splits into representations of the principal continuous series only.

PROOF. It suffices to verify that \jjx(g) can be expanded in zonalspherical functions of the corresponding irreducible representations. We

may limit our attention to the matrices g =(c°s, r s mt ' ) . For these

Vsinhfcoshr/matrices we have i|?Hg)= COsh 2*4-1 Y*' F u r t n e r m o r e > w e know that the

zonal spherical functions of the representations of the principal continuousseries have the following form:

cpi-npteHP-i-ip (cosh 20,2

where 1 - i p is the Legendre function.

Let X > 1; then (-^Tf) is square integrable on [1, °°) and can there-

fore be expanded in an integral of functions P-i-iP (x) (the Fock-Mehler2

transform). Thus, we have

o

The coefficients ax(p) in this expression can be calculated by theinversion formula for the Fock-Mehler transform; we obtain

1 ( ) ( ! )"K (P) = - s j r f27X] P t a n h

Now let 0 < X < 1. It is known that the zonal spherical function= ^-x/2(c°sh 2t) of the representation Tx of the supplementary series

has the following asymptotic form:


It follows that the function

2 W2 " ^ l 1 " ! " ) p_w{x)

is square integrable on [1, ©°) and can therefore be expanded in an integralof functions P - i - i P (x).

2

In what follows we say that the canonical representation Lx for0 < X < 1 is congruent to the representation Hx of the supplementaryseries modulo representations of the principal series.

The explicit separation of the component of the supplementary series inLx will be carried out a little later (see Theorem 1.3).

In conclusion we give another two expressions for the kernel^N>i, y2) = tHgigi1), where yx = yogu y2 = yog2.

From the definition of \p(g) it follows easily that

^CVij yi) ~ cosh~xp(yly y2),

where p(yi, y2) is the invariant metric on the Lobachevskii plane.We suppose further that the Lobachevskii plane Y is realized as the

interior of the unit disk \z\ < 1 in the complex plane, where G acts by

fractional linear transformations: z-+az _ . It is easy to verify that in this

realization ^ x has the following form:

(1-1*1 I2) (1-1*2 I2) "W2

We observe that the invariant measure on the unit disk is ( 1 - \z\2)~2dz dz.Thus, in the realization on the unit disk the norm in the space of thecanonical representation has the following form:

(5)i-i*i i2) ( i - i

_| i _ Z l z 2 | 2

2b. A SECOND METHOD OF SPECIFYING A CANONICAL REPRESEN-TATION. Suppose that the Lobachevskii plane is realized as the interior of theunit disk \z\ < 1. Then the norm in the space L\ of a canonical represen-tation is given by (5). If we now go over from the functions f(z) to thefunctions (1 — |z|2)"*"x~2/(z), then we obtain a new and very convenientrealization of a canonical representation.

In this realization the space Lx of a canonical representation is thecompletion of the space of finite (that is, vanishing close to the boundary)continuous functions in the unit disk | z | < 1 with respect to the norm

22 A. M. Vershik, I. M. Gel'fand, and M. I. Graev

(6) | | /1 |2 = j \i-ziz2\~Kf{zi)f(z2)dzidzidz2dz

The representation operators act according to the formula

We note that for finite continuous functions /(z) in the disk | z | < 1the norm ||/ | |2 can be written in the following convenient form:

oo

Vu/ 11/11 j/J l

m, n=0

where

Cmn 00 = —

In particular, if/ depends only on the modulus r, then the norm ||/| |2

takes the following simple form:

To derive (8) it is sufficient to represent the kernel | 1 — zxz2 \~x as theproduct (1 - z1z2)"x/2(l - z1z2)"x/2, and then to expand each factor in abinomial series.

This space is very interesting. We shall see now that it contains a largestore of generalized functions.

We consider the space K of test functions 0(z) that are continuous and infi-nitely differentiable in the closed disk |z| < 1. Let /(0) be a generalized func-tion, that is, a linear functional on K. From / we construct a new functional inthe space Ko of functions /(z) that are infinitely differentiable and finite inthe disk |z| < 1 (that is, vanish near the boundary):

(l,f)= 2 cmn.(K)l(zmzn) j f(z)zm~zndzdz.m,n—0 |z'l<l

If this series converges absolutely and | (/, / ) | < c \\f\\, then the functional(/, f) can be extended to a continuous linear functional in Lx and there-fore specifies an element / in Lx.

We claim that the delta function £x = 5(z) concentrated at the pointz = 0 belongs to LK. In fact, since £x(l) = 1 and £K(zmzn) = 0 form 4- n > 0, we have

(9) (6x, /) = (6, f) = j f(z)dz dz.


Consequently, by the definition of the norm in Lx, I £\, / ) I ^ 11/11 andhence 5(z) G Lx.

It can be shown that £x is a canonical vector in Lx, that is,

This is easily derived from (9) if we replace / by T(g)fn, where fn is asequence of test functions converging to £x, and then proceed to the limit.1

It is not difficult to verify that Lx contains not only 5(z), but also all

its derivatives, 6(m> n)(z)= °m " i . In particular, all derivativesdzm dzn

g(2n+i> ^ __ dr2nX lie in the subspace of Lx consisting of functions2* that

depend only on the modulus r.We now look at the generalized functions / = a(z)5(l - | z | 2 ) , where

a(z) is a continuous function on the circle | z | = 1 (that is,

1

We can obtain this result formally by substituting in (9)

' The functions 6 ^ \z) form an orthogonal basis in Lx. Thus, each element / of this space can bewritten in the form

7__ v u dm+nS(z)^ dzm dzn

m, n—0

with

!IZH2= S ( m ! r l ! ) 2 %nW|Vn | 2 .m, n=0

In particular, in the subspace of functions depending only on the modulus r there is an orthogonal

basis consisting of the functions fi<2n+i> (r) _ 2n+i • Thus, any element of this subspace can be

-_. d2n+16 (r)written in the form / = >j bn 2n+\ ' , with


LEMMA 1.3. The functions I = a ( z )5 ( l - \z\2) belong to the space Lx

for 0 < X < 1.PROOF. Since

2JI

) = 1 j a (eif ) «*<*-»>' dt = am_n,

we have

\(l,f)\<m,n=:0

Hence, by the Cauchy inequality

| ( J , / ) | < ( 2 CmnM | CCm-n|2)1/2 || / | | .

m, n=0

Thus, it remains to prove the convergence of the series

(10) 2 C m n(X)|am.n |2= S | a , | 2 ( S cmn(X)).m, n=0 ft=-oo m-n=fe

To do so we use the following estimate for the coefficients

If follows from this estimate that for 0 < X < 1 the series 2 cnn (X)n

converges. On the other hand, it is not hard to see that 2n-0

m—n=k

< 2 Cnn(h) and, therefore, 2 cmn(X)<6'1, where Q does not dependn=0 m—n=h

oo

on /:. Since 2 I a^ |2 converges, the convergence of the series (10) follows.^fc0

^ We note that the function 6(z) can also be obtained by a limit passage. Let us consider thesequence of functions fn G L\ of the form

cn for

0 for — < | z | < l ,

where cn > 0 is defined by the condition II /„ II = 1. It is easy to verify that this is a fundamentalsequence in L\ and that its limit is equal to 6(z). Similarly, we can obtain 6(1 - |z|2) as the limit of afundamental sequence of functions of the form

cn, for

where cn is determined from the relation \ fn (z) dz dz = n.


We state without proof two simple propositions on the functions, = 5(z) and / = / ( z ) 5 ( l - | z | 2 ) .

LEMMA 1.4. Let lx = fx(z) 5 (1 - | z | 2 ) , l2 = / 2 ( z ) 5 ( l - | z | 2 ) , then2n2n

(11) l ) (J i , « x = 2 - - j j | sin — 2 —0 0

(12) 2) (6(z), 6 ( l - | z | 2 ) ) 3

From (11), it follows, in particular, that

L E M M A 1 . 5 . 77ze representation operator T(g), where g = ( ? _ ) , fate' P a '

e function / ( z ) 5 ( l - Iz I2) i/ifo / ( ^ £ ± J ) | p2 + a |*-26(1 - \z |2).

The next result follows immediately from Lemmas 1.4 and 1.5.THEOREM 1.3. For 0 < X < 1 the subspace Hx C Lx generated by the

functions / (z)5(l - |z|2)/s invariant and the representation of G acting init is the representation Tx of the supplementary series.

In what follows we find it useful to know the projection of the canonicalvector £x onto H\. This projection is obviously invariant under the maxi-mal compact subgroup and is consequently proportional to 5(1 — |z|2). Let

where c(X) = ||5(1 — Izl2)!!"1. Then, according to Lemma 1.4, 2) we have(£\> V\) = KC(\); consequently the projection of the canonical vector %x

onto the subspace Hx is equal to 7rc(X)rjx, where c2(X) = || 5 (1 - |z |2) | |"2

= 2X7T"3/2 r ( 1 - X / 2 ) / r ( | ( l - X ) ) .The next result follows easily from standard estimates for F(X).LEMMA 1.6. (J x , T?X) = 1 + 0(X2) as X^ 0.COROLLARY. As X -• 0, the distance from the canonical vector £x in

Lx to Hx is 0(X2).3. Theorems on tensor products of representations of G. THEOREM 1.4.

Let Tx and Tx be two representations of the supplementary series. IfXt + X2 < 1, then Tu ® Tk2 = TXi+X2 0 T, where T splits into represen-tations of the principal continuous and the discrete series only.

If Xt + X2 ^ 1, then TXi (g) 7\2 splits into representations of theprincipal continuous and the discrete series only.

The tensor product of two irreducible unitary representations of whichat least one belongs to the principal continuous or the discrete series splitsinto representations of the principal continuous and the discrete series only.

For the proof see [12].


We now let Gn = G x . . . x G. Since G is of type 1, it is standardn

knowledge that any irreducible unitary representation T of Gn can beobtained as follows. We are given irreducible unitary representationsr(1), . . ., r(w) of G, acting in Hilbert spaces Hu . . ., Hn, respectively. Arepresentation T of Gn acts in the tensor product Hi ® . . . ® Hn accord-ing to the following formula:

(14) T(gu . . ., gn){lt ® . . . ® En) = (Tto(gl)ll)<8> ' ' ' ®(T<n)(gn)tn).

Here two representations T' and T" of Gn are equivalent if and only if allthe corresponding representations T'^ and T"^ of G are equivalent,i = l , . . . , n.

We say that a representation T of Gn of the form (14) is purely of thesupplementary series if all the T^ are representations of G of the supple-mentary series. In this case, if T^ = TXi, i = 1, . . . , « , then the corre-sponding representations of the group Gw are denoted by 7\w>. . . , \ n , andthe representation space by / /X i > . . . , \ n .

The next theorem follows from the one stated above.THEOREM 1.5. Let T^,...,^ and Tx"v...xn be two representations ofGn

purely of the supplementary series: and let \\ + X{' < 1, / = 1, . . ., n. ThenT\'v...xn ® ^ j , • • • Xn ~ ^M+î'• • • ' n+rn ® ^ ' where in the decompositionof T into irreducible representations there are no representations purelyof the supplementary series.

In what follows we find it useful to specify explicitly an embedding ofH\l + \2{\ + X2 < 1) in the tensor product//"^ 0 Hkr

Let G be defined in the first form. Then H\i + \a is the completion ofthe space of finite continuous real functions fix) with the norm

-f-oo -J-oo

j j— oo — oo

Now //"M ® H%2 is the completion of the space of finite continuous func-tions Fixi, x2) of two variables with the norm

x2-x'2 \-** F(xu X2)F(x[, x'2) dxi dx2 dx\ dx2.

In Hu (g) HX2 there are many generalized functions. The precise meaning ofthis statement is the following.

Let liF) be a linear functional on the space of infinitely differentiablefunctions P(xu x2) such that \F(xu x2)\ < C(\ + x\Y^12 0 + xlr*J2,By means of / we construct the following linear functional T on the spaceof finite infinitely differentiable functions F(xi, x2): (/, F) = /(F), where

OO + C X ,

\Xi-x[\-^\x2-x'2\-^F{x[, x'2)dx[dx2.+

= J— oo — o o

Representations of the group SL(2, R) 27

If the functional (7, F) is defined and continuous in the norm ofHki <g> Hk2,then we identify the generalized function / with the vector I 6 H^ <S> Hu-

LEMMA 1.7. / / \ > 0, X2 > 0, \ + X2 < 1 and I is a generalizedfunction having the form I = ftx^dixx - x2) (f(x) finite), that is,

l(F)= \ F(x, x)f(x)dx,— oo

then the functional (7, F) is continuous in the norm of Hu ® HX2.This lemma is proved by standard calculations involving the Fourier

transform, and we omit the proof. Next we can establish the followinglemma.

LEMMA 1.8. Let 71 and T2 be defined by the generalized functionsh = /i(*i)5(*i ~ x2) and l2 = f2(x1)5(x1 - x2). Then

+ OO+OO

(h, h)= j J \Xl-X2\~Xi~k2 fl(xl)f2(x2)dxldx2.— (XI —OO

THEOREM 1.6. / / Xj > 0, X2 > 0, Xx + X2 < 1, then the mappingdefines an isometric embedding of HXi+x2 in Hkl <g) Hk27 consistent withthe action of G on these spaces.

This theorem follows immediately from the lemmas stated above.We need a somewhat more general theorem, which can be proved

similarly.THEOREM 1.7. Let \ u X2, . . ., X* > 0 and X! + . . . + X* < 1. Then

the mappingf(x) H-* fifo — xh, . . ., xA_i — xh)f(xk)

defines an isometric embeddingHu+...+xk-+- HM 0 . . . <g> H%h,

consistent with the action of G.The meaning of the concepts and mappings introduced is the same as

that explained earlier for k — 2.The mappings indicated above are consistent; namely, if

i i

then the mappingi, 3

is the composition of the mappings

and H%. -v (g> H^...

§2. Construction of the multiplicative integralof representations of G = PSL(2, R).

Let G = PSL(2, R) and let X be a compact topological space with agiven measure m. We define a group operation on the set of functions


g: X -• G as pointwise multiplication: (g\g2)(x) = gi(x)g2(x). We defineGx as the group of all continuous functions g: X -• G with the topologyof uniform convergence.

We give here a construction of an irreducible unitary representation ofthe group Gx, which we call the multiplicative integral of representationsof G.

Following the definition of the integral as closely as possible, we replaceGx by the group of step functions and define an integral on it as a tensorproduct of representations. As we decrease the length of the intervals ofsubdivision and simultaneously allow the parameter on which the represen-tations in the tensor product depend to approach a certain limit, we obtaina representation of Gx', which we call the integral of representations. It isremarkable that the integral of representations is an irreducible representation.

We now proceed to precise definitions.1. Definition of the group G°. For every Borel subset Xf C X we denote

by Gx' the group of functions g: X -» G that are constant on X' andequal to 1 on the complement of X'. It is obvious that there exists anatural isomorphism GX' = G.

A partition v: X = U Xj of X into finitely many disjoint Borel subsetsis called admissible.

On the set of admissible partitions we define an ordering, settingvx < v2 if v2 is a refinement of vx. It is obvious that the set of admissiblepartitions is directed (that is, for any p1 and v2 there exists a v such thatVi C v and v2 < v). h

For any admissible partition v: X = [} Xt WQ denote by Gv the group

of functions g: X -> G that are constant on each of the subsets Xt. It isobvious that

Gv = GXi X . . . X GXj}.

Observe that for vx < v2 there is a natural embedding: GVi -+ Gv^. Wedefine the group of step functions G° as the inductive limit of the Gv\

G° = l im Gv.

In this section we construct a representation of G°. We make the tran-sition from this representation to a representation of Gx in §3.

2. Construction of a representation of G°. Let m be a positive finitemeasure on X, defined on all Borel subsets of X. We always assume thatm is countably additive.

Let us consider the Hilbert spaces in which the representations 7\ ofthe supplementary series act. In § 1 we denoted these spaces with theaction of G defined on them by HK, 0 < X < 1. Next, we denote by i/0

the one-dimensional space in which the identity representation of G acts.

Let v: X = \j Xt be an arbitrary admissible partition such that


X,- = m{Xt) < 1. We set

In SBv we define a representation of the groupGv = GXi x . . . X GXfe,

supposing that GXi = G (i = \, . . ., k) acts in HXi.The representation of Gv in G^V is irreducible (see §1.3); we have

agreed to call such representations purely of the supplementary series.Note that since GVi C GVt for vx < v2, a representation of each of the

groups Gv', v < J>, is also defined in SBv •LEMMA 2 A. If v i < v2, then S£V2 splits into the direct sum of sub-

spaces invariant under GVi: SBX2 = SBVi 0 SB' where SB' does not containinvariant subspaces in which a representation of GVi purely of the supple-mentary series acts.

k

PROOF. Let v2 > vu that is, vx\ X = U Xh v2\ X = U Xih where

Xi = U Xtj. We set \ t = m(Xt), \tj = m (Xtj)\ thus, SBVl = ® HK.,

SBV = 0 H^... We also set SB\ = 0 ^ . . ; then SBV = 0 dî . It is evidenti , j lJ i 2 i

that Gxt acts diagonally in c^v2 = 0 ^^ i ; (that is, acts simultaneously oneach factor HXij). Thus, the representation of Gxt — G in <^J is a tensorproduct of representations 7\l7 of the supplementary series.

From this it follows that SB\ = H%i 0 H\, h = l!i *>u, w h e r e H\i i s

the space in which the representation TXi of the supplementary series acts,and H*x splits only into representations of the principal, the continuous,and the supplementary series (see §1.3). Forming the tensor product ofthe spaces SB\2 and bearing in mind 0 Hj,. = SBVl^nd GXi x . . . X ^xh = GVi,we obtain: SBV2 = SBVi® SB', where %SB' does not contain representationspurely of the supplementary series.

THEOREM 2.1. There exist morphisms of Hilbert spaces

defined for each pair vx < v2 of admissible partitions of X satisfying thefollowing conditions:

1) h2vv commutes with the action of GVi in S£Vi and SBV2;2) h]v2 ° U%vx = h,Vl for any vx <v2 <v3.These morphisms are determined uniquely to within factors cViVi(\cViVi\ = 1).PROOF. From Lemma 2.1 it follows that for each pair vx < v2 there

exists a morphism /V2vi: 3£\i -+ 3£\2 that commutes with the action of Gvând that this morphism is uniquely determined to within a factor. Weclaim that the morphism /V2vi can be chosen so that condition 2) is satisfied.


Let v2 > vu that i s ^ : X = (J Xt, v2: X = U. xih where Xt = [) Xu.

set \f = m{Xi), \if = m(Xz/); thus, 38Vi = O # v < va = 0 Hk..i=l i, j

For each / = 1, 2, . . ., k we define a mapping i / \ { ->- (g> H%..ri ii, i

î = 2 Xiy, compatible with the action of Gxt — G, just as this was done3

in §1.3. These mappings induce the mapping

7*V2Vi : $£vi ~*~ 3&V2 ,

which is compatible with the action of Gv in these spaces.From the definition of the mappings H^—*- <g> HK.. it follows easily

3 lj

that the mappings / ^ so defined satisfy the compatibility requirement 2)of the Theorem.

REMARK. Another method of specifying the compatibility of the system ofmorphisms / ^ will be given in §3.

DEFINITION. We assign to all possible pairs p1 < v2 of admissiblepartitions of X the morphisms /^ ^ : 3£Vi ->- SSV2, which commute with theaction of GVi in S£Vi and 3£V2 and satisfy the compatibility conditionKV2 ° iv2vx

= U^v, f°r v\ <V2 < P3- W e introduce the spaceS£° = lim $gv,

and let S£ be the completion of $£° in the norm defined in 3£°.Then there is a natural way of defining a unitary representation £/ of

G° incf/A(Specifically, we have the natural embeddings Gv cz^ G°, SSV c_> ^ ° .

Let g G G° and g £ J^°. Then there exists an admissible partition v suchthat g E Gy, J G G$?V7 and we set C/jJ = T(g)£, where T is a represen-tation operator of Gv in 3£v. It is evident that this definition does notdepend on the choice of v. The unitary operator U% on $g° so con-structed can be extended by continuity from 38° to its completion 3£.)

LEMMA 2.2. The representation Ug does not depend on the choice ofthe morphisms j v ^ .

PROOF. Let z ^ : S£Vi -> ^ v a be another system of morphisms satis-fying the conditions of the definition, and let 38'° = lim 38v be the

inductive limit constructed with respect to this system of morphisms. Weclaim that the representations of G° in 36* and 3£'° are equivalent.

By Theorem 2.1. we have / ^ = cv%vjViVx, where cViVi are numericalfactors satisfying cv^cv%Vx = cv^x for vx < v2 < v2. It follows from thiscondition that cv^ = cVjJ,o C^VQ , where v0 is a fixed admissible partitionO0 < vx < v2). For ^ > o w e specify isomorphisms of the spacesa%v->- 38x

in the following manner: £H->CVV0H. Since / ^ = c^^c^J^^, they take


jv%Vx into jliVi and consequently induce an isomorphism of the spaces St°and SB'0 compatible with the action of G°.

THEOREM 2.2. The representation U of G° in SB is irreducible.u

PROOF. Let v\ X = U Xt be an admissible partition such that

X,- = m(Xi) < 1. We restrict the representation of G° in SB to Gv.k

An irreducible representation of Gv acts inc^ v = <8> H%. . By Lemma 2.1i=l l

SBV occurs with multiplicity 1 in each SBV> for v > v. Since d^v isirreducible, it follows that it occurs with multiplicity1 in the whole space SB.

We now suppose that SB splits into the direct sum of invariant subspaces,$£ = SB' © Si". Then Stv is contained in one of the summands, forexample, inS6'. Now let v > v. Since S£v> => S£v and an irreduciblerepresentation of Gv> acts ino^V, we have SBV> cz SB'. Consequently, SB'contains all the subspaces SBV', y' >.v, and therefore coincidesThis completes the proof.

THEOREM 2.3. Let mx and m2 be two positive measures on X,and U^ representations of G° defined on these measures. If mx =fc m2,then the representations U^ and £/(2) are inequivalent.

PROOF. We denote by SB0* and St(2) the representation spaces of ifi*and U^2\ Since mt ¥= m2, there exists an admissible partition

v: X = U Xt such that X^1} = m^Xf) < 1, X[-2) = m2{X{) < 1 for

/ = 1, . . ., k and miiXi) ^ m2(X{) at least for one /.We claim that the representations of Gv C G° in SBO) and ^ ( 2 ) are

inequivalent. It then follows that the representations of the whole groupG° in these spaces a fortiori are inequivalent.

Let us first consider the spaces

SB^ = I Hiso andSB™ = ®H%<»,

in which the irreducible representations of Gv act. Since X} =£ Xf for atleast one /, the representations of Gv in SB™ and SB™ are inequivalent.

Furthermore, ( 'v2'} = SB™ 0 cî, for every v > ^, and Mv does notcontain representations purely of the supplementary series of Gv, thereforedoes not contain representations equivalent to SB™ (see Lemma 2.1).Consequently the whole space SB™ does not contain representationsequivalent to d^^. But then c "(2> also does not contain representations ofGv equivalent to SB™. Since obviously SB™ cz SBa\ the representations ofGv in SBO) and S£{2) are inequivalent.

32 A. M. Vershik, I. M. Gel'fund, and M. I. Graev

§3. Another construction of the multiplicative integral ofrepresentations of G = PSL(2, R).

The concept of the multiplicative integral of representations ofG = PSL(2, R) introduced in §2 can also be obtained starting out fromthe canonical representations of G. Here we explain this second method.It is surprising that although the representations in the product aresignificantly more "massive", their product turns out to be the same asbefore.

1. Construction of the representation. As before, let X be a compacttopological space on which a positive finite measure m is given, defined onall Borel subsets and countably additive.

We consider the canonical representations of G in the Hilbert spaces Lx

introduced in §1.2. Next we denote by Lo the one-dimensional space inwhich the identity representation To of G acts.

We recall that in each space Lx we have fixed a cyclic vector £x whichwe have called canonical. For this vector

where \jj(g) is the function defined in §1.2.h

With each admissible partition v\ X = U Xh we associate a Hilbert

space

where A, = m(Xi). We define in Xv a unitary representation of

Gv = GXi <g> . . . <g> Gxh,

assuming that each group GXi = G acts in Lx. [n accordance with thecorresponding canonical representation, and trivially on the remainingfactors LXj, j = /. We observe that since Gv C Gv for vx < v2, an actionof each of the groups Gv>, v < v, is also defined in Xv -

kFor each admissible partition v\ X = U I j we specify a vector ?v 6 Xv\

?v = hi ® - • • ® &tft»where £Xl- is the canonical vector in LXi. It is obvious that £„ is a cyclicvector in Xv,

LEMMA 3.1. For any pair of partitions vx < v2 the mapping £vi I-> £v2

can be extended to a morphism/v2vi • %vi -+ XV2,

which commutes with the action of GVi.PROOF. It is sufficient to verify that

for any gVi G GVi, where the parentheses denote the scalar product in the


corresponding space.h

According to hypothesis we have v^ X= \J Xt, v2: X= U Xtj, wherei = l i, j

Xt= U Xu. Let gVl G G^, that is, gVi = gx . . . gn, where gt E Gx. ~ G.j

Then ft fe

(i) (T (gVi) HVI, ^ ^ = _ n (T (gt) hr ih)Lt=.n tf> (gt),

where X,- = m(Zz). Similarly, let gVj E G^, that is, gV2 = ]f ^ . , wherei,3

gij E G^. s G. Then

(2) (r (gv2) ^v2, ?v2)^V2 = JI. ^" (gu),

where X// = m(Xij).If now g,,2 = gVi, this means that g^ = gj for every i = \, . . ., k. In

addition, since Xt = 2 ô"' f° r a ny z = 1, • • •, ^ we havej

Ijip^j (g/<7-) = ip^K^)- Consequently the expressions (1) and (2) are the3

same and the lemma is proved.It is obvious that the morphisms jVj Vi satisfy the compatibility con-

dition jv^2 ojv^ = j ^ V i for vx <v2 <v3-DEFINITION. We denote by X° the inductive limit of the spaces

%v, X° = lim §CV and by X, the completion of X° in the norm defined inX°. - *

There is a natural way of defining in X a unitary representation of thegroup G° = lim Gv, which we denote by Ug .

—>2. Definition of a vacuum vactor in X. Let K be a maximal compact

subgroup of G, and let K° C G° be the subgroup of step functions on Xwith values in K. Vectors in X of unit norm that are invariant under K°are called vacuum vectors.

We claim that a vacuum vector exists in X.Let /„ be the natural mapping # v -> X. It follows from the definition of

£y that their images /„£„ in X coincide, that is, there exists a vector £0 G %

such that £0 = ]v\v for any admissible partition v.Since each vector £„ G <£v is invariant under .£„ C Gy, it follows that

So = lim Sv is invariant under i£° = lim Kv, so that it is a vacuum vector

in X.~* ~În addition, since each £„ E < v is a cyclic vector in < v under Gv,

l0 = lim Ev is a cyclic vector in < under G° = lim Gv.

In §3.3 we shall prove the uniqueness of the vacuum vector in X(Theorem 3.2).

Let us calculate the spherical function of the representation U^:


where g = # ( • ) E G°.

LEMMA 3.2.

where \p(g) is the function on G introduced in §1.2.PROOF. For any g = g(») G G° we can find an admissible partition

ftv. X= U Xt of X such that g E Gv, that is, g = gx . . . g^, where

i = l

Si G GXi = G. But then, setting X/ = m(Xt), we have

(Z1 (S) iv, ?v)^v

n *** (^)=exp ( 2 i

= exp ( f In (if (g (x)) dm (x)

3. Equivalence of the two constructions of the representations of G°.We claim that the unitary representation of G° in X constructed here isequivalent to the representation in Si constructed in §2.

LEMMA 3.3. There is an embedding morphism Si -> X that is com-patible with the action of G°.

PROOF. In §1.2 we have shown that for 0 < X < 1 the canonicalrepresentation Lx of G is congruent to the representation Hx of thesupplementary series modulo representations of the principal continuousseries; in other words, Lk == Hk ® Lk, where L'x can be expanded in anintegral over representations of the principal continuous series.

k

Hence it follows that the representation space Xv = <g> L^. ofk ^ l

Gv = GXi X . . . X GXk, where v: X = [} Xh X,- = m{Xt) < 1, splits

into a direct sum of invariant subspaces:

k

where Siv = (8) HK., and X'v for any v < v does not contain represen-i=i l

tations purely of the supplementary series of Gv\This also shows that under the morphism ]v<iVx: XVi -*- Xv.21 v{ <. v2, the

subspace 3£Vi cz XVI is mapped into SSV2t a XV2. Consequently,X° = lim Xv contains Si0 = lim Sis* Going over to the completions, wesee that X contains 36.

REMARK. We have incidentally constructed a compatible system ofmorphisms jVjVi: SiVi->- 3£V2, Vi <Cv2, that commute with the action of GVr


In §2 we have given another method of specifying such a compatiblesystem of morphisms. However, the method given there has the advantagethat it does not make use of the concept of a vacuum vector and doesnot depend on the choice of a maximal compact subgroup.

THEOREM 3.1. The morphism SS -> X defined above is an isomorphism.Thus, the representations of G° in S£ and X are equivalent.

PROOF. Since the vacuum vector l0 6 X is cyclic in X, it is sufficientfor us to verify that £ belongs tod^7. To do this we find the projection of£0 onto each 3£v.

kLet v. X ~ (J 1^ be an arbitrary admissible partition such that

i=i k

X; = m(Xi) < 1. Then we represent £0 as an element of Xv = <g> L^ inh *=* l

the form %0 = (g) | ^ , where £Xl. is the canonical vector in L\v

According to § 1.2b the projection of i*x ^ L\ onto Hx C LK is equalto cxrix, where r?x is the fixed unit vector of HK explicitly constructedthere; here 1 - cx = O(\2) as X -• 0.

k k

Hence it follows that the projection of i0 = 0 t%. onto S£v= (g) H^.»=i l i=i l

is equal tok

Tiv = ( U %) T]v,

ft

where r^ = 0 %. is a vector of unit norm.

From the estimate cx = 1 + O(K2) it follows that for an indefiniterefinement of v, as max X,- tends to zero, the norm of r\v tends to 1,hence that y\v itself tends to £0, and the theorem is proved.

THEOREM 3.2. The vacuum vector in X is uniquely determined towithin a factor.

PROOF. Let £o and £o be two vacuum vectors in X, and let r\v and 77"ft k

be their projections onto S8V = ® - "x where v: X = U X/, Xz- = mCX,) < 1.i=i 1= 1

Since Jo and go a r e invariant under Kv = Kxx X . . . X Xfe c ^^5 theirprojections 17[, and 97[I onto ^Kv have the same property. But in 3£v thereis, to within a factor, only one vector that is invariant under Kv\ conse-quently, r)'v and 17[I are proportional.

On the other hand, since $£° = lira 3&\ is everywhere dense in X (byTheorem 3.1), the vectors r\v and 77 converge, respectively, to £0

a nd Jowhen v is refined indefinitely provided that max Xz- -* 0. Consequently,since r\v and r\l are proportional, so are the limit vectors Jo and £{J. Thiscompletes the proof.

REMARK. We emphasize that the vacuum vector £0 is contained ineach subspace Xv whereas it is not contained in any of the subspaces $£?.

36 A. M. Vershik, I. M. GeVfand and M. I. Graev

4. A representation of Gx. So far we have constructed a representationUg of the group G° of step functions X -* G. We now show how arepresentation of the group Gx of continuous functions on X ->• G can bedefined in terms of this representation. Namely, we claim that the represen-tation U-g of G° (second construction) can be extended to a representationof a complete metric group containing both G° and Gx as everywheredense subgroups. This then defines an irreducible unitary representation ofGx.

For simplicity we assume further that the support of m is the wholespace X.

We first construct a certain metric on G°. Let p(yi, y2) be the invariantmetric on the Lobachevskii plane Y = K\G. We define on G a metricd{g\, £2) invariant under right translations and such that

d(gu £2) > Phfogu yog2)

for any gt, g2 €E G, where j>0 is the point on the Lobachevskii plane Ythat corresponds to the unit coset. (Such a metric exists; for example, wemay set d(gl9 g2) = p(y081, yogi) + piy\g\, yigi)> w n e r e ^1 ^ JV) Wenow introduce a metric 5 on the group G° of step functions, setting

Completing G° in this metric we obtain a complete metric group Gx,consisting of all m-measurable functions #(•) for which

f d(g(x), e) dm (x)< 00.

Observe that the completion of G° in the metric 5 contains, in par-ticular, the group Gx of continuous functions; Gx is everywhere dense inthis completion.

We claim that the representation 1/% of G° constructed above can beextended to a representation of Gx.

For this purpose we consider the functional ^ on G° introduced earlier:

where 5o 6 < is a vacuum vector. It was shown above that

(3) T(i) = e x p ( jln^(x))cfoii(«)) •

LEMMA 3.4. The functional ^(g) can be extended from G° to a con-tinuous functional on the whole group Gx.

PROOF. It is sufficient to verify that ty(g) in (3) is defined and con-tinuous on the whole group Gx.

We use the following expression for i//(g) introduced in §1.2a:\jj(g) = cosh'1 p(yog, g).


From this expression it follows that | In \jj(g) | < piyog, ô) ^ dig, e)>

therefore the integral In \p(g(x))dm(x) converges absolutely.

The continuity of ^(g) follows immediately from the following estimate:

| j lny(gl(x))dm(x) - J lnq(g2(x))dm(x) | < 6(^(0, &(•))•

We shall prove this inequality. We set 77 Qt) = piyogjix), y0) and use the

bound I Incosh T2

l n ilj(gl(x))dm(x) - J i n- J 'We have

In

•J l .= In dm(x)

§ digx(x), g2(x))dm(x) = fifeO), &(•)).

THEOREM 3.3. TTie representation^^ °f G° can be extended by conti-nuity to a unitary representation of Gx.

This follows immediately from the preceding lemma and the followingproposition.

LEMMA 3.5. Let G be a topological group satisfying the first axiom ofcountability, G° C G a subgroup everywhere dense in G, and T a contin-uous unitary representation of G° in a Hilbert space H with a cyclic vectorJ. Further, let ®(g) = (T(g)£, £), g G G°. If $(•) can be extended to acontinuous function on G, then the representation T of G° can be extendedby continuity to a unitary representation of G.

PROOF OF THE LEMMA. We first show that for any r\u r\2 £ H thefunction ^ ^ ( g ) = (T(g)rii, r)2) can be extended by continuity from G°to G.

Let HQ be the space consisting of finite linear combinations of vectorsT(g)%, g £ G°. Since £ is a cyclic vector, Ho is everywhere dense in H.

It is evident that if r?1? 7?2 G Ho, then $ ^ ^ ( 0 can be extended bycontinuity from G° to G. For if r?j = 2a,T(g{)S, T?2 = EbjT(gj% then

We now let r}1, r\2 be arbitrary; without loss of generality we maysuppose that H77JI = ||r?2|| = 1. For any r?i, T?2 £ H, \\rj\\\ = ||i?2|| = 1, wehave

I < i w (g) - ^ t , ; (g) l < I N i - TI;H +11 ri2 - -niii-

Hence the family ^^^(g) is equicontinuous in 971? T?2, and since it can beextended to all g E G for an everywhere dense set of vectors 1715 r?2, thisproves that ^ ^ ( O c a n be extended to G for all 7?l5 r?2.

We claim that the operators T(g), g E G, so obtained are unitary, that is,


(T(g)£, T(g)£) = (£, £) for any { G E Let {gn} be a sequence of elementsof G° that converges to g.

For any e >> 0 there exists an TV such that for ra, n > N

(4) I W s J E , r(firn)|) — (g, 0 | < e

(since (7Xsm)f, rfew)£) = CTfe,^ )£, £), hence converges to (£, £)).On the other hand, we can find m and w, greater than N, such that

(5) | (T(gm)h T(gn)l) - (T(g)l, T(gn)l) | +T(gn)l) - (T(g)t, T(g)l) | < e.

Comparing (4) and (5), we see that

\(T(g)l, T(g)t)-(Z, I) | < 2 e ,

which proves that Tig) is unitary.From the fact that the Tig) are unitary and weakly convergent it follows

that \\T(gn)$ - Tig)i\\ -> 0, as gn -+ g, gn G G°. From this it followsautomatically that Tigt)Tig2)£ = Tigxg2)^ for all gl9 g2 G G.

Thus, we have constructed an irreducible unitary representation of thecomplete metric group Gx in X. Restricting it to the everywhere dense sub-group Gx of Gx, we obtain the required irreducible unitary representationof Gx mX.

Since Gx is dense in Gx, the vacuum vector £0 is also cyclic relative toGx, and for every gG Gx we have

(6) (U~t0, go) = exp ( j In ^{g{x))dm{x)) .

We see that the metric 5 does not figure in (6). Hence our representa-tion of Gx does not depend on the choice of the metric 5 in theconstruction.

5. The representation U^ commutes with the transformations of X thatpreserve the measure m. We consider continuous transformations a: x H-* XQ

of X. They induce automorphisms of Gx:

I = S(') »-* ?a = £a('), where g%r)

If t/^ is the representation of Gx constructed in this section and o is anarbitrary continuous transformation of X, then we can define a new represen-tation U~ of Gx by setting U~ = U%a.

THEOREM 3.4. The representation U~ is equivalent to the representationof Gx defined in terms of the measure ma on X, where ma(Xr) = m(X'a)for any measurable subset X' C X. In particular, if o preserves m, thenU~ and Ug are equivalent.

PROOF. Let U~ be the representation of Gx defined in terms of themeasure ma on X. We compare the spherical functions (Uô, £0) and

o)» where £0 is the vacuum vector. On the one hand,


(U^to, £0) = exp ( Jin cosh"1}fo(x))dm°(x)) .

On the other hand,

(U-to, to) = (UTU, g0) = exp ( j In cosh"1 i / / ^ " 1 ) ) ^ * ) ) =

= exp ( f In cosh"1 \p(g(x))dma(x)

Thus, «7~£o, £o) = ( ^ ? o , So) for any g E Gx, hence Uj and Of arei lequivalent.

6. Invariant definition of a canonical representation. In § 1 a canonicalrepresentation of G was defined constructively. We wish to demonstratethat its connection with the representation of Gx in X we have con-structed is not accidental.

There is a natural embedding of G in Gx. When we restrict ourrepresentation of Gx in X to G, we obtain a certain representation of G. Welook at the vacuum vector Jo in X, that is, the vector £0, ||£0|| = 1, thatis invariant under Kx. We consider the minimal G-invariant subspace thatcontains £0 and denote it by Lx. It follows from the construction performedin §3.1 that Lx is a canonical representation of Gx with X = m(X). Wedraw attention to the following interesting fact.

If in X we consider the restriction of the representation of Gx to G,naturally embedded in Gx, then for X = m(X) < 1 there is precisely onerepresentation of the supplementary series Hx, with X = m(X), that occursas a discrete component in the decomposition. The orthogonal complementto this space splits into representations only of the principal continuousand the discrete series. For m(X) > 1 this representation of the supple-mentary series is absent. This follows from the construction of the rep-resentation of Gx carried out in §2.

§4. A representation of Gx associated with the Lobachevskii plane.

Here we give an explicit form of the multiplicative integral of represen-tations of G = PSL(2, R).

1. Construction of a representation of Gx. Let I be a compact topo-logical space, m a positive finite measure on X defined on all Borel subsetsand countably additive. For simplicity we assume that the support of m isthe whole space X.

Let 7 be a Lobachevskii plane on which the group of motions G actstransitively. We consider the set Yx of all continuous mappings y = y(*):X -+ Y. We introduce the linear space $£°, whose elements are formalfinite linear combinations of such mappings:

2 U o y t , Xt 6 C , y t e Yx,

In other words, 36° is a free linear space over C with Yx as a set ofgenerators.


We introduce a scalar product in $£°. Let pOi , y2) be the invariantmetric on the Lobachevskii plane. For any pair of mappings in Yx,Vi = J>i(0 and y2 = y2(') we set

($1, y2) = exp/l In cosh^pOiOO, y2(x))dm(x)\

and then extend this scalar product by linearity to the whole space St°.The Hermitian form so defined on S6° is positive definite (for a proof seethe end of §4.2 below). Let S£ be the completion of S£Q in the norm

n\\2 - «, *).We define a unitary representation £/~ of Gx inSS. For this purpose we

observe first that an action of Gx on the set Yx of continuous mappingsX -> Y is naturally defined. Namely, an element g = #(•) G Gx takesy = y(-) into yg = î(-) , where ^ ( x ) = y(x)g(x).

We assign to each g E Gx the following operator £/~ in c^.°:

THEOREM 4.1. T/ze operators Ug are unitary on 36° and form a rep-resentation of Gx.

PROOF. The fact that the operators U^ form a representation is obvious.That they are unitary follows immediately from the invariance of p(yi, y2)on Y.

Since the operators U~ are unitary on SS°, they can be extended tounitary operators in the whole spaced^. So we have constructed a unitaryrepresentation of Gx m&8.

2. Realization in the unit disk. We provide explicit expressions for thescalar product in $6° and for the operator U^ when Y is realized as theinterior of the unit disk \z\ < 1.

fa p.\Let G be given as the group of matrices g= I s - I, let Y be the

interior of the unit disk \z\ < 1, and let G act in the unit disk in the

following manner: z -> zg'1 = ^ z + j .

Then c^° is the space of finite formal linear combinations

where z(«) are continuous mappings of X into the unit disk \z | < 1. For a pair ofmappings zx{>) and z2(«) the scalar product in (00 has the following form:

(1) (Zl(.), «,(-)) = ex

The representation operator f/_, g=r=/îl^LU ^, takes z(-) into

z ( . ) r = , , ( . ) , where ^ ^ ^ ^(2) oWi

We indicate another convenient realization of the representation (1). (Itcan be obtained from the first by the transformation z(-) -> X(z(«)) ° z(-),


where X(z(-)) = exp j In (1 - \z(x)\2yldm(x).)In this realization, as before, the elements of 3£° are formal finite linear

combinations of continuous transformations of X into the unit disc \z\ < 1:

but the scalar product has the simpler form:

(3) ( Z l ( . ) , z 2 ( . ) ) = ex

The representation operator Ug is given by the formula:

*V(-) = exp( j In |^)z(x) + ^ ) ' | - 1 ^ (

where zt(x) is defined by (2).In conclusion we verify that the Hermitian form introduced in 36° is

positive definite. It is simplest to confirm this for the Hermitian formgiven by (3).

We introduce the following notation:

zi(x) for n>0,f zi(x)<i zfl(x)

fi(x, n) =i f l ( ) for

hFi(xi, .. ., xh; nu . . .,nh)= [\ ft (xs, ns), where i = 1, 2.

It is not hard to check that the scalar product (3) can be represented inthe following form:

oo

(4) (Z l(.), 2 s ( . ) ) = S 2 ITKUiX

X I ^1(^1, . • .,xk; nu . . ., nh)F2(xi, . . ., xh; nu . . ., rcft) dm(xi) .. . dm(xk).

To obtain this expression from (3) we have to expand first the functionIn 11 - z1(x)z2(x)\~1 in a series:

In I 1 — zi (x) z2 (x)l"1 = j 2 J ^ J ) /1 (^, /*) /2 (^, n) dm (x).

Then we expand exp 1/ in a power series, where

(J' n)h(x, n)dm(x),

and obtain the required expression (4).It is evident that each term in (4) gives a positive definite Hermitian

form on 36°; consequently, the Hermitian form given by (4) for any pairof mappings ZiO) and z2(«) is positive definite.

3. Equivalence of the representation constructed here with the precedingones.

THEOREM 4.2. The representation U~ of Gx in 3£ is equivalent


to the representation constructed in §3. Hence it follows, in particular,that U-g is irreducible.

Let >o be the point of the Lobachevskii plane Y = K\G (where K is afixed maximal compact subgroup) corresponding to the unit coset. Wedenote by y0 = yo(*) the mapping that takes X into y0. The vector y0

belongs toS£, and it is clear that

for every g G Kx. Thus y0 is a vacuum vector in SB.LEMMA 4.1. The vector y0 is cyclic in SB-PROOF. It is sufficient to verify that as g ranges over Gx, U^Vo ranges

over the whole of Yx.It is known that the natural fibration G -> Y = K\G is trivial, hence

there exists a continuous cross section s: Y -> G. Now s induces themapping Yx -• Gx under which y = >>(•) G F x goes into if = g(.) G Gx,where g(x) = s[y(x)]. It is also clear that >>(•) = J o C*)- This completesthe proof.

Let us find the spherical function (U~y0, y0), where y0 is a vacuumvector. Since U~yQ = yog'1, we obtain by the formula for the scalar pro-duct in SB

In cosh"1 p(yog""1 (*)» fa)

= exp J In cosh~lp(y0g(x), y0) dm (x) = exp \ In ty(g(x)) dm (z).

We proceed now to the proof of Theorem 4.2. In §3 the representation£/- of Gx was defined in the Hilbert space X with the cyclic vacuumvector £0. It was also established that (U~%0, £0) = exp f In \p(g(x))dm(x).

So we see that (Uô, %Q)X = (Ugy0, ^ 0 ) c ^ ' S m c e t h e vectors g0 and y0

are cyclic in their respective spaces, it follows that the mapping |0«—*• 1}Q

can be extended to an isomorphism X-+ SB that commutes with theaction of Gx in X a n d ^ . This proves Theorem 4.2.

§5. A representation of Gx associated with a maximalcompact group K C G

1. Construction of a representation of G. We take G to be the group of/a p\

matrices Ig- — I, | a |2 — | p |2 = 1. As before, let X be a compact topo-

logical space with positive finite measure m. For simplicity we assume thatthe support of m is the whole space X and that m(X) = 1. Henceforth wewrite dx instead of dm(x).

Although the method of construction that we use here for the rep-resentation of Gx is cumbersome, it has the advantage that all the formulaecan be written out explicitly and completely and are to some extent


analogous to the expression of representations of the rotation group bymeans of spherical functions.

We suggest that on first reading the reader should omit the simple buttedious proof in the second half of §5.1 of the fact that the formulaegives a unitary representation.

Formulae for representations of the Lie algebra of Gx are given in twoforms at the end of this section.

We introduce the Hilbert space S£ whose elements are all the sequences

*• = (/<>, fu • • ., h, . . . ) .

where f0 E C, and fa for k > 0 are functions X x . . . x X

X Z X . . . x Z->- C, satisfying the following conditions:1

ft

1) fa(*i, • • • > xk\ n\> • • • > nk) is symmetric with respect to permuta-tions of the pairs (*/, «/), (XJ, rij)\

i, . . ., xk; nu . . ., nk) |n .= 0 =

ft-ife, . . . , # * , • • ., ^ ; ^ i , . . ., w f , . . ., nh) (i =

( 1 ) 3 )

2 2 fcT | nx ... nh | J l ^ ^ i ' • • • ' • r ^ nu • •ft=l ni, . . . , nft

(n i ^ 0)

REMARK. Nothing would be changed in the definition of S£ if we wereto assume that all the integral indices nt are non-zero. Then, of course,condition 2) is unnecessary, and the norm, as before, is given by (1).

We construct a unitary representation of Gx inSS. First we introduceon G functions Pmn(g) and pn(g). We define Pmn(g) for n > 0 as the

coefficient of zm in the power series expansion of/ ?z _ \n, wherery ft ^ P Z + a '

For n < 0 we define Pmn(g) by:

x,-, «,• indicate that the corresponding variables are omitted.

44 A. M. Vershik, I. M. GeVfand, andM. I. Graev

From this definition it follows that 1) P-m-n(g) = Pmn(g); 2) Pmn(g) = 0if mn < 0; 3) Pmo(g) = 1 for m = 0 and Pm0(g) = 0 for m =£ 0;

4) Pon(g) = ( x ) n for n > 0 and POn(g) = (-J-)W for « < 0.

Next we set pwfe) = (~|-)nfor n > 0, pnfe) = ( - | " ) ' n | for « < 0,

= I-

Let ? = ( a - \ be an arbitrary element of Gx. We associate with itVP(-) «(•) /

the operator f/~ in ^f that is given by the formula

U7{h) = {<!*),where

(2)

S { n *-,«,<*<*«»* 2 2i l Z

(*i ¥= 0)s

x J I I Pij(S(fj))fh+a{xif . . . , x h , t u . . . , t a ; m u . . . , / T i k , Z l f . . . , Z 8 ) ^ i . . .

and

(3) ¥(g)-exp j l n ^ 2 ^ ^ ) ) ^ .

Here \p(g) denotes lal"1.THEOREM 5.1. The operators £/~ are unitary and form a representation

of Gx, that is,{UtFu U7F2) = (Fu F2)

for any g G Gx and Fx, F2e\S6 and

TJ~U~F= U~~F

for any *gl9 g2 G Gx and F ^S£.We verify these relations for the vectors F of a certain space 3£° every-

where dense in SB, which we now introduce.We denote by M the set of sequences of the form

F = (1, fu fzf . . ., /A, • • 0,where

fk{xu . . ., xh\ nu . . ., nk) = w^ , w4) . . . w(a;A, rcA),

and w(x, «) is a function continuous in x for any fixed n such thatM(*, 0) = 1 and

Represen tations of the group SL (2, R) 45

(4) 2 f i r J i»(*.») !•<**< oo.We denote by 3t° the space of all finite linear combinations of elementsof M.

We must verify that Si0 a SB. Now it is evident that the vectorsF G M satisfy conditions 1) and 2) in the definition of SB. Furthermore,if

(5) F = (1, u(x, n)9 . . ., u(xi9 n), . . ., u(xk, nk), . . . )

is a vector of M, then its norm \\F\\ can be represented in the form:

(6) | | * T = ex

Consequently, by (4), F also satisfies condition 3), and hence F 6 SB. Weobserve that if F G M, that is, if it has the form (4), then the expressionfor UgF reduces to the following simple form:

(7) UgF = X(g, M ) ( 1 , y(a:, » ) , . . ., v{xu n ^ , . . . , v ( x h , n h ) , - - .)»

where

(8) X(?f ")

(9) v(x,n)=2Pmn(g(x))u(x,m).m

LEMMA 5.1. The space SB0 is everywhere dense in SB.PROOF. We assume that all the indices «,- are non-zero (see the Remark

on p. 115). Suppose that SB0 is not dense in SB, hence that there exists anon-zero vector F = {/&}, orthogonal to SB0. We consider in SB the vectorsof t h e f o r m F% = {khfk}, w h e r e f0 = 1, f k ( x u . . ., x k 9 ; n l 9 . . ., nk) == u(xl9 nx) . . . u(xk, nk) for k > 0, u(x, n) is a continuous function,and X is an arbitrary number.

Since Fx £ SB0, we have (Fx, F) = 0, that is,CO

2 ~w ( 2 I wi • • •n* I"1 x

X \ /k (î, • • •, %h; ^ i , . . . , fe) fu {xi, . . . , Xh; 724, . . . , rih) dxx .. . dxk 1 = 0

f o r a n y X. H e n c e i t f o l l o w s t h a t f o r e v e r y k = 0 , 1 , . . . w e h a v e t h er e l a t i o n

| î . . . nh I"1 xni , . . . ,n f t

(n-^0)

4, . . ., nh)fh(xu ...,xk; nt, .. . ,^)rfx1 . .. da;fc = 0

46 A. M- Vershik, I. M. Gel'fand, and M. I. Graev

or(H) 2 \ni--nh\'1 X

X \ u {xi, Hi) ... u (xu, nh) fh (xu . . ., xh; nil . . ., nh) dxx . . . dxh = 0

for any continuous function u(x, n) such that

2J I n I ] I u\xi n) I "^ <C oo.

Since f°k(xu . . ., x^; nXi . . ., % ) is symmetric under permutations of thepairs (x/, nt) and (x;-, «7-), it follows from (11) that /£ = 0 (fc = 0, 1, . . .)•Thus, F = 0, in contradiction to the hypothesis.

To prove Theorem 5.1 we need certain relations for the functions Pmn{g)and piig):

a) (~V n Pmn(g)^- {7.!}m Pnm(g) for m ^ 0, ^ ^ 0;

b) for any compact subset V C G there exist constants C > 0 and r,0 < r < 1 , s u c h t h a t l ^ f e ) ! < l l | |

S mm> (g2) - Pd)

e)

2- 1

Ml

' Pmn (g) Pm-n (*) =

- 2 I n

I "i ri6mm» for m ¥= 0, m' ¥= 0,

- ( - * ) - | . » | - ' M g ) for m ^ 0, in '= 0,— ( —I)m'|m'|-1pm,(g-)for m = 0, m' ^ 0,

— 4 In i|?(g) for m = m' = 0

(5wm ' is the Kroneker delta).1)LEMMA 5.2. The function \(g, u) defined by (8) satisfies the following

functional relation:

(12) \(gl9 v)\(gu u) = X&gi, u)>

whereV{X, Tl) = S ^ m n f e ^ ) ) ^ , m).

' We can derive a) from the relation Pmn(g) = I (-5^—1L j z~m~1 dz, n > 0; the bound

b) for Pmnig) follows from the fact that the radius of convergence of the series ^ j Pmn (§) zm is greaterTrt

than 1; the relation c) follows immediately from the definition of the functions Pmn', d) follows easilyfrom the definition of Pmn and p\ and a).


PROOF. It follows from the definition of X that

) exp ( 2 2 "TTT^ J Pi (Si (*)) pmi

We sum over /, apply d),1 and obtain

)u(x, m)dx

LEMMA 5.3. £7- ^ = U^^F for any F G M and any gu g2 e Gx.PROOF. Let '

F = (1, w(a:, 72), . . ,, Mfo, «i), - . . , u(xk, nk), . . .)•

Then

6 7r / = X(g2, M)(l, I;(O:, »), f . . , vfo, ni) . . . v (^ , nk), . . .),

where

M ) ( 1 , W{X, n), . . ., w(xu nt), ., .,w(xh, nk), . . .)»

where

M; (x, 72) = 2 ^rn'n (#1 (^)) y tei ^ ' ) = 2 Pm'n (gi ($)) Pmm* (#2 (x)) U (x, m).m' m', m

It follows from Lemma 5.2 that \(gi, v)\(g2, u) = ^digi, u)- On t n e

other hand, by c) for Pmn(g), we have w(x, n) = 2 Pmn((jgig2)(x))u(x> m)-

Consequently, UgU^F = U^F.COROLLARY.1 The operators Ug form a representation of Gx in &£?.LEMMA. 5.4 . (0-^ , U?F2) = (Fl9 F2) for any Fu F2 G M and g G Gx.PROOF. Let

F i = ( 1 , M i t e , » ) , . . . , _ .^ 2 = ( 1 , ^ t e r W)» • • •» ^ 2 t e l , « l ) , . . . , M 2 t e f t » ^fe)» • • • ) •

T h e n / V 1 f \

On the other hand, using the expression (7) for UgF we have

(U~Fi, U~F2) = exp J4 j In a|) (gr (x)) dx +

— —— I T)l(cj('r\\il fry 7\ y7^, I \TJ

M l J i v » / / ^j

J7J S J P™n (S (XV Prn'n (g (x)) Ut (x, Hi) U2 (x\Q '

Reversal of the order of the summations is permissible in view of b).


In the last expression we sum over n under the exponential sign, thenuse e) and Uf(x, 0) = 1, and obtain (UgFl9 UgF2) = (Fl9 F2) after someelementary simplifications.

COROLLARY. The C7~ are unitary operators in SB0.2. Irreducibility of the representation Ug. The representation operators

Ug assume a specially simple form when restricted to the subgroup ofmatrices

<P(-)/2 0

\ 0 c-iV(-

Namely,

\2/-

(13) U~ {/„} = {fk},ft

where / i ^ , . . . ,a : A ; 7it, . . . , 7ife) = exp ( i " 2 rcs<p(3s))/fc(£i, . . . , £ * ; / i j , . . . , n f t ) .8 = 1

From this expression it is clear that the family of commuting operatorsUg has a simple spectrum in SB. The vacuum vector in SB is

lo = (*! /i(*» n)i • •

where fk(xl9 . . ., xk; nl9 . . ., nk) =* 0 / / K l + . . . + \nk\ > 0.THEOREM 5.2. 77ze representation U^ of Gx in S£ is irreducible.PROOF. Let A be a bounded operator ind^ that commutes with all the

operators Up in particular, with the U% of the form (13). Since the familyof operators Up has a simple spectrum, an operator A that commutes withthem has the form

(14) Mh} = W*}>

where fljt(î» • • •>** ; wi> •• •» wi) a r e measurable functions (satisfying thesame relations as the fk). Let £0 be the vacuum vector in SB defined above,It follows from (14) that A£o

= 0o£o-We apply to ^0, the operator Ug, g G G, where

/ cosh r sinh r\ , o

\sinhr cosh r/

and r does not depend on x. We obtain

k

where fk(xl9 . . ., xk\ nx, . ..., %) = cosher ff tanh'^'r.i

From U-gh£Q = A Ufa it follows that ao{/ft} = {ahfk}, hence ^0A = "kfk-Consequently, since the fk do not vanish, a^ = a0 for every /:, that is, Ais a multiple of the unit operator.

3. Equivalence of the representation Ug to representations constructedearlier. THEOREM 5.3. The representation Ug of Gx in SB is equivalent to


the representations constructed in the preceding sections.PROOF. It follows easily from the definition of Ug and the scalar

product in SB that the spherical function corresponding to the vacuumvector £0 has the form

=exp(2

So we see that this spherical function coincides (with suitable agreementof measures on X) with the spherical functions of the representations ofGx constructed in the preceding sections. The theorem follows immedi-ately from this fact and the irreducibility of Uj.

4. Infinitesimal formulae for the representation. We give formulae for therepresentation operators of the Lie algebra of Gx in SB (that is, of thealgebra (&x of continuous mappings X -> @ of X to the Lie algebra <& ofG with the natural commutation relations).

We take the following matrices as generators of ©:

\ it '

where <p(-)> r(#) a r e continuous mappings X -* R. We denote the Lieoperators mSB corresponding to these elements by A^, A~ and At~

Expressions for A%, A~, and A{~ are easily obtained from the formula(1) for the representation operators U^ of Gx. Namely,

(A^f)k (xt, ...,#&; nu . . . , nh) = i ( S ns^ (*s)} h(xu . . ., xh; nt, . . . , nk);

(A~f)h (xi, . . . , xu; nu ..., nu) =h

j , . . . , zh i i; nu . . . , nh,

~/)k (a?i, . . . , xh; nu . . . , wft) =

ft

= * 2 ^sT(^s)(/fe(x1, . . .,a:ft, «!, . . ., n« +

+ fh(xu ...,xh,nu.. ., ns — 1, . . .,

It is convenient to go from Af and ^4^ to


The operators A~ and A~ act in the following manner:

(4ffh (*i, • • •, Xh\ nu . . . , nh) =

— I . . . , s f t , *; / i t , . . . , nkf 1 ) dt,

f /)ft (î, • • •, xh; nu . . . , nk) =

- k

= ^ ^ s T ( ^ s ) / f e ( ^ ! , . . ,,xk; nu . . . , ns — 1 , . . .,nh)

Since ^1~, ^4~, and A~ form a representation of the algebra @z, thesame commutation relations hold for them as for the corresponding elementsof the Lie algebra @z, namely,

U+ A\] = -iA%r U r , A%] = iAr^ U+ ^-] = - 2 ^ ^ 2 .

5. Another method of realizing the operators A~, A*~ and A~. Themethod of realization proposed here seems very interesting to us. Wespecify the elements of SS not as sequences of functions of xl, . . ., x^,Hi, . . ., nk, but as sequences of functions of the x-parameters alone.

Let us consider, for example, the function f3(xl, x2, x3; 2, 1, - 4). Weassign to it the function /3,4(*i, xl9 x2; y\, yi9 y\, yt). More generally,If a function fk(xl9 . . ., x^; nl9 . . ., n^) is given, we first discard thezeros among the numbers nl9 . . ., n^. We then pick out the positivenumbers among the nt and denote their sum by m and the sum of theabsolute values of the negative nt by n. xx is then repeated \nx\ times,x2 is repeated \n2\ times, etc. Because of the symmetry of / in the pairs(Xj, rii) we can write down first all the arguments X( with positive «,-, thenall those with negative «,; we denote the resulting function by

. . . , î , . . . , Xk, • •

I n l I I n k f

Now we give a precise definition that does not depend on these argu-ments.

We introduce the space S£°, whose elements are infinite sequences/ = {/mn},m,n=o, i, ..., where U 6 C, fmn: X x . . . X X -> C for m + n > 0,

m-[-w

satisfying the following conditions:1) the functions / m n(î , . , ., xm ; yl9 . . ., yn) are continuous;


2) the functions /mw(x l5 . . ., xm ; yl9 . . ., yn) are symmetric in thefirst m arguments and in the last n arguments:

3) l|/||2= 2 21 {| » |

7711

i, .. .dxhdyi . ..

where the inner sum is taken over all partitions m = m1 + . . . + mk,n = n t + . . . + n k , a n d \ m \ = m l 9 . . ., m k , \ n \ = n l f . . ., n k . ( F o r

m = Owe take |m| = 1.)An isometric correspondence between 38° and the previous space is given

by

{fmn) *-+ {fh},where

fh (xu ...,xp,yl,...,yq;mi,...,mp,ni,..., nq) =

= fmn (* i , ...,xl,...,xp,...,xp; y { , ..., y u ..., y q , . . . , y q )

mi mp |m| | n ^ |

. . + ^ , TI = | n^ \ -f . .. + | nq\).

Now ^4^, A^f, and >1~ act in 36° according to the formulae:

, Xm\ yu

S5 = 1

i, n (î, • • •, xm, xs; yu . . . , yn) 4-

— J T ( 0 / m + i , n ( ^ i , • • - , x m , t; y u . . . , y n ) d t ,

., xm; yu . . ., yn) =m

2 J T ( ; r * ) / m - i , n ( ^ i , • • . , ^ s , • . . , x m ; y u . . . , y n ) —

— j


§6. Another method of constructing a representation of Gx

1. The general construction, a) Construction of the representation space.Let G be an arbitrary topological group and H a linear topological space inwhich a representation T(g) of G is defined. Further, Let X be a compacttopological space with a positive finite measure m. As before, we assumethat m is a countably additive, non-negative Borel measure whose supportis X.

We suppose that in H there is a linear functional / (/ =£ 0) invariantunder T(g). Then we construct a representation £/~ of the group Gx ofcontinuous mappings X -> G from the representation T(g) of G and thefunctional /.

We denote by Hx the set of all continuous mappings / = /(*) : X -* Hsuch that /(/(») does not depend on x and l{f{x)) - 0.

We introduce a new linear space 36° whose elements are formal finitesums of elements of Hx:

Here we set XJ + \2f = (kt + X2)/ if \ t + X2 =£ 0, and / + ( - / ) = 0.We emphasize that if fx (x) and f2 (x) are not proportional, thenf — f\ + Ii a n d f\ + A are regarded as distinct elements.

In <2%>° operations of addition and multiplication by a factor X E C aredefined in the natural way. Namely, the product o f / = / ( • ) by X G C isdefined as X o /(JC) = \f(x) if X =£ 0, and 0 o /(*) = 0. As a result, SBbecomes a linear space.

b) Action of the operators in 38°. We define a representation £/~ of Gx

inc^0 by

(2) (£/;/)(*) = %T(g[x))f(x),

Where X(f, / ) is a function of £ and / such that X(g, cf) = \(g, f) for anyc =£ 0, and we extend L^ by additivity to all elements (1).

LEMMA 6.1. The operators £/~ form a representation of Gx if and onlyif the function \(g, f) satisfies the following additional condition for anySu 82 e Gx and f G Hx: X(gu A)X(f2, / ) = X g ^ , / ) where

/ / f/ze weaker relation

the f/~ /orm <3 projective representation.The proof is obvious.c) Construction of a unitary representation. Let Ho C i / be the set of

elements £ such that /(£) = 0. By the invariance of /, Ho is an invariantsubspace of H. Suppose that an invariant positive definite scalar product(£i> £2) is defined in H.

Representations of the group SL{2, R) 53

We construct a scalar product in the space S£Q from the scalar product(£1, £2) in H. To do this we first fix a vector £0

e H such that /(£0) = 1-For any pair of elements / i = / iO) and / 2 = /2(#) of i / x we define ourscalar product as follows:

(3) <£, f2> = z(7i)7(^j exP ( — L = , \ (/;

where /?(*) = ft(x) - /(/p£o are elements of Ho for any x G Z. (We recallthat l(f(x)) is independent of x. Instead of l(f(x)) we write /(/), where

We extend this scalar product to the whole space 36° by linearity.LEMMA 6.2. The Hermitian form on S£° defined by (3) is positive

definite.PROOF. We choose arbitrary elements/j, . . .,fn of Hx and prove that

the matrix (ft, fj) is positive definite. We have

(4) ^JJL =2 J ^ ( _ _ 4 = f (/;(*), f-(x))dm(x))n.Hfi)l(fj) nZo V Hfi)l(fj) J '

Since a,-/ = ~X ~ yJlix), fj(x))dm(x) is positive definite, by Schur'

lemma each term of (4) is positive definite, and the lemma is proved.Thus, U^ acts in a pre-Hilbert space. Let us see how the multiplier

Mg, / ) can be chosen so that the representation is unitary.For this purpose we first construct from £0 a function /3(g) with values

in H: Q(g) = T(g)%0 - £0- Since / is invariant we have /(/3(g)) = l(T(g)%0) -= 0, that is, P(g) G Ho for any g G G. It is not hard to see that

is a cocycle with values in Ho, in other words, it satisfies the relation0fei) + T(gl)(3(g2) = P(glg2) for any gl9 g2 G G.

We observe that j3(g) depends on the way we have fixed the vector

LEMMA 6.3. / / we set

(5) M?,fi =

/'(x) = fix) - l(f)Zo, \c(g)\ = 1, then the operators Uj defined by(2) are unitary and form a projective representation. Specifically,Uz> UZ = c « , & ) £ f c j r , , where

(6)

X exp (i j Im (71 (g (i)) p ( f t (*)), p(gl (x))) dm (a:)) .The proof comes from a direct verification.


REMARK. This condition on X(g, f) is also necessary.We now state our final result. A linear topological space H is given and

also a representation T(g) of G in H. A linear functional / is given in Hthat is invariant under the action of G, that is, KT(g)%) = /(£) for anyg E G and £ E H. We define a scalar product (£1? £2)

m the subspace Ho

of elements £ such that /(£) = 0.A representation of Gx is constructed as follows. We consider continuous

mappings / = /(.)'• X -> H such that l(f(x)) = const =£ 0. Weintroducethe space S£° whose elements are the formal sums fx + . . . + fn withthe relations \J + X2f = (X1 + X 2) / if Xi + X2 =£ 0, / + ( - / ) = 0.We construct the scalar product:

f (/; (x), f'2(x))dm(x)\J /

^ 5l(fi)l(f2)

where fl(x) = ff(x) — /(/j-)£0> an(i £o *s a fixed vector in H such that

(£o) = 1- This scalar product is then extended to the whole spaced0.We denote by S£ the completion of $£° in this scalar product.

The operators Uj are defined by the formula (Ugf)(x) = X(g, f)T(g(x))f(x),where

c ( | ) e x p ( ^

kg) = ng)%0 - g0, \c(g) i = i,and are extended by additivity to sums of the form (1) and then to the com-pletion. These operators are unitary and form a projective representation of Gx,namely, £/~ Ug = c(gig2)U^, where c(gl5 g2) is defined by (6).

2. Construction of a representation of Gx, where G = PSL(2, R). Wenow apply the general construction described above to the case of thegroup G = PSL(2, R), given in the second form.

We define H as the space of all continuous functions on the circleIJI — 1 in which the representation acts according to the followingformula:

(7) !p

and the invariant linear functional / is

0

In the subspace Ho of functions /(?) for which l(f) = 0 we specify ascalar product as follows:

or, in integral form, 2n2n

(9)

It is clear from (8) that this scalar product is positive definite, and from


(9) that it is invariant under the operators T(g) of the form (7). (We recall

that

We now construct ^°.We fix in H the function £0 = /0(f) = 1. Then

P(g) = T(g)f0 - / 0 ; hence, P(g, f) = + al"2 " 1, where g=(? -) .

Note that j3(g, f) takes only real values. We examine the set Hx of con-tinuous functions f(x9 f) satisfying the following condition:

2n

-^ [ /(a;, elt)dt = l for all ^ 6 ^ -

The elements of 3£° are all possible finite formal linear combinations ofsuch functions: ^]hofi(^, £)> * e C. A scalar product is defined for anypair of functions fx = fx(x, f) and f2

= fi(x, f) in Hx by the formula

where f. = f. — \9 a nd | s then extended by linearity to the whole spaceThe representation operator U^ is defined by the formula

feHx, where

*, /) = exp(— c f In s i n X

/ ' — f — 1, and is then extended by linearity first to the whole space 3£° 9

and then to its completion S£mWe note that in the case considered here the scalar product of the vec-

tors T(g1(x))@(g2(x)) and (l(gi(x)) is real. Therefore, by Lemma 6.3 (see theexpression for c(gl9 g2)), the operators Ug form a representation of Gx.

We make here an essential remark. We can take for H the subspace offunctions on the unit circle | f | = 1 that are boundary values of analyticfunctions analytic (or anti-analytic) in the interior of the unit disc. Thenwe obtain other representations of Gx, which are projective.

3. Another construction of a representation of G x , where G = PSL(2, R).It is sometimes convenient to define representations of G not on functionson the circle but on functions on the line. Let G be given in the firstform.

We consider the space H of all real continuous functions that satisfy thefollowing condition: f{t) = O(t~2) as t -> ± <»; and we define a represen-


tation of G in H by the following formula:

We further define in H a G-invariant linear functional /(/):

*(/)= ] f(t)dt.

Let HQ C / / be the subspace of functions on which /(/) = 0. In Ho

there is an invariant positive definite scalar product+00+00

(/i, /1) = — J J In I *!-<— 00 —00

J t 1

We now fix the function fo(t) = ,1 + 2 in H, for which l(f0) = 1. Wealso set 0fe, f) = (7W 0 ) ( r ) - / 0 ( f ) .

We proceed to the construction of 3£°- We consider the set Hx of con-tinuous functions f(x, t), x G X, t E R, satisfying the following conditions:

1)/(X, 0 = 0 ( r 2 ) as t ~> ± oo;

2) J /(x, r)t/r = 1 for any x G X.

The elements of SS° are all possible formal linear combinations of such

functions: 2îo/i(s» *)» ^ 6 C.A scalar product is defined for any pair fi(x, t), f2(x, t) by the follow-

ing formula:

(—j In

where / - ( ^ 0= / f (^ , 0

The representation operator acts as follows:

where

f In I ^ —

REMARK. If we use instead of / / only the subspace H+ (or H~) offunctions which are boundary values of analytic functions in the upper (orlower) half-plane, then we obtain other (projective) representations of Gx.

4. Equivalence of the representation Ug to the representation constructedin §5. THEOREM 6.2 The representation Ug of Gx, G = PSL(2, R) con-structed here is equivalent to the representation constructed in §5. Hence

Representations of the group SZ(2, R) 57

it follows that Ug is irreducible.PROOF. For the proof it is sufficient to construct an isometric mapping

of SS° into the representation space of § 5 that commutes with the actionof Gx in these spaces.

Let us examine the construction of the representation in §6.2. In it 3£°consists of formal linear combinations of functions f(x, f), |?| = 1, such

that -L f f(x, eil)dt = 1 for any x G X. We expand f(x, eif) in a Fourierb

series in t: f(x, eu)= 1 + ân(x)eint. We associate with f(x, t)£3t°n=£0

an element of the space SSy constructed in §5: F = (1, u(x, n), . . .,. . ., u(xu nx\ . . ., u(xk, nk), . . .), where u(x, n) = (- \fan{x). Weextend this to a linear mapping of the whole space S£° onto $£\. Fromthe definition of the norm in these spaces it follows easily that themapping so constructed is an isometry. Furthermore, it can be shown thatit commutes with the action of Gx in these spaces.

§7. Construction with a Gaussian measure

1. To explain the construction of this section we find it convenient tomake some modifications in the general constructions in §6.1.

Let G be a topological group, E a real Hilbert space, and suppose thatan orthogonal representation T(g) of G in E and a cocycle with values inE are given, that is, a function ]3: G -* E satisfying the relation

From the part (T, 0) we construct a new unitary representation of G.In what follows we change the notation for the group and write T in placeof G, because in the examples F can be both G and Gx.

First we construct a new (complex) space M° whose elements areformal finite linear combinations of elements £,- G E:

(l) Xi ° *i + . . • + xw o £„, x, ec.

In contrast to the preceding section, X o £ and X£ are now regarded asdistinct.

Thus, the original space E has a natural embedding in 36° as a subset(not as a subspace!).

We now define a scalar product in 36°. Namely, for elements £1? £2 e H

we define a scalar product by the formula <£x, £2^ = e xP (£i> £2)* where

the round parentheses denote the scalar product in E, and then we extendthis scalar product by linearity to all formal linear combinations like (1).It is easy to verify that the Hermitian form so introduced is positivedefinite (see the proof of Lemma 6.2).

Let SS be the completion of S^ in the scalar product just introduced.We define a representation Uy of F in 36°. The action of operators Uy,


7 E r , on elements £ E E is given as follows:

*7v-iE = exp ( — i || P(V) | | a - ( r v 5 , P(v))

and then we extend these operators by linearity to the whole space d&°-It can easily be established that the U7 are unitary operators, hence can

be extended to the whole space $£, and that these operators form a rep-resentation of F. Later we shall see how this construction is related tothat of Araki and Streater.

2. Let us consider two examples.a) Let T = PSL(2, R). Let T(g) be the representation of PSL(2, R) con-

structed in §6.3 (or §6.2), and j3(g) the cocycle defined there. We denoteby E the real subspace of Ho also given there. From it we construct SSand the representation Ug.

THEOREM 7.1. The representation so constructed coincides with thecanonical representation introduced in §1.

b) Let T = (PSL(2, R))x. We consider the space of all mappingsf\ X -• E that are measurable on X and satisfy the condition

f (x)\\* dm (x)<oo.

This is a real Hilbert space. We define in it a representation T of T by theformula T(g)f = rfe(x))/(x)5 where T is the representation of example a).Next, we introduce a cocycle J3, setting (f(g) = P(g(x)), where ($(g) is thecocycle of example a), and we construct from the pair (Z1, j3) a represen-tation Up by the procedure indicated above.

THEOREM 7.2. The representation so constructed coincides with therepresentations £/~ constructed in §§2—6.

3. We explain briefly another method of constructing a representation ofthe group, which can be specialized to yield the construction explained atthe beginning of this section.

Let K be a real Hilbert space and K' its dual space. We consider thefunctional x(£') = e"11*'"* .It is normed, continuous in the norm, andpositive definite on K\ as on an additive group. Therefore it is the Fourier

transform of a weak distribution in K: %(£>')= \ £*<£'. S> dv(k). (A weakK

distribution is a finitely additive, normalized, non-negative measure definedon the algebra of cylinder sets in K, that is, sets of the level of Borelfunctions of finitely many linear functionals.)

It is known that v can be extended to a countably additive measure inan arbitrary nuclear extension K of K. We call this the standard Gaussianmeasure, and we quote two properties of this measure that we shall need.

1) The standard Gaussian measure in K is equivalent (that is, mutuallyabsolutely continuous) to its translations by elements of K.

Representations of the group SZ(2, R) 59

2) Every orthogonal transformation of K can be uniquely extended to alinear and measurable transformation in K that is defined almost every-where and preserves the Gaussian measure.

We now consider the space E, introduce the pair (T, j5) (See §7.1), andchoose the standard Gaussian measure /z in some nuclear extension E of E.We examine the space L2(E, n) of all square integrable complex-valuedfunctional on E (more precisely, of classes of functional that coincidealmost everywhere). From the pair (T, 0), we construct in L2(E, JU) a rep-resentation Ug:

4-"4"l\&8)\I2-<T(gMg), q» , o / \\= a 2 F (21 (g) cp + P (g)),

where F G L2(^, p), <p G E.The following theorem can be proved:THEOREM 7.3. The correspondence g ^ Ug is a unitary representation

of G in L2(£, M).

REMARK. If there is another cocycle ff in E' = E, then we can con-struct the more general representation:

4. We indicate briefly the connection between the representation con-structed here and that constructed in §7.1. Let £ G E. To £ we assign thefollowing function in L2(E, JJL): g •—• /'(cp) = ce(£, <P), where the number c is

determined by the condition c2 f e2^^ d\i = 1, c > 0 .

It is easy to verify that c = coe~llv?" , where c0 is an absolute constant.Let Fx and F2 correspond to the elements £t and £2 °f E. We compute

It is easy to see that (Fl9 F2) = ^(^'^)9and thus is the isometric mappingofd^ into L2(E, JU) given in §7.1. It can be verified that this mappingis an isomorphism. An elementary calculation shows that for a given T(g)the representations in SS and L2(E, (i) are equivalent.

It is well known that the space L2(E, /x), where M is the standardGaussian measure, can be represented in the form

00

L*(E, n) = e x p f f s ^ ®-^FH® • • • ® H >n

where H is the complexification of E' and H 0 . . . 0 H is the subspace ofn

generalized Hermite polynomials of degree n. Hence the preceding investiga-tions show that our representation of Gx is realized in S£ = exp H bymeans of the cocycle j3 (See §7.1). This realization coincides with thegeneral scheme of Streater and Araki, which they have examined, however,

60 A. M. Vershik, I. M. Gel'fand, andM. L Graev

only for certain soluble groups. In the terms used here the problem of theconstruction of a representation reduces to that of the discovery of thecocycle j3 and to the proof of the irreducibility of the representation of Gx.We have done this in the present paper for G — PSL(2, R).

References

[1] I. M. Gel'fand and M. I. Graev, Representations of the quaternion groups over locallycompact fields and function fields, Funktsional. Anal, i Prilozhen. 2 (1968) no. 1,20-35. MR 38 #4611.

[2] I. M. Gel'fand and I. Ya. Vilenkin, Nekotorye primeneniya garmonicheskogo analiza.Osnashchennye gilbertory prostranstva, Gos. Izdat. Fiz.-Mat. lit. Moscow 1961. MR26 #4173.Translation: Generalized Functions, Vol. 4: Applications of harmonic analysis,Academic Press, New York and London 1964. MR 30 #4152.

[3] I. M. Gel'fand, M. I. Graev, and I. Ya. Vilenkin, IntegraVnaya geometriya i svyazannyes nei voprosy teorii predstavlenii Gos. Izdat. Fiz.-Mat. lit. Moscow 1962. MR 28 #3324.Translation: Generalized Functions, Volume 5: Integral geometry and representationtheory, Academic Press, New York and London 1966. MR 34 # 7726.

[4] H. Araki, Factorizable representations of current algebra, Publ. Res. Inst. Math. Sci.5 (1969/70), 361-422. MR41 #7931.

[5] I. Dixmier, Les C*-algebres et leurs representations, second ed., Gauthier-Villars,Paris 1969. MR 30 # 1404, 39 # 7442.

[6] A. Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes inMathematics, Springer-Verlag, Berlin-Heidelberg-New York 1972.

[7] D. Mathon, Infinitely divisible projective representations of the lie Algebras, Proc.Cambridge Philos. Soc. 72 (1972) 357-368.

[8] K. R. Parthasarathy, Infinitely divisible representations and positive functions on acompact group, Comm. Math. Phys. 16,148-156 (1970).

[9] K. R. Parthasarathy and K. Schmidt, Infinitely divisible projective representations,cocycles, and Levy-Khinchine-Araki formula on locally compact groups, ResearchReport 17, Manchester-Sheffield School of Probability and Statistics, 1970.

[10] K. R. Parthasarathy and K. Schmidt, Factorizable representations of current groupsand the Araki-Woods embedding theorem, Acta Math. 128, 53-71 (1972).

[11] K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensorproducts, and central limit theorems of probability theory, Lecture Notes in Math-ematics, Springer-Verlag, Berlin-Heidelberg-New York 1972.

[12] L. Pukanszky, On the Kroneker products of irreducible representations of the 2 x 2real unimodular group, Part I, Trans. Amer. Math. Soc. 100 (1961) 116—152.

[13] R. F. Streater, Current commutation relations, continuous tensor products, andinfinitely divisible group representations, Rend. Sci. 1st. Fis. E. Fermi, 11 (1969),247-263.

[14] R. F. Streater, Continuous tensor products and current commutation relations, NuovoCimentoA53 (1968), 487.

[15] R. F. Streater, Infinitely divisible representations of lie algebras, Z. Wahrscheinlichkeits-theorie und Verw. Gebiete 19,1971, 67-80.

Received by the Editors,Translated by W. J. Holman 15 June 1973

To the memory of SergeiVasil'evich Fomin

REPRESENTATIONS OF THE GROUP OFDIFFEOMORPHISMS

A. M. Vershik, I. M. Gel'fand and M. I. Graev

This article contains a survey of results on representations of the diffeomorphism group of a non-compact manifold X associated with the space Y% of configurations (that is, of locally finite subsets) inX. These representations are constructed from a quasi-invariant measure n on Y%. In particular, necessaryand sufficient conditions are established for the representations to be irreducible. In the case of thePoisson measure ju a description is given of the corresponding representation ring.

Contents

Introduction 62§0. Basic definitions and some preliminary information 63§ 1. The ring of representations of Diff X associated with the space

of finite configurations 66§2. Quasi-invariant measures in the space of infinite

configurations 72§3. Representations of Diff X defined by quasi-invariant measures

in the space of infinite configurations (elementaryrepresentations) 82

§4. Representations of Diff X generated by the Poisson measure . 89§5. The ring of elementary representations generated by the

Poisson measure 94§6. Representations of Diff X associated with infinitely divisible

measures 95§7. Representations of the cross product S - C°°0Y)-Diff X . . . 100

Appendix 1. On the methods of defining measures on the configur-ation space Fx 102

Appendix 2. S^-cocyles and Fermi representations 104Appendix 3. Representations of Diff X associated with measures in

the tangent bundle of the space of infinite configurations . . . . 108References 109

61

62 A. M. Gershik, L M. Gel'fand andM. I. Graev

Introduction

This article is a survey of results on the representations of the groupDiff X of finite diffeomorphisms of a smooth non-compact manifold X. Asfor many infinite groups, it is rather difficult to see what the completestock of irreducible unitary representations of this group might be. There-fore, it is of some interest to single out certain natural classes ofrepresentations.

We consider the space Tx of infinite configurations (that is, locally finitesubsets) in X on which the group Diff X acts in a natural way. If n is aquasi-invariant measure in Tx and p is a representation of the symmetricgroup Sn(n = 1, 2, . . .), then we construct a unitary representation ofDiff X from JU and p, which we call elementary. There is, therefore, a closeconnection between the theory of elementary representations of Diff X andthe theories of quasi-invariant measures on Tx and representations of thesymmetric groups. We note that quasi-invariant measures on Tx are studiedin statistical physics (Gibbs measures and the simplest of them - thePoisson measure) (see, for example, [12]); and in the theory of pointprocesses (see, for example, [17] and elsewhere). The space of infinite con-figurations Tx is, in its own right, a very important example of an infinite-dimensional manifold, and its study is one of the interesting problems oftopology, analysis, and statistical physics.

Representations of Diff X that are of finite functional dimension, thatis, representations associated with the space of finite configurations, wereconsidered in [8] and [9]. In §1 we incidentally prove by a new methodthat the representations of a wide class are irreducible. However, we arebasically interested in representations of infinite functional dimensionassociated with Fx; they can be regarded as limits of "partially finite"representations.

In this paper necessary and sufficient conditions are obtained forelementary representations to be irreducible. In the case when JLX is thePoisson measure it is proved that the set of elementary representations ismultiplicatively closed, that is, the tensor product of two elementaryrepresentations splits into the sum of elementary representations, and thestructure of the corresponding representation ring is described.

An important property of Diff X, which distinguishes it from locallycompact groups and which will become apparent in the situations we dis-cuss, is that to a single orbit of Diff X in Tx there is no correspondingrepresentation; however, one can construct a representation from a measureon Tx that is ergodic with respect to the action of Diff X. More interest-ing and more widely studied is the class of representations associated withthe Poisson measure on Fx (see §4). The representation of Diff X in thespace L^(TX), where ji is the Poisson measure, arose (as an N/V limit) in[15]; however, the role of the Poisson measure was not noted here. It is

Representations of the group of diffeomorphisms 63

remarkable that this same representation can be realized in a Fock spaceas EXPp T, where T is a representation of Diff X in L^ (X) and j3 is acertain cocycle (see §4). This circumstance links the theory that wediscuss here with [1] and [2];

Representations of Diff X associated with the Poisson measure on Fx

are studied by another method in [22]. As far as we know, up to now,no measures, and in particular no Gibbs measures apart from the Poissonmeasures, have been discussed in connection with representations of Diff X.

Representations of the cross product of the additive group C°°(X) andthe group Diff X are investigated in a number of very interesting physicspapers (see [15] for a list of references; see also [19] and [20]). All therepresentations of Diff X discussed in this article extend to representationsof the cross product C°°(X)*Diff X\ the mathematical part of the resultsof [15] is contained in this paper.

§0. Basic definitions and some preliminary information

1. The group Diff X. Everywhere, X denotes a connected manifold ofclass C°°. Diff X denotes the group of all diffeomorphisms \p: X -> X thatare the identity outside a compact set (depending on i//). The group Diff Xis assumed to be furnished with the natural topology: a sequence 4/n isregarded as tending to \p if i// and every \jjn is the identity outside a certaincompact set K and if \pn, together with all its derivatives, tends to \puniformly on K. If Y C X is an arbitrary open subset, then Diff Y denotesthe subgroup of diffeomorphisms \p € Diff X that are the identity onX \ Y.

2. The groups S°° and S^. We denote by S°° the group of all permuta-tions of the sequence of natural numbers, by S^ C S°° the subgroup of allfinite permutations, and by Sn the group of all permutations of the numbers1, . . . , n (n = 1, 2, . . .). We regard the Sn as subgroups of S^; thus,Sx C . . . C Sn C . . . and S^ = lim Sn. In what follows, So is understood

to mean the trivial group.3. The configuration spaces Fx and Bx. Any locally finite subset of X

is called a configuration1 in X, that is a subset ) C I such that y n K isfinite for any compact set K C X. By this definition, any configuration iseither a finite or a countable subset of X\ if X is compact, then all con-figurations in X are finite.

Let us denote by Fx the space of all infinite and by B^ the space ofall finite configurations in X. The group of diffeomorphisms Diff X actsnaturally on Fx and Bx. The space of finite configurations B^ decomposes

Sometimes a configuration is defined differently, allowing points x e X to be included in 7 withrepetitions; with such a definition a configuration is not a subset of X.

64 A. M. Vershik, I. M. Gel'fand andM.L Graev

into a countable union of subsets that are transitive under Diff X:Bx = U B*fP\ where B^0 is the collection of all «-point subsets in X. We

note that B ^ consists of a single element — the empty set 0.For any subset Y C X with compact closure the space Tx splits into

the product Tx = BY X TX\Y- Consequently, since By = U B(y \ we have

Tx = LJ B(y *X r x x y , and all the subsets in this decomposition are invariantn > 0

under Diff Y.4. The space X°° and the topology in Tx. Let us consider the infinite

oo

product X°° = II X/? X{ = X, furnished with the weak topology. The group

S°° acts naturally on X°°. We define the subset r C I°° as the set of allsequences (xl9 . . . , xn, . . .) E I ° ° such that: 1) xt ¥= Xj when / = / and2) thejequence x l 5 . . . , xni . . . has no accumulation points in X. Thespace X°° is invariant under the action of Diff X and S°°, and the S°°-orbitof any point of X°° is closed.

There is a natural bijection X^/S00 -• F^ . We introduce the correspond-ing quotient topology in Tx; this topology is Hausdorff and metrizable.Similarly, the bijections Xn/Sn -> B^ } , where

Xn = {(*!, . . ., xn) 6 Zn; ^ =^=^ when i ^=/},

and 5W acts on Xn as the permutation group of the coordinates, give thetopology on B(

Fw)0i = 1, 2, . . .) and hence on BY = LJ ( )

It is easy to see that F^ is, as a topological space, the projective limitof the spaces B^. Namely, F^ = lim (B^-, irKK>)9 where K runs through

the open submanifolds in X with compact closures, andirKK>: BK -> BK{K' C K) is the restriction of the configuration 7 £ B^ toK\ that is, ITKK> y = y n Kf.

5. Quasi-invariant and ergodic measures. Let G be a group acting on aspace y. A measure JU given on some G-invariant a-algebra in Y is said tobe quasi-invariant under G if the inverse image of any measurable set ofpositive measure, under any transformation g: Y -» Y with g £ G, haspositive measure. If /z is quasi-invariant, then the measures n and gii (wheregfx is defined as the image of fi, that is, gfi(Q = Kg"1 Q) are equivalent;the density of gn with respect to /1 at a point y € 7 is denoted by

). The class of a-finite measures equivalent to /z is called the type of

A quasi-invariant measure jit in Y is said to be ergodic with respect to theaction of G if every measurable set A C Y such that ix(gA A A) = 0 for


any g G G is either a null set or a set of full measure.We discuss measures on Fx, and other spaces connected with Fx, that

are quasi-invariant under the action of Diff X, and we construct from thesemeasures unitary representations of Diff X.

6. Measures in the configuration space Fx. We define,1 as usual, thea-algebra $ (F^) of Borel sets on Fx. Henceforth, when we talk ofmeasures on F^,we mean2 complete, non-negative, Borel, normalized,countably-additive measures ju. Since the structure of a complete metricspace can be introduced in Fx, for any Borel measure /z the space (Fx, 11)is (after taking the completion of the a-algebra $ (Tx) with respect to M)a Lebesgue space [11] , and the technique of conditional decomposition(conditional measures, and so on) can be applied. The same applies to otherspaces and fibre bundles over Fx that occur in this paper.

Many measures on Fx arising for various reasons in statistical physics andprobability theory turn out to be quasi-invariant and ergodic under Diff X.The following example is classical.

POISSON MEASURE. Given any positive3 smooth measure m on a mani-fold X, we consider the union Ax - Bx U F ^ of all configurations on X.We define the measure of each subset { ) G Ax; | 7 O U | = n } by

where X > 0 is fixed. By Kolmogorov's theorem there exists a uniquemeasure on $[(FX) defined by these conditions. It is called the Poissonmeasure with parameter X (associated with the measure m on X).

Let us note the following important properties of the Poisson measure 11,which follow immediately from its definition.

1) When m (X) < °°, the measure fx is concentrated on the set Bx offinite configurations, and when m (X) = °°, it is concentrated on Fx.

2) Suppose that the manifold X = Xx U . . . U Xn is split arbitrarilyinto finitely many disjoint measurable subsets, thatAx = Ax X . . . X Ax is the corresponding decomposition of Ax into adirect product, and that jz,- is the projection of the Poisson measure JJL ontoAx. (i = 1, . . . , n). Then (x = nxX . . . X fin. This property of the Poissonmeasure is called infinite decomposability.

3) The Poisson measure is quasi-invariant under Diff X and invariantunder the subgroup Diff(X, m) C Diff X of diffeomorphisms preserving m.Here,

1Note that VL(rX) is a-generated by sets of the form CUn= {yerx;\y n U\= n), where [/runs

over the compact sets in X (n = 0, 1, . . .)•

In statistical physics a measure JU on Tx is usually called a state (see, for example, [12]) and in prob-ability theory and the theory of mass observation it is usually called a point random process (see, forexample, [17]).

By a positive smooth measure we mean a measure with positive density at all points x GX.

66 A. M. Vershik, I. M. Gel'fand and M. I. Graev

dutyiy) _ yy dm (ijr1*)^ > d\x.{y) 1 1 dm{x)

x£y

(the product makes sense, because by the finiteness of \p9 almost all thefactors are equal to 1).

4) If m(X) = oo5 then the Poisson measure JU is ergodic with respect toDiff X. Furthermore (see §4), if dim X > 1, then the Poisson measure isergodic with respect to Diff (X, m).

Any measure in Bx that is quasi-invariant under Diff X is equivalent toa sum of smooth positive measures on B*£\ In particular, any two quasi-invariant measures on B ^ are equivalent. Let us note that for any F C J ,where Y is compact, the projection of any quasi-invariant measure in Tx

onto By = LJ B& is non-zero for all n.

§ 1. The ring of representations of Diff X associated withthe space of finite configurations

We discuss here the simplest class of representations of Diff X. Theserepresentations have finite functional dimension; from the point of view oforbit theory they have been discussed in detail by Kirillov [9].

1. The representations Vp. We associate with each pair (n, p), where pis a unitary representation of the symmetric group Sn in a spaceW(n = 0, 1, . . . ) , a unitary representation Vp of Diff X. The constructionof Vp is similar to Weyl's construction of the irreducible finite-dimensionalrepresentations of the general linear group.

Given a positive smooth measure m on X, we define mn in Xn to bethe product measure: mn = m X . . .X m. We consider the spaceL2 (Xn, W) of functions F on Xn with values in the representation space

n

W of p such that2 = J i, . - . , xn)\\lvdm(xl)...dm(xn)<oo.

A unitary representation Un of Diff X is given on L2m (Xn, W) by the

formula

(1) {Un 0 0 F)(xit...,Xn)= [\ 4 / 2 (Xk)

where Jîx) = . Let us denote by Hn the subspace of functionsdm(x) >p

dm(\p lJC)

dm(x)F G L2

mn(Xn, W) such that F(xa(l), . . . ,xa(w)) = p"1 (a)F(xl5 . . . ,xn) for any

a G 5W. It is obvious that Hn p is invariant under Diff X.We define the representation Vp of Diff X as the restriction of Un from

/ . * ( * » , W)to// f l > p .


In the particular case when p is the unit representation of Sn, then Vp

acts by (1) on the space of scalar functions F(xly . . . , xn) that aresymmetric in all the arguments.

It is obvious that if m is replaced on X by any other smooth positivemeasure, each Vp is replaced by an equivalent representation.

Le t us construct another realization of Vp, which will be useful later on.Let Xn C Xn be the submanifold of points (xi9 . . . , *W)JE Xn with pair-wise distinct coordinates. We consider the fibration p of Xn by the orbitsof Sn, p: Xn -> B(^\ Note that p o \p = \p o p for any \p G Diff A\Suppose that we are given any measurable cross section s\ B ^ -> Xn.Obviously, for any \jj G Diff X and 7 G B*^ the elements ^(^"17) andi//~1(57) lie in the same fibre of p, and we define a function a onDiff I X B ^ with values in Sn by the formulasW'iy) = [^(sy)] o(\p, 7 ) , where 1 (x l 9 . . . , x n)o = (x O{1)9. . . , ^ f f ( n ) ) .

Let /i = pmrt be the projection onto B^* of the measuremn = m X . . . X m on Xn. We denote by ££(B^°, V) the space of functionsF on B^z) with values in W such that

\\F ||2 =

We define the representation F^ of Diff X in LjCB^0, PV) by

(2) {

It is not difficult to check that this representation is equivalent to theone constructed earlier. To see this it is sufficient to consider the maps*: HntO-» Ll(B^\ W) induced by the cross section5, ((s*F)(y) - F(sy)). It is easy to verify that s* is an isomorphism andthat the operators Vp(\p) in Hn p go over under 5* to operators of theform (2).

In the particular case when p is the unit representation of Sn, thenVp acts on L^(B^) according to the formula

2. Properties of the representations Vp. From the definition of Vp weobtain immediately the following result.

PROPOSITION 1. For any representations pj and p2 ofSn (n = 0, 1, . . .) there is an equivalence Vp**p* = FPi © Vp*.

DEFINITION (see [18]). The exterior product pxo p2 of representationsPi of Sn and p2 of Sn^ is the representation of Sn^n^ induced by the

a is a 1-cocycle of Diff X with values in the set of measurable maps B^ l ) -> Sn (see Appendix 2).

68 A. M. Vershik, I. M. Gel'fand andM. I. Graev

representation px X p 2 of Sn^ X Sn^\ p{ o p 2 = Ind5Wl

x ^2 (pt x p 2) . We are

assuming that Sn and £„ are embedded in Sn + n as the subgroups ofpermutations of 1, . . . , nx and of nx + 1, . . . , nn + n2, respectively. Note(see [18]) that exterior multiplication is commutative and associative. Thefollowing fact parallels standard results about representations of the classicalgroups in the Weyl realization.

PROPOSITION 2. For any nlt n2 = 0, 1 , 2 , . . . and any representationsPi and p 2 of Sn and Sn , respectively, there is an equivalenceypx»fh ^ ypi 0 yp2t

COROLLARY. The set of representations Vp is closed under the operationof tensor multiplication.

THEOREM 1. \) If p is an irreducible representation of Sn, then therepresentation Vp of Diff X is irreducible. 2) Two representations VPi andVPi, where px and p2 are irreducible representations of Sn and Sn ,respectively, are equivalent if and only if nx = n2 and px ~ p2.

PROOF. We consider P^", where pn is the regular representation ofSn (n = 0, 1 , 2 , . . .). It is easy to see that VPn is equivalent to therepresentation in <§) L2

m (X) given by

(vPnwF)(Xi, ...,xn) = ft J ^ / 2 ^ ) , P ( r ^ ! , . . . , r 1 ^ ) .Results of Kirillov ([9], Theorem 4) imply that the VPn are pairwise dis-joint and that the number of interlacings of VPn is «!, that is, equal tothe number of interlacings of pn. Hence and from Proposition 1 theassertion of the theorem follows immediately.

When dim X > 1, a stronger assertion is true, which we prove independ-ently of the results in [9]. Namely, let m be an arbitrary smooth positivemeasure on X such that m(X) = «>. We denote by Diff(Z, m) the subgroupof diffeomorphisms \p G Diff X that leave m invariant.

THEOREM 2. If dim X > 1, then the assertion of Theorem 1 is truefor the restrictions of the Vp to Diff(X, m).

The proof will depend on the following two assertions.LEMMA 1. For any natural number n and any set of distinct points

xl9 . . . , xn in X there exist neighbourhoods Ox, . . . , On, correspondingto xi9 . . . , xn> with the following properties'.

1) the closure Oj of Ot is C°°-diffeomorphic to a disc, Ot dOj = 0 when^ 0 0

2) for any permutation (kx, . . . , kn) of 1, . . . , n there is a diffeo-morphism \p G Diff(X, m) such that \p(Ot) = Ok.(i = 1, . . . , « ) .

PROOF. It is sufficient to consider the case when X is an open balland m is the Lebesgue measure in X. In this case it is easy to check thatfor any xt and Xj, i =£ /, there is a diffeomorphism \pfj G Diff(T, m) withthe following properties:


1) for any sufficiently small 8 > 0 we haveîjD%i = Dz

x., i>ipx- = D%p where Dex is a disc of radius e with centre at

x G X\2) the diffeomorphism i//zy is the identity in neighbourhoods of xk for

which k =fc /, /.Hence the assertion of the lemma follows immediately.LEMMA 2. For anyôpen connected submanifold Y C X with compact

closure, the subspace L^ (T) C L^ (Y) of functions f on Y such that

\ f(y)dm(y) = 0 is irreducible under the operators of the representation ofV

Diff(7, m): (U(^)f(y) = ft^y).PRCKDF. First we claim that for any non-trivial invariant subspace

X C LîY) and any neighbourhood 0 C Y, where 0 is C°°-diffeomorphicto a disc, there is a vector / G X , / =£ 0, such that supp / C O . For letus take an arbitrary vector / (1 )G «£ , /(1> =£ 0. Since Z^ 1 ^ const on Y,there is a y0 G 7 such that / ( 1 ) ^ const in any neighbourhood 0' of >vConsequently, there exists a diffeomorphism \jj E Diff(y, m) such thatsupp i// C o ' and /(1)(i//y) ^ / (1 )(y). We put f&y = fa)(\py)-f(1)(y).Then /<2> G X, /(2) ¥= 0 and supp / ( 2 ) C 0 ' . If the neighbourhood 0' issufficiently small, then, by Lemma 1, there is a diffeomorphismi//i G Diff(y, m) with ^ 0 ' C 0 that carries / ( 2 ) into a vector / withsupp / C 0. ^

Let us suppose that L^(Y) -%i® ^2»where « x and %2 are non-zeroinvariant subspaces. We fix neighbourhoods 0 and 0 ' in 7 such that 0and 0 ' are C°°-diffeomorphic to discs, 0 Pi 0 ' = 0, and m(0) = m(0').From what has been proved, there are f( E X (, ft = 0, such thatsupp/;- C 0 (i = 1, 2).

It is obvious that we can find a neighbourhood 0X C 0, where Ox isC°°-diffeomorphic to a disc, and a diffeomorphism \J/ G Diff(0, m) such

that \ fi(4iy)f2(y)dm(y) =£ 0; without loss of generality we may assumeithat \p = 1. For any € > 0 we can write_0 = O± U 0 8 U(0 \ ( 0 ! U 08)),where O8 is C°°-diffeomorphic to a disc, Ox O 0 e = 0, andm(0 \ (0! U 08)) < 8. It is not difficult to prove that there is adiffeomorphism \//e G Diff(T, m) that is the identity on 0 t and such thatm(\pe Oe \ O') < e (see, for example, [3], Lemma 1.1). Since 0 O 0 ' = 0,we have

0\(0iU08)


Consequently, because Xi and X* are orthogonal,

\fi{y)hW)dm{y)+ j fd^)h^)dm(y) +01 Oe\$Q1Ot

Since the second and third terms in this equation can be made arbitrarily

small, we have \ fi(y)f2(y)dm(y) - 0, which is a contradiction.

PROOF OF THEOREM 2. Let us realize the representation Vp = Vn>p

of Diff X as acting on the subspace Hn p C L2m (Xn, W), where W is the

space of the representation p of Sn ( for the definition of Hn , see §1.1).In this realization the operators of the representation of Diff(Z, m) havethe following form:

(yn' P (^)F)(xXi . . ., xn) = Fiq^Xi, . . ., if-^n), if 6 Diff (X, JII).

Let Ol, . . . , Ow be arbitrary disjoint neighbourhoods in X satisfyingconditions 1 and 2 of Lemma 1. We denote by H~'p the sub-

" i , . . . , Ufi

space of functions of Hn p that are concentrated onU (0fr X . . . X 6k ) C Xn where (A:l5 . . . , kn) runs over all permuta-

(klt...,kn) l n

tions of (1, . . . , « ) ; obviously there is a natural isomorphism

We consider the subspace

Hno\l..,on^Zl(Ox) 0 . . . ® Li(On) (g) PF,

where L^ (O,-) C L2m (Ot) is the orthogonal complement to the subspace of

contsants. From the definition it follows that H^fP Q is invariant under

under the subgroup Go ^0 of diffeomorphisms \p E Diff(Z, m) such

that MOi U . . . U ^ ) ' = O 1 U . . . U O r t . W e denote by V£p Q the

restriction of the representation Vn>p of GQ 0 to HQ>P Q

Note that the subgroup GJ? n C Gn n of diffeomorphismsult... ,un ulf... ,un

that are the identity on (9^ . . . , Ow acts trivially on Hg>p Q and that

i * • • • ' / I

by Lemma 1 the factor group Gn n IG% n is isomorphic to theult... ,un ul,. .. , un

cross product of Diff(0!, m)X . . . X Diff(Ow, m) with *Sw. The assertionhow follows easily from this and from Lemma 2.

The representations V^tP Q ofGQ o , where p runs over the

i > • • • > fi i » • • • J pj

inequivalent irreducible representations of Sn, are irreducible and pairwise


inequivalent.We now claim that the representation V^>p

Q ofGQ Q occurs in

Vn>p with multiplicity 1 and not at all in representations Vn>p , wherep + p, nor in representations Vn>p>, where ri <^n.

For let H' be the orthogonal complement to H^>p Q ini/wp-We

split H' into the sum of subspaces that are primary with respect toDiff(Olf m)X . . . X Diff(Ow, m). It is not difficult to see that in each ofthese subspaces at least one of the subgroups Diff(6^, m) (/= 1, . . . , « )acts trivially. But the representation of each subgroup Diff(0,-, m) inHQP

O is a multiple of a non-trivial irreducible representation. Conse-quently, the representations V£>p

Q are not contained in H', nor for the

same reason in Hn> p , n < n.From the properties of V^>p

Q we have just established it follows

immediately that the representations Vp of Diff(X m) are pairwise in-equivalent. We claim that they are irreducible.

Let X C Hn p be a subspace invariant under Diff(X, m), X =£ 0. Thenfor any collection Ou . . . , 0W of disjoint neighbourhoods satisfyingconditions 1 and 2 of Lemma 1 either H"tP C ^ , or^ ult... ,unHo[P...,o n X = 0. It is not difficult to see that the spacesHg'p 0 generate Hn p, therefore, H£p

Q C X for some collectionOu . . . , On. But then, by Lemma 1, X contains the whole ofH^jp

0 and hence coincides with Hn . The theorem is now proved.

REMARK. Let us denote by 21 the group of all (classes of coincidingmod 0) invertible measurable transformations of X that preserve the mea-sure m (the dimension of X is arbitrary); we furnish W with the weaktopology. The representation Vp of Diff(X m) C 21 extends naturally toa representation of 21 and the resulting representation Vp of 21 is con-tinuous in the weak topology. It is easy to show that in the weak topologyDiff(X, m), for dim X > 1, is everywhere dense in 21. This makes itpossible to prove Theorem 2 anew, reducing its proof to those of theanalogous assertions for 21, which are easily verified. On the other hand,this path enables us to establish Theorem 1 for any weakly dense subgroupof 21, that is, to prove the following proposition.

THEOREM 3. The assertions of Theorem 1 are true for the restrictionsof the representations Vp of 21 to any subgroup G C 21 that is weaklydense in 21.

3. The representation ring 'M . We consider the free module !J9 over Zon the set of all pairwise inequivalent irreducible representations Vp ofDiff X as basis. By the propositions in §1.2, the tensor productj/Pi 0 yPi of irreducible representations VPi and Vp* decomposes into asum of irreducible representations Vp and therefore is an element of Jl'.

72 A. M. Verskik, I. M. Gel'fand andM. I. Graev

In this way a ring structure is defined in 91, where multiplication is thetensor product.

Let us introduce another ring R(S) associated with the representations ofthe symmetric groups Sn (see [18]). We denote by R(Sn) the free moduleover Z on the set of pairwise inequivalent irreducible representations ofSn (n = 0, 1, 2, . . . ) as basis (where R(S0) - Z). We consider the Z-module

R(S) = © R(Sn) and give a ring structure to R(S) by defining multiplicationn — 0

as the exterior product. From the propositions in §1.2 we obtainimmediately the following result.

THEOREM 4. The ring ffi generated by the representations Vp ofDiff X is isomorphic to R(S).

For the map p -> Vp, where p runs over the representations ofSn (n = 0, 1, . . . ) extends to a ring isomorphism R(S) -*~ ffi.

REMARK. There exists a natural ring isomorphism

0: R(S) -+• Zlfli, a2, . . . ] ,

where an is the n-th elementary symmetric function in an infinite numberof unknowns, n - 1, 2, . . . ; for the definition of 6 see, for example, [18].By the theorem we have proved, there is a ring isomorphism

X -> Z[al9 a2, . • • ] , where to each representation V9 there correspondsthe symmetric function 0(p).

These symmetric functions in an infinite number of unknowns have theusual properties of characters: each representation Vp is uniquely deter-mined by its symmetric function, on adding two representations theircorresponding symmetric functions are added, and on taking the tensorproduct they are multiplied.

§2. Quasi-invariant measures in the space of infinite configurations

Before turning to the discussion of representations of Diff X associatedwith the space of infinite configurations Tx, we ought first of all to studyin detail measures in Vx, and in fibrations over it, that are quasi-invariantunder Diff X. We have already recalled that there are many such measureswith various properties (see §0.6); these measures arise (in anotherconnection) in statistical physics, probability theory, and elsewhere.

The ergodic theory for infinite dimensional groups differs in many waysfrom the theory for locally compact groups (see, for example, [6]). Inparticular, the action of Diff X in Yx is such that in Tx there is noquasi-invariant measure that is concentrated on a single orbit.1 In addition,care is needed because an infinite-dimensional group can act transitively, butnot ergodically, on an infinite-dimensional space [13]. This explains the

I

For locally compact groups such a measure exists and is equivalent to the transform of the Haar measureon the group.

Representations of the group of diffeomorphisms 7 3

somewhat lengthy proof of the lemma in §2.1, which at first glance wouldappear obvious.

1. Lemma on quasi-invariant measures on B ^ X Yx y.* LEMMA 1. LetY C X be a connected open submanifold with compact closure, let fin bea measure on B^^X Fx_Y that is quasi-invariant under the subgroupDiff Y, and let iin and ju ' be the projections of (xn onto B ^ and ^x_r

respectively. Then fxn is equivalent to n'n X JUJJ (n = 0, 1, 2, . . . ).REMARK. If ixn is the restriction to B(

yw) X Tx_Y of a fixed quasi-invariant

measure JJL on Tx, then the measures ju ' on Fy_ r are, generally speaking,not equivalent. It is easy to show that the equivalence of the measuresfx'n' on rx_Y {n = 0, 1 , 2 , . . . ) corresponds precisely to the equivalence ofthe measures [i and ju' X // ' on Tx, where n' and /z" are the projectionsof ix onto BY and I ^ y 1

First we prove the following geometrically obvious proposition.PROPOSITION \. In Diff Y there is a countable set of one-parameter

subgroups Gl such that the group G C Diff Y generated by them actstransitively in B ^ (n = 1, 2, . . . ) .

PROOF. We suppose first that dim Y = 1. We specify in Y a countablebasis of neighbourhoods Ur, Ur C Y, that are diffeomorphic to R1. We fixfor each r a diffeomorphism $/. R1 ->• Ur. Under $r the group of translationson R1 goes over into a one-parameter group of diffeomorphisms x -+ ft(x)on Ur (-°° < t < °°), which acts transitively on Ur. The map <pr can al-ways be chosen so that the diffeomorphisms x -* ft(x) on Ur extendtrivially to a diffeomorphism on the whole of Y. It is not difficult tocheck that the sequence of groups {Gt} constructed in this way satisfiesthe required condition.

Now let dim Y = p, where p > 1. We specify in Rp a countable set ofone-parameter subgroups Hx C Diff Rp such that the group generated bythem acts transitively in Rp; the construction of such a family presents nodifficulty.

Now we take a countable basis of neighbourhoods Ur in Y, diffeo-morphic to Rp, and fix diffeomorphisms \pr\ R

p ->- Ur. Let us denote byGlr the image of Hj under <pr. The elements of Glr can be extendedtrivially to diffeomorphisms over the whole of Y, and so Glr can beregarded as a one-parameter subgroup of Diff Y.

For a fixed r, the subgroups Glr generate a group which acts transitivelyin Ur and leaves the points of Y \ Ur fixed. Hence it is obvious that thegroup G C Diff Y generated by all the Glr acts ^-transitively inY (n = 1, 2, . . . ), and Proposition 1 is proved.

The following proposition is concerned with the theory of measurablecurrents of a quasi-invariant measure.

When ju is the Poisson measure, the equivalence n ~ ju' X n" is a direct consequence of the property ofbeing infinitely decomposable. The assertion of the lemma in this case is trivial.

74 A. M. Verskik, I. M. Gel'fand and M. I. Graev

PROPOSITION 2. Suppose that R1 acts measurably1 on the Lebesguespace (X, n) with quasi-invariant measure JU and that f is a measurablepartitioning of (X, /x) that is fixed mod 0 under R1. Then for almost allC G f the conditional measures iic on C are quasi-invariant under R1.

PROOF. For any t G R and C G f we put

q(t, C) = iA

where the inf is taken over all subsets A C C, with MCC4) = 1- We alsouse the notation qo(t, C) = q(T, C)q(-t, C). Obviously, the conditionyC ^ Tl[ic is equivalent to qo(t, C) = 1.

Since the action of R1 on (X, JU) is measurable and f is a measurablepartitioning, q{t, Q, and hence also qo(t, C), are measurable as functionson R1 X JT (Xt = X/Z).

Since JJL is quasi-invariant, for any fixed ( G R 1 we have ixc ~ Tx\f foralmost all C E f with respect to the measure JU? on X^ (fx^ is the pro-jection of pi); hence qo(t, Q = 1 almost everywhere with respect to /i? onX^. Hence, by Fubini's theorem for (R1 X X^, m X )uf), where m is theLebesgue measure on R1, for almost all C G f with respect to pif we have:m { t e R1; (jfo(r, C) =£ 1 } = 0.

On the other hand, the set of t G R1 for which JUC ~ ^rMC (for a fixedC) forms a group. Thus, for almost all C G f the set { t E.Rl\qQ{t, C)= \)is a subgroup of R1 of full Lebesgue measure. But every subgroup of alocally compact group with full Haar measure coincides with the wholegroup [4]. Consequently, for almost allC G f { t G R1; qo(t, Q = 1 } = R1, that is, for almost all C G f themeasure JJLC is quasi-invariant under the action of R1, and Proposition 2 isproved.

PROOF OF LEMMA 1. Let {GJ be a countable set of one-parametersubgroups of Diff Y such that the group G generated by them actstransitively in B ( ^ ; such a set exists by Proposition 1.

Since Gt = R1, it follows from Proposition 2 that for almost all, (in thesense of Mn')> configurations 7 G Tx Y the conditional measure JU on Bîs quasi-invariant under each Gl (/ = 1, 2, . . . ), hence also under thewhole group G generated by them.

On the other hand, the measures on B(rw) that are quasi-invariant under

G are all equivalent to each other, and consequently to fi'n. For on B^\ ason every smooth manifold, there is, up to equivalence, a unique measurethat is quasi-invariant under a group of diffeomorphisms acting transitively,namely, the smooth measure with everywhere positive density.

That is the map R1 X X -* X ((g, x) -* gx) is measurable as a map between spaces with measuresm X M and /u respectively, where m is the Lebesgue measure in R1.


Thus, for almost all 7 G TX_Y (m *ne s e n s e °f Mp the conditional mea-sure y?n on B ^ is equivalent to ixn, and the lemma is proved.

2. Measurable indexings in Tx. We say that / is an indexing in Tx if foreach configuration 7 6 ^ there is a bijective map i(y, -): y ^ N,N= {1, 2, . . .}.

We denote by Fx 1 the subset of elements (7, x) G F^ X X such thatx G 7, and we associate with each indexing / a bijective mapFx 1 ->• F^ X iV, defined by (7, x) -* (7, /(% x)). If this map is measurablein both directions (with respect to Borel a-algebras on F^ l and F^ X AO,then the indexing / is called measurable.1

Let / be a measurable indexing. We introduce a sequence of measurablemaps ak\ Tx ->• X (k = 1, 2, . . . ) defined by the conditions:ak(y) G 7, /(7, ^ (7) ) = k (that is, 0^(7) is the k-th element of the con-figuration 7). We associate with / a cross section s: Fx -> X°, definedby 5(7) = (a 1(7), . . . , an(y), . . . ). It is not difficult to verify that the set5 Fjf is measurable and that the bijective map F^ -> sFx is measurable inboth directions. ^

For any ^ G Diff X and 7 G F^, the elements s^'1 y) e X°° andijj'1 (sy) G X°° belong to the same ^-orbit in ^°°. We define a mapa: Diff XX Tx -> 5°° by î / /"^) = [i//"1 (57)] a(0, 7); the notation heremeans (xl9 . . . , xn, . . . ) o = (xa(1), . . . , xa(wj, . . . ).

Let us now introduce the idea of an admissible indexing. We are givenan increasing sequence X{ C . . . C Xk C . . . of connected open subsetswith compact closures such that X = U A^.

DEFINITION. We say that a measurable indexing / is admissible (withrespect to the given sequence Xx C . . . C Xn C . . . ) if the mapa: Diff XX Fx -> *S°° defined by it satisfies the following condition: ifsupp \// C Xk and | 7 O Xk \ = n, then o(\p, 7) G ^ (k = 1, 2, . . . ;* = 0, 1, . . . ) .

In particular, o(\p, 7) G 5^ for any \// G Diff X and 7 G F x .It is not difficult to construct examples of admissible indexings. For

example, the following indexing, which was proved to be measurable in[17], is admissible.

Let a continuous metric be given on X. With each positive integer k weassociate a covering (Xkl)l= ^ 2j of X by disjoint measurable subsets withdiameters not exceeding \/k, satisfying the following two conditions.

1) the partitioning X = Xkl U . . . U XkiU . . . is a refinement ofX = XlU(X2\X1)U .. .U(Xn\Xn^)U .. . ;

2) Each set Xn is covered by finitely many of the sets Xkl.

1If a measure JI is given in Tx, then indexings need be given only on subsets of full measure in r x , and

we make no distinction between indexings that coincide mod 0.

76 A.M. Vershik, I. M. Gel'fand and M. I. Graev

It is obvious that such a covering exists. We number its elements so thatif Xki C Xn and Xkj C X \ Xn, then i < j (n = 1, 2, . . . ). For any x E Xand ^ E i V w e put Z OO = /, if x E X^. The correspondencex -* (/i(x), . . . , fkix), . . . ) is a morphism from X to the set of allsequences of positive integers. We define an ordering in X by puttingx ' < x " if ( / A x 1 ) , .. . f k ( x ' ) 9 .. . ) < ( f l ( x " ) , . . . , f k ( x n ) , . . . ) i n t h elexicographic ordering.

For any 7 E Tx and x E 7, the_set { x ' E 7 ^ ' -< #} is finite, becauseit is contained in the compact set I n U . . . U XXj^xy Consequently, for any7 E Fx, the set of elements x E 7 is a sequence with respect to theordering introduced in X; we denote by z(7» x) the number of elementsx'^y in this sequence. The map (7, x) -> /(% x) so constructed is ameasurable indexing (see [17]). It is not difficult to prove that it is alsoadmissible.

3. Convolution of measures. DEFINITION. The convolution Mi * M2 (see,for example, [17]) of two measures Mi and ji2 on the space of all con-figurations Ax = FÛ Bx is defined as the image of the product measureMi x M2 o n &x x ^x u n de r the map (7 i , y2) ^ Ti u 72-

REMARK. This definition agrees with the usual definition for the con-volution of two measures in the space $F (X) of generalized functions onX(AX is embedded in 2F{X) by 7 -* S 5X), because the union of (disjoint)

configurations corresponds to the sum of their images in IF (X).It is obvious that ^(Mi* M2) = ^Mi * ^M2 for any \jj E Diff X, where

\j/[i is the image of M under the diffeomorphism \p. Hence the convolutionof quasi-invariant {under Diff X) measures is itself quasi-invariant.

Note that / / Mi and \i2 are Poisson measures with parameters \x and X2,respectively, then their convolution Mi * M2 is the Poisson measure withparameter Xj + X2. (This fact follows easily from the definition of thePoisson measure).

Later on we shall be interested in the case when one of the factors is aquasi-invariant measure concentrated on Fx, and the second is a smoothpositive measure mn concentrated on B^x\n = 1, 2, . . . ). Since all smoothpositive measures mn on B ^ } are equivalent, the type of the measureH*mn depends only on the type of M and on n.

Let us agree to call the type of JJL * mn on Yx the n-point augmentationof M and to denote it by n o M- Thus, with each measure JJL on Tx thereis associated a sequence of measures 0 o M ~ M> 1 °M> • • • »n ° M> • • • definedup to equivalence. Note that nxo («2° M) ^ («i +^2)°M f ° r anY î a n ( i

Here we establish the following properties of the operation °.1) For any quasi-invariant measure M on Tx there exists a quasi-invariant

measure (if such that 1 ° M' ~ M-2) / / (2 measure M 0/7 F ^ w ergo die, then 1 ° M W 0&O ergodic.


To prove this we give an admissible indexing / on Fx and lets: Fx -> X°° be the cross section defined by this indexing (see §2.2). Wedenote by JTS the image of Fx under s and by A the minimal ^-invariantsubset of X00 containing Y^ obviously, As is the disjoint unionAs = U Yso. Since Ys C X°° is measurable and S^ countable, As is a mea-

surable subset of X°°. Since the indexing i is admissible, it follows that As

is invariant under Diff X.Let jit be a measure on Fx that is quasi-invariant under Diff X. Let c be

an arbitrary positive function on S^ such that 2 c(o) = 1; we introduceo£5 M

a measure M on X°° by the formula:

? = S c(a)(sfi)o,

where (5/x)a is the image of // under the map y -* (.?7)a. In other words,for any measurable subset A C Z°°

(1) ? ( 4 ) = 2 c(a)fx[p(^n^scr)]

where p is the projection X°° -^ Fx.Obviously, M(AS) = 1. Note that the choice of the positive function c on

S^ does not play a role in defining £, because the measures on X°° con-structed from two such functions are equivalent. From the definition of Mit follows easily that:

a) pfx = ix where pjl is the projection of £ onto Fx.b) the measure £ on X°° is quasi-invariant under both Diff X and S^.We cite without proof two further simple assertions.PROPOSITION 3. If a normalized measure fxx on X°° is quasi-invariant under

Diff X and if p\x.\ = M and Mi(A5) = 1, then Mi ~ M-PROPOSITION A. If a measure ii in Fx is ergodic, then ]1 is also ergodic

with respect to Diff X.oo

Let us decompose the space X°° = U Xi9 where Xt = X, into the directOO i = 1 r^s ^/

product X" = X X n Xj and consider the induced map h = X X X°° -* X°°/ =2

(that is, /i(x; {xft} ^=1) = ( {xfh} %=l;x[j=jc, x'k =xk_x when k > 1)).

PROPOSITION 5. /*(m X JLI) ~ m * jit, where m is an arbitrary smoothpositive measure on X.

PROOF. Consider the diagram

x x r Y — - > r Y

78 A. M. Vershik, I. M. Gel*fand andM. I. Graev

where px = Id X p, h(x,^y) = 7 U {x}. Obviously, this is commutative,and (p o h) (m X /I) = (h ° px) (m X £) = m * fi. Further, the measurehim X p) is quasi-invariant and concentrated on As (since the indexing /is admissible, the point h(x, 57) belongs to the same 5^-orbitâs5(7 U {x} )). Consequently, by Proposition 3, him X pt) ~ m * /x.

The proof of the following assertion is similar to that of Lemma 1 in §2.1.PROPOSITION 6. Every measure /x in X°° that is quasi-invariant under

Diff X is equivalent\Jo the product mn X \xn of its projections in thefactorization X°° = Xn X X™+1; moreover, mn is equivalent to a positivesmooth measure on Xn, and fxn is quasi-invariant under Diff X.

COROLLARY. A quasi-invariant measure ]x in X°° is ergodic if and onlyif it is regular (that is, satisfies the 0 — 1 law).

PROOF OF PROPERTY 1). Let n be a quasi-invariant measure on Fx.By Proposition 6, JJL ~ him X p^), where m is a smooth positive measure onX, /*! is a quasi-invariant measure on X°°, and h: X X X°° ->• X°° is the mapinduced by the direct product (see above). Since / is admissible, /xl5 like JJL,is concentrated on As; consequently, by Proposition 3, /xi ~ /x' is a quasi-invariant measure in P^. By Proposition 5, /I ~ /z(m X JU') ~ m * JU';consequently, /x ~ m * ju'> as required.

PROOF OF PROPERTY 2). Ifjhe measure /x in F^ is ergodic, then byProposition 4, the measure /x in X°° is ergodic; consequently, by the corol-lary to Proposition 6, £ is regular. Obviously, m X H is then also regularand therefore ergodic. Consequently, the measure m * ju ~ him X /I) isalso ergodic and hence, so is its projection m * ju.

DEFINITION. We say that a quasi-invariant measure \x is saturated if1 o /x ~ jLt (and consequently, « o /x ~ /x for any n).

It is not difficult to verify that the Poisson measure is saturated (thisfollows from the property of being infinitely decomposable).

We now give a criterion for a measure fx to be saturated. The mapT: X00 -• X00, defined by (Tx)t = xi+l(i ^ 1 , 2, . . . ) is called lefttranslation in X°°. Obviously, the subset X°° is ^-invariant.

PROPOSITION 7. For a quasi-invariant measure on^Tx to be saturatedit is necessary and sufficient that the measure £ on X00 corresponding to it(defined by means of a fixed admissible indexing) is quasi-invariant underthe left translation T.

PROOF. From the definition of the left translation T it^follows thatpt ~ h(m X J/x). On the other hand, by Proposition 3, 1 ° JJL ~ h(m X /x).Hence it is obvious that the condition 1 ° ju- ~ /x is equivalent to ju ~ T]x.

An example of a non-saturated measure JU will be given in Appendix 1.4. The space Yx n and Campbell's measure on P j n . We consider the

Cartesian product Yx X Xn (n = 1, 2, . . . ) and denote by Tx n the set ofelements (7; xu . . . , xn) G F x X Xn, where 7 G r j ( x,- G X, such thatxt G 7 (/ = ! , . . . , « ) and xz- =£ x;- when / =£ /. Further, we put Tx 0 = Fy.


Obviously, Tx n is closed in Tx X Xn.Now Fx n can be regarded as a fibre space, TT: VX n -> F ^ , whose fibre

over a point 7 E F ^ is the collection of all ordered «-point subsets in 7.Let us denote by Wn the cr-algebra of all Borel sets in Tx n. We associate

with each subset C G I n a function on F ^ : ^ ( 7 ) = {the number of points(x l 9 . . . , xn) E Xn such that (7: xl9 . . . , *„) E C } .

From the continuity of IT it follows that *>c is a Borel function.DEFINITION. Let pi be a measure on F ^ . The Campbell measure on

TXn associated with /x is the measure # on 2In defined by

= JrA Campbell measure /I induces on the fibres of the fibration

TT: Tx n -* Fx a uniform measure, which is 1 at each point of the fibre.We define in F ^ n the actions of the groups Diff X and Sn:

a: (Y; xr, . . ., xn) - * (y; xo{1), . . ., xo(n)).

Obviously, \p and a are continuous and \// o <? = o ° 4? for any \p E Diff Xand o € Sn. The next result is easy to establish.

LEMMA 2. 772e Campbell measure ^x on Tx n corresponding to a measureJJL on Tx is invariant under Sn. If the measure /i on Tx is invariant underDiff X, then the Campbell measure /I is also quasi-invariant under Diff X,and _

d\i (-c) c=(v; xi, . . ., xn)

Now let / be a measurable indexing in Fx. We denote by Nn the set ofall ^-tuples of natural numbers (il9 . . . , /„), where ip =£ iq when p ^ q(p, q = 1, . . . , « ) . We define a map

(2)

by(Y;

This map is bijective, measurable in both directions, and carries^ theCampbell measure M £ n Fx n into the measure /z X v on F ^ X Nn, wherev is the measure on Nn

9 that is equal to 1 at each point on Nn. Thus, thespace (Tx n, JU) can be identified with (F^ X TV", pi X v).

Under this identification the actions of Sn and Diff X go over from^x,n t 0 rx x ^ 1 : Tt i s n o t difficult to verify that the action of thesegroups on Tx X Nn are given by:

a: (Y, a) H^ (Y, aa),

op: (Y, a) »-> (tp-^, a(a|),


where a = (il9 . . . , / „ ) ; aa = ( i a ( 1 ) , . . . , ia(n)) a G Sn;

aa = (aO'i), . . . , o(in)) o G S°°; and a(#, 7) is the function onDiff X X Fx with values in S°° defined by i (see §2.2).

5. The map Fx X Xn -> Fx n. Let us consider the spaces Yx X Xn andFx n together with their a-algebras of Borel subsets (see §2.4). Let M be aquasi-invariant measure on Fx, and mn a smooth, positive measure on Xn.In Tx X Xn we specify the measure /i X mn and in Fx n the Campbellmeasure fT°~ix corresponding to n o ^ ~ M * mn on Yx.

A map a: F^ X X" -> Fx n is given by the following formula:

a(v; *lt . . ., *„) == (v U {«i, • • ., *n}; *i, • • ., xn);

a is taken to be defined on the subset of elements(7: * ! , . . . , xrt) G Tx X Xn for which 7 O {;cl3 . . . , xn } = 0; it is notdifficult to verify that this subset and its image in Tx n are sets of fullmeasure. Nor is it difficult to check that a is measurable in both directionsand commutes with the action of Diff X on both Tx X Xn and rx n.

THEOREM. The image a(ju X mn) of the measure \x X mn on Tx X Xn

under a is equivalent to the Campbell measure n © JU.PROOF. We carry out the proof for the case n = 1; the arguments for

arbitrary n are similar.We define maps 0 : Tx X X -+ Tx and a2: Tx x -> Fx by

«i(7» *) = 7 u { } , ^2(7. ^) = 7- It is obvious that the following diagramcommutes:

\T

The image of the measure fxXmonTxXX under e^ = OL2OL is 1 ° /x-Hence it follows that a(/x X w) •< 1 © pi. It remains to prove thata(ju X m) > 1 © JU.

First we construct a measurable indexing in Tx in the following way. Wefix a continuous metric p in X and a point x0 G X. We consider the sub-set of IV

(3) {Y £ IY, p(x0, x) =^=p( 0, a;') for any x ¥= x' in 7}and the preimage of (3) under a2 in F^ x. As is easy to see, these subsetsare of full measure in F^ and F^ x, respectively, and it is to be under-stood in what follows that it is these subsets which are meant by Tx and

We prescribe an ordering on each configuration 7 G Fx, putting x < x'for any x, x G 7, if p(x0, x) < p(x0, x'). For any (7, x) G Fx x wedenote by i(y, x) the number of the element x G 7, as given by the


ordering on 7. It is not difficult to see that / is a measurable indexing. Wenow introduce a sequence of measurable maps ak: Tx -+ X (i = 1 ,2 , . . . ) ,where ak(

>y) is the A>th element in the configuration 7.Now let C C Tx x be an arbitrary measurable set of positive Campbell

measure: f^jxiQ > 0; we have to prove that then (ju X m){a~l C) > 0.Since Tx ± splits into the countable union of subsets{(7, ak(y)); 7 £ Tx} (k = 1, 2, . . . ), we may assume without loss ofgenerality that C C {(7, ak(y))} for some k. We introduce the notationCn = {(y, x) 6 Tx X X;yÛ {x} e a2C, an(y U {*})) = x} (n = 1, 2, . . .

Note that the condition 1 o /i(C) > 0 is equivalent to

(4) (\JL X m){(7', x)£Tx X X; y' [j {x) £ a2C} > 0.

In its turn (4) is equivalent to the existence of a natural number /, forw h i c h (ju X m) { ( 7 ' , x) G Tx X X, 7 ' U {x} E a2C, a^y'U {x})=x}>0,

that is, (M X m) (C/) > 0.On the other hand, since C C {(7, ^ ( 7 ) ) } , it follows that

OL-^C = {(y\ x) 6 Tx x X; 7' U W 6 aaC, afc(v' U W ) = ^},

that is, a~lC = C^. Thus, the proof of the lemma reduces to proving thefollowing assertion: for any natural numbers k and I the conditions(ju X m) {Ck) > 0 and (ju X m) (Cz) > 0 are equivalent.

Let us prove this assertion. We write Xr - {x E X\ p(x0, x) < r},where r > 0 is an arbitrary rational number (To = 0). We fix a positiveinteger n > max(A:, /) and introduce the following subsets in Tx X X:

V l ) | = 1 when i ^ p ,

where rx, . . . , rn are rational numbers such that0 = r0 < rx < . . . < rn (p = 1, . . . , « ) . It is obvious that the setsU, , cover C ; consequently, the condition (fx X m) (C,) > 0 amounts tothe existence of some U such that

1» • • • ' n

(5) ( x m) {(/, a;) 6 ^ l f ..., rn; 7' U W 6 «2C, a, (7' U {x}) =

We no te tha t Ulr C (B(

FW~1} X L V ) X I and make use of the fact

M» • • • >rn

A r x^xn

that by Lemma 1 of §2.1 the restriction /zw-i of M to B^~1} X Txxxrn rn

is equivalent to the product m X . . . X m X JLIJ2'_1 , where IJL'^ is the

projection of nn_l onto ^X\X . So we obtain that (5) is equivalent to the

following condition:


(6) (m x . . . X m X \in-i) {(*!, • • ., xn\ y') 6 l n X

Thus, (JU X m) (C/) > 0 amounts to the condition that (6) is satisfiedfor some collection of rational numbers 0 = r0 < rx < . . . < rn^ Butfrom the same arguments it follows that the condition (JU X m) (Ck)> 0also is equivalent to (6). Consequently, the conditions (/x X m) (C{) > 0 and(JU X m) (Ck) > 0 are equivalent, as required.

§3. Representations of Diff X defined by quasi-invariantmeasures in the space of infinite configurations

(elementary representations)

1. Definition of elementary representations. Let fx be a quasi-invariantmeasure in the space of infinite configurations Tx. We introduce a series ofunitary representations of Diff X associated with JU. First we consider thespace L^(TX). In it a unitary representation U^ of Diff X is defined by1

We do not study the properties of U^ separately, but examine straightawaya wider class — the elementary representations. For the Poisson measure JUthese representations are additive generators in the representation ringdetermined by U^ (see §4).

Although the proof of the irreducibility and other properties of U^ aresimpler than in the general case, we prefer to study all the elementaryrepresentations simultaneously.

DEFINITION. A representation of Diff X is called elementary if it is ofthe form U^ 0 Vp, where U^ is the representation in L^(TX) given by(1), and Vp is the representation defined in §1.

Thus, each elementary representation is given by a quasi-invariant measureix on F^ and a representation p of the symmetric group Sn (n = 0, 1 ,2, . . . ) .

THEOREM \. If ix is an ergodic measure on Tx and p is an irreduciblerepresentation of Sn, then the elementary representation U^ ® Vp ofDiff X is irreducible.

REMARK 1. The converse assertion is obvious.REMARK 2. Another convenient formulation of Theorem 1 is: When ix

is ergodic, then U^ is absolutely irreducible, that is, remains irreducible aftertaking the tensor product with any irreducible representation Vp.

Essentially, the whole of §3 is devoted to a proof of Theorem 1. Butfirst we construct some other useful realizations of elementary representations.

If ju is concentrated not on rx, but on B^\ then (1) gives the representation VPn (see § 1), wherep° is the unit representation of Sn.


2. The representations Up. Let p be a unitary representation of Sn in aspace W (n = 0, 1, . . . ). We consider the space L^(TX „, W) of functionsF on Tx n with values in W such that

I * 1 1 1 = J \F(c)\\lrdi!i(c)<oo;xX,n

ft is the Campbell measure on Tx n corresponding to the measure ix on Tx

(see §2.4). A unitary representation U of Diff X is given in L^{TX n, W)by

We denote by H^ n p the subspace of functions F G L~ (F^ w, JV) suchthat F(7, xa(1), . . . , xa(w)) = p"1 (a)F(7, xx , . . . , xn) for any o E Sn.Obviously, H^ n is invariant under Diff X

DEFINITION. The restriction of the representation U of Diff X from*<l<Fx,n. W) t 0 ^M,«,P i s denoted by 0£.

In the particular case when p is the unit representation of5 0 , tf£ = f/ , where ^ is the representation in ^ ( r x ) defined by (1).

REMARK. If in this construction of U£ the space Tx of infinite con-figurations is replaced by the space B*^ of Appoint configurations (k > n),

then we obtain instead of UP the representation Kpopfe-" defined in §1,where p£_w is the unit representation of Sk_n.

THEOREM 2. Up 0 Vp s C/POM, where p is a representation of £„.(For the definition of n ° pi, see §2.3).

PROOF. In §2.5 we have established an isomorphism between spaceswith measures

(2) (Tx X Xn, fi X mn) -> (Tx,n, /7^1)

(n o ju is the Campbell measure on F^ rt corresponding to /? ° JU), whichcommutes with the action of Diff X. Let us consider the isomorphism ofHilbert spaces Ll^{YXn, W) -> L^(FX) ® ^mn(^"> ^ ) induced by (2),

where W is the space of the representation p of Sn. It is easy to verifythat the image of #M „ p C L ? _ (p^^ , ^ ) i sL^(F x ) ® #WfP, where

//w p C L^ (Xn, W) is the subspace of the representation Vp (see §1) andthat the operators Uno^) in H^np go over to U^) <g> Fp(i//).

COROLLARY 1. 77ze cte5 o/ //z^ representations Up is the same as thatof the elementary representations U^ (g> Vp.

For on the one hand, C/M 0 Fp s /7pOjli; and on the other hand, for

any quasi-invariant measure JU on F^ there is another quasi-invariantmeasure n' such that ju ~ n o /x' (see §2.3) and, consequently,


up s u^ <g> vp.COROLLARY 2. If n is a saturated measure (that is, 1 o /z ~ /z) and,

in particular, if fx is the Poisson measure, then Up = U^ 0 Vp.THEOREM 3. If n is an ergodic measure on Tx and p an irreducible

representation of Sn, then the representation Up of Diff X is irreducible.We note that Theorem 1 follows immediately from Theorems 2 and 3.

For let p be an irreducible representation of Sn and JU an ergodic measureon Vx. Since n o /z is also ergodic (see §2.3), U^ ® Vp = Up

noyL isirreducible by Theorem 3.

3. Another realization oiJJp. Let p be a unitary representation of Sn

in W. We consider the set Nn of all ^-tuples a = (/l9 . . . , in) of naturalnumbers, where / =£ iq, when p =£ q. We define an action of 5"°° on Nn

by a -> aa = (aO^), . . . , JJ( /W ) ) , cr G 5°°. We denote by l2(Nn, W) thespace of functions y on Nn with values in Pi such that

I I ^ l l 2 ^ 2

We consider in l2(Nn, W) the subspace Hp of all functions <p G /2(NW, W)such that ^0 a ( 1 ) , . . . , ia(n)) = p'l(a)^(ii, . . . , /„) for any o e Sn.Obviously, Hp is invariant under the action of S°°. Now let i be anarbitrary measurable indexing in Yx and a the map Diff XX Yx -* S°°defined by it.

LEMMA 1. The representation £/£ of Diff X is equivalent to therepresentation in L^(TX)<S) Hp, defined by

(3) <ff{l

PROOF. In §2.4 an isomorphism was established between spaces withmeasures:

(4) (Tx X N\ \i X v)->(r x > n , ]I)

(v is the measure on Nn that is 1 at each point). We consider the iso-morphism of Hilbert spaces Ll(TXn, W) -* I j ( r ^ ) ® /2(iv«, H )induced by (4). It is easy to verify that the image of the subspaceH^np C Ll(TXn, W) on which C/jJ acts is Ll(Tx) 0 ^pand that theoperators U^(\jj) in H^n>p go over to operators of the form (3).

4. The decomposition of the space of the representation £/£ of Diff Xinto a sum of subspaces that are primary with respect to the subgroupDiff Xk. Let

(5) Xi c= . . . c= Xh a . . .

be an increasing sequence of open connected subsets with compactclosures such that X = U L .

it

We fix an admissible indexing / (with respect to (5)); let a(i//, 7) be the


map Diff XX Tx -> S^ defined by it. By Lemma 1, the representation£/£ may be realized in Ll(Fx) <g> H; here H = Hp is the space offunctions \p on TV" with values in the space W of the representation p ofSn such that II <£ ||2 = S II <p(a) II2, < °° and <p(ao) = p" 1 (a)^(fl) for any

a^Nn

o€iSn. The representation operators are given by (3).We decompose L^(TX) <S> H into a direct sum of subspaces that are

primary with respect to the subgroups Diff Xk C Diff X (that is, thosethat are the identity on X \ Xk) (k = 1, 2, . . . ).

First we decompose Fx into a countable union of spaces that areinvariant under Diff Xk:

(6) Tx = U BQ (L^ ( B ^ x r^NZfe) 0 # ) ,

where /xr is the restriction of JU to the subset B ^ X T]CKX C T^. It remains

to decompose each term in this sum into a direct sum of invariant subspacesthat are primary with respect to Diff Xk.

Next we split H into the direct sum of subspaces that are primary with respectto the symmetric group Sr C S^ . This decomposition can be presented in thefollowing way: H = ® (Wl

r 0 C}), where Wlr are the spaces in which the

irreducible and pairwise inequivalent representations p\ of Sr act; C\ is thespace on which Sr acts trivially.

As a result, we obtain a decomposition into the direct sum:

All the terms of this decomposition are invariant under Diff X. For sincethe indexing is admissible, it follows from \p E Diff Xk and) G B ^ X 1]^^ that cr(tf/, 7) G ^ . We claim that these subspaces areprimary and disjoint.

We denote by \ir and JJL" the projections of nr onto B ^ and Fxxx ,(\respectively. By Lemma 1 of §2, the measure \xr on B^} X Fx^x is equivalent

to the product (i'r X fi" of \xr and 11". Consequently, there is an isomorphism

xr: L\T ( B ^ x

defined by

86 A. M. Vershik, L M. Gel'fand and M. I. Graev

We denote by the same letter the trivial extension of rr to an isomorphism

We denote the elements of B*J? by 7(r) and define a mapAk

or\ Diff J ^ X B<£} -> S, by or(\p, y(r)) = o(\jj, y), whereK

y G B<£} X r ^ ^ , 7 O Xfc = 7(r). This is well defined, because if7 O Ifc = 7' D I f c , then o(\p, y) = o(\p9 y'). Immediately from thedefinition of rr we derive the next result.

LEMMA 2. Under the isomorphism Tr the operatorsU(\l/) = tf£(\(0, ^ G Diff Xfc ô over to operators TrU(\l/)T^ of thefollowing form: TrU(\p)r~1 = Ul

r(\jj) (8) /, where I is the unit operator inL\ (Txxx) <8> C\ and £//(0) is an operator in L% (B£> ) 0 W//in the space of functions on B ^ with values in Wl

r defined by

The representation Uj. of Diff X^ is equivalent to. VPr defined in §1.1.By Proposition 3 of §1.2, all the representations VPr of Diff Xk areirreducible and mutually pairwise inequivalent. Therefore Lemma 2 has thefollowing corollaries.

COROLLARY 1. The representations of Diff Xk in the subspacesLlr^xk

x r * \ ; ^ ® Wr ® C'r (r = 0, 1, 2, . . . \ i = 1, 2, . . .) are

primary and disjoint.COROLLARY 2. Any invariant subspace under Diff Xk

X\, r a (L^(B^) (g) Wl) 0 (Ll;(Tx^xk) 0 C*)

is of the form XlK T = (L2^ (B^ ) 0 Wl

r) 0 D/, w/zê

COROLLARY 3. >4w subspace X C I^(F Z ) 0 ^ that is invariant underDiff Xk splits into the direct sum X = © ^ i , r, vv/zere

r, i

(7) Xlr = X() (Llr (B^ x rZNxA) <g> t n 0 C).5. Proof of Theorem 3. We use the notation and results of the preceding

subsection. We denote by L~(TX) the space of essentially bounded functionson Tx with respect to fx. Now L™(TX) is a ring with the usual multiplication.Further, if / G L\ (T^) 0 # and <p G I^ ( r^ ) , then ^ / G l j t ^ ) 0 i/.

LEMMA 3 . 7 / ^ ^ 2 ^ ) % H is invariant under Diff X, then X isinvariant under multiplication by elements of L~(TX).

PROOF. We denote by [Xx the projection of ji onto Bx , and byK K

IT (Bx ) the space of essentially bounded functions of B^ with respectv-Xfr k k


to the measure \ix (k = 1, 2, . . . ). We identify each space L~ (Bx )k »xk k

with its image under the natural map L~ (Bx ) -* L~(TX) (that is, the

space of essentially bounded functions on Fx that are constant on thefibres of the fibration FY -+ BY ). We consider the union U L~ (BY ).x xk k »xk

v xk

It is obvious that for any / G ^ ( r x ) <g) / / and i / by elements</>'/, where / S U T (Bx ). Therefore, to prove the lemma it is

k **k k

sufficient to check that X is invariant under multiplication by elementso f l ^ (BXk)(k= 1 , 2 , . . . ) .

We fix k and denote by L~>n (B^}) the subspace of functions inLuv ($xJ t h a t a r e concentrated on B^ } x TY Y , (here B(

Fw) is the sub-

xk k k A~Ak Akspace of w-point subsets in Xk) (n = 0, 1, . . . ). Obviously, for any/ G L2(TX) (8) H and i p G T (B^ ) the product \pf is approximated in

Xk k

Ll(rx) ® H by f i n i t e s u m s o f elements (prt/, where yn G £~» (B^}). Thus,the proof of the lemma reduces to the following assertion:

If X C Ll(Fx) <g> H is invariant under Diff Xk, then X is invariantunder multiplication by elements of L~< (B^w)) (n = 0, 1, 2, . . . ).

Let us prove this assertion. We use the decomposition X = 0 X\ r,r, i

where

(8) 45i, r = ^ fl (^^r ( B ^ x Tx^Xh) 0 Wj 0 C\).

It is sufficient to prove the assertion for each subspace X\y r separately.Note that when r ¥= n, the supports of the functions inU\ir (Bx x rx\x ) a n d i n LM'n(Bx}) d o n o t i n t e r s e c t - Therefore, it is onlynecessary to consider the case r = n.

Let Xk, n be the image of Xl, n under rn. By Corollary 2 to Lemma 2,Xln has the form Xi,n = ^ ; ( B ^ ) <8) Wl

r (g) i)J.

Hence it is clear that J£ft,n is invariant under multiplication by functionsin L< ( B ^ ) . Since the corresponding elements in X\ n

a nd Xln differ

only by the factor (—^—-) j the space < ft, n is also invariant under

multiplication by elements of U"> ( B ^ ) , and the lemma is proved.Mn AkWe consider the space H - Hp, a factor in the tensor product

) ® H. A representation R of ^ is defined in H by

8 8 A. M. Vershik, I. M. Gel'fand and M. I. Graev

(R(o)ip)(a) = Mo-1 a).Note that R = Ind|~xsn (p X I), where SZ is the subgroup of finite

permutations leaving 1, . . . , n fixed and / is the unit representation ofSZ • Hence the next result follows easily.

LEMMA 4. The representation R of Sx is irreducible.LEMMA 5. Every subspace X C L^(TX) 0 H that is invariant under

Diff X is also invariant under the operators I 0 R(o), o E S^, where I isthe unit operator in L^(TX).

PROOF. Since S^ = lim Sp, it is sufficient to prove that X is invariant

under the operators / 0 R(o), o £ Sp (p = 1, 2, . . . ).We use the notation and results of §3.4. Let p be any fixed positive

integer. We consider the subspace

where Xh, r is defined by (8).Clearly the union (J XhtVi$ everywhere dense in X. Therefore, it is

sufficient to prove that each space Xl, r, r > p, is invariant under/ <g) R(o).

We consider the image X\, r of X\,r under Tr. By Corollary 2 toLemma 2 Xl, r and hence its preimage X\,r is invariant under /.(g) R(o),a £ Sr. Since p < r, we have S^ C ^ , and so X\%T is also invariantunder the operators / 0 R(o), o £ Sp. The lemma is now proved.

LEMMA 6. Every subspace X C L£(TX) 0 # invariant under Diff Xw o/ /7ze form X = L^{A) <g> H, where A C Tx is a measurable subset.

PROOF. It follows from Lemmas 4 and 5 that

X = E 0 H,

where E C L^(TX), and from Lemma 3 that E is invariant undermultiplication by functions from L~(FX). Consequently, E = L^(A),where A is a measurable subset in TXi and the lemma is proved.

The assertion of Theorem 3 follows immediately from Lemma 6. Forlet X C L^(TX) <S> H be an invariant subspace with respect to Diff X.Then from Lemma 6 we have X = L^(A) <g> H, where A C Tx is ameasurable subset. Consequently, since ju is ergodic, either X = 0 or# = I ^ ( r x ) 0 //. This proves Theorem 3, and with it Theorem 1.

REMARK 1. The assertion of the theorem remains true for any subgroupG C Diff X satisfying the following requirements:

1) G acts ergodically on Tx;2) for any open connected submanifold Y C X with compact closure,

Theorem 1 of § 1 about the representations Vp of Diff Y remains true for


the restriction of the Vp to G O Diff Y.REMARK 2. The proof of Theorem 3 can be simplified considerably in

the case n = 0, that is, for the representation U^ in L2l(Tx). In this case

it reduces to proving Lemma 1 of §2.1 and establishing a functional versionof the 0 - 1 law. In the simplest case when JU ~ ixx (k = 1, 2, . . . )

(V 9 i" are the projections of/i onto Bx and Tx_x , respectively), this lawIf fc ^

consists of the following. Let X be a subspace of L2t(Tx). If in terms of

the decomposition L2(TX) s L2> {Bx ) 0 Lj. ( r z-^ , ) t h e subspacexk k xk k

X C L^(TX) for any k has the form# s LJL (Bz ) (g) Cfc, Cft c L*. (r z-zk) ,

*fe xh

then1 either ^ = 0 or X = Ll(Tx). An extra difficulty comes from thefact that there is no equivalence /i ~ ii'x X nx , generally speaking, andonly the weaker relation \i ~ 2 Mr x M" is true (for the definition of n'rand /x" see p. 25). r=0

§4. Representations of Diff X generated by the Poisson measure

1. Properties of the Poisson measure. Let X be a non-compact manifoldwith a smooth positive measure m, m(X) - °°, and let \i - JJLX be thePoisson measure on Fx with parameter X corresponding to the measure mon X (for the definition of the Poisson measure, see §0.2).

Some basic properties of Poisson measure were stated in §0.6.LEMMA 1. If dim X > 1, then for any two [i-measurable sets

Ai, A2 C Fjr with positive measure there exists a diffeomorphism\P e Diff (AT, m) such that2 fi(Al O $A2)>±ii(A1MA2).(Diff(X, m) is thesubgroup of diffeomorphisms preserving m.)

PROOF. First we recall some definitions and facts. By a cyclindrical set inTx we mean a set A C Tx, of positive measure, of the form A = irÂ', whereY is a compact set in X, Ar C BY is a measurable subset and TTY is thenatural map Tx -* BY i^y 7 = 7 ^ Y)\ Y is called the carrier3 of ASince the Poisson measure JU is infinitely decomposable, it follows that ifthe carriers of two cylindrical sets A1 and A2 intersect in a set of measure0, then [x(Ax n A2) = JU(^1)JU(^42).

We recall that any /x-measurable set C CTX can be approximated bycylindrical sets (that is, for any e > 0 there is a cylindrical set A such thatH(CAA)< e).

We recall that in these terms the usual 0 - 1 law would be formulated as follows. Let / e L2 (Tx). If/ = lfe 0 /fe for any k, where lfe is the constant in L2> (B^- ) and fk e L^» ( r x _ x ), then/ = const. fe k

For X = R1 the lemma is false, because in this case Diff(Jf, m) is trivial.Of course, the carrier of a cylindrical set is not uniquely defined.


Hence it is clear that it is sufficient to prove the assertion of the lemmafor cylindrical sets Ax and A2. Without loss of generality we may supposefurther that X = Rn, n > 1.

Let us establish the following property of Diff(X, m): if Yx and Y2 aretwo compact sets in X, then there is a diffeomorphism \p E Diff(X, m)such that Yx n \p Y2 = 0.

For if Ylt Y2 are compact in X = Rn and m is the Lebesgue measurein R", then there exists a disc containing Yx and Y2 and a rotation ^ ofthe disc (this preserves rn) such that Y1 n i// F2

= 0; this rotation i// canbe extended beyond the boundary of the disc to a finite diffeomorphism ofR" preserving rn. If now m is an arbitrary smooth positive measure in R",then it is sufficient to use a lemma (see [21]), which states that any openball in Rn, n > 1, with a smooth measure m can be mapped diffeomorphi-cally onto itself so that m goes over to the Lebesgue measure.

Let Yx and Y2 be the carriers of Ax and A2. By what has just beenproved, we can find a diffeomorphism \p E Diff(Z, m) such thatYx O \//72 = 0. But then / i ^ n \J>A2) = M ( ^ I ) M ( ^ 2 ) = M(Î)M(^2) ? andhence /x( 4 j O \jjA2) > y MW I )/x( 42 )• The lemma is now proved.

THEOREM l . / /d imX > 1, then the Poisson measure /x in Yx is ergodicwith respect to Diff(X, m).

PROOF. Let A C Tx, ix(A) > 0, be a subset that is invariant mod 0 underDiff(X, m); we must prove that fi(A) = 1. Suppose the contrary: thatJJL(TX \A) > 0. Then by Lemma 1 there is a \jj E Diff(X, m) such thatju(rx \i4) O \//^) > 0; hence, since A is invariant, ix((Tx \A)C\A)>09

which is false. This proves the theorem.2. The representation of Diff X generated by the Poisson measure. Let

(i = /xx be the Poisson measure on Tx with parameter X > 0. In §3 wehave associated with each quasi-invariant measure /x on Tx a unitaryrepresentation U^ of Diff X in L^(TX) defined by

and also a set of elementary representations £/jj). For the Poisson measureJUX the theory of such representations can be advanced considerablyfurther than in the general case. In particular, it is possible to describe thecorresponding representation ring. Furthermore, for jix the representationsUp can be realized in the form EXP^ T (see [1] and [2]).

In what follows we write Ux (instead of U^ ) for representation

generated by the Poisson measure nx.Since nx is ergodic, by Theorem 1 of §3, Ux is irreducible.3. The spherical function of the representation Ux. Let us assume that

dim X > 1. We consider the subgroup Diff(X, m) C Diff X of diffeomor-phisms preserving the measure m; for us this subgroup will play a rolesimilar to that of maximal compact subgroups in the theory of representations


of semisimple Lie groups.Since /x is invariant under Diff(Z, m) and Ux is infinitely decomposable,

the restriction of Ux to Diff(X, m) is given by

In view of Theorem 1 there is in L^ (Px) one, and up to a multipli-cative factor, only one vector that is invariant under Diff(X m), namely,/o = 1-

DEFINITION. The following function on Diff X is called the sphericalfunction of Ux:

where the brackets denote the inner product in L2^ (Tx).

Let us find an explicit form for the spherical function. Letsupp \p C y, m(Y) < oo. We denote by /Ix the projection of JUX onto B r

and by Jxx the restriction of /Ix to B ^ . Then

) = JB

So we obtain

(2) i*x W = e x

where J . (x) = . Since f0 is defined invariantly in the repres-^ <im(jc)

entation space L2^ (Tx) and since, by (2), ux ¥= ux^ when \x ^ X2, weobtain the following theorem.

THEOREM 2. The representations UXi and UX2 of Diff X (dim X > 1)are HO^ equivalent when \ l ^ X2.

4. The Gaussian form of the representation Ux of Diff X. Let us con-sider the real Hilbert space H - Lm (X), where m is a smooth positivemeasure on X. A unitary representation T of Diff X is given in H by

), where J ^ (X) = ^ / } . A 1-cocycle


j3: Diff X -> H is given by [M)]{x) = J^2(x) - 1.In accordance with [1] and [2] this is a way of constructing a new

representation & = EXP^ T of Diff X.We denote by jl a measure in the space $F (X) of generalized functions

on X given by its characteristic functional:

(3)

where ||*|| is the norm in L2m(X). We call # the standard Gaussian measure

in tF (X).The representation U = EXP^ 71 is given in Z,~( jF (X)) by

(£/ (i|>) (D) (F) = ei{F> m)® (T* (yjp) F),

where the operator T*(\p) is defined by <T*(\p)F, f> = <F, T()p)f).LEMMA 2. The vector >0 = 1 in Ll{ & (X)) is cyclic with respect to

Diff X.PROOF. Since (U(\p)<bQ)(F) = e

KFM)\ the assertion of the lemma isjust that the set of functional £?/</7'W», \p G Diff X, is total inLl(tF (X)), that is, the minimal linear subspace X <Z Ll{JF (X)) contain-ing them is Li (2F (X)) itself. It is known [ 1 ] that the functional of theform

w o ^ f i <*•,/«>,where / l 5 . . . , / „ are smooth finite functions on I (n = 0, 1, 2, . . . )form a total set in L2( ^ (X) ) ; therefore, it is sufficient to prove that Xcontains all functional of the form (4).

Let / be any smooth finite function on X satisfying j f(x)dm(x) = 0,let T be any real number such that 1 - rf(x) > 0 for all x G X. Ameasure mT in X is given by dmT(x) = (1 - Tf(x))dm(x). The measures mand mT coincide outside a compact set Y D supp /, and m(Y) = mT(Y).Therefore, by a theorem of Moser [21], there exists a diffeomorphism\jj e Diff X carrying m to mT, that is, J ^ (JC) = 1 - r fix). But thenM) = VO ~ r / W ) ~ *> a n d h e n c e t h e functional e»<^ Vd-r/lx))-i>belongs to <£ for any sufficiently small r. Hence all the terms in theexpansion of el<F> v(i -r/(*))-1> a s a p O W e r series in r also belong to L.

The coefficient of r71 in this expansion is cn(F, f)n, cn ¥= 0, apart fromk

terms of the form II (F, /}>, k < n. Therefore, by induction on n, we/= l

can verify that X contains all functional of the form (F, f)n, where

\ f{x)dm{x) = 0, and hence those of the form <F, />", where / is an

arbitrary smooth finite function. Since the functional (4) can be presented


as linear combinations of the (F, f)n, they also belong to X, and thelemma is proved. ^

If in the definition of U we replace the 1-cocycle j3 by sj3, where s isany real number, we obtain a one-parameter family of unitary representationsof Diff X: Us = EXPs/3r.

It is obvious that the assertion of lemma 2 remains true for allrepresentations Us, s ¥= 0. ^

THEOREM 3. If s ^ 0, then Us = Usr, where Ue is the representationof Diff X generated by the Poisson measure with parameter s2 (see ^ 4.2).

PROOF. We compute the matrix element (Us()p)$0, <£>0>, where <J>0 = 1.From formula (3) for the characteristic functional of Jt we immediatelyobtain

(Us (i|>) O0, Oc> = exp (s2 j ( j ; / 2 (x) -1) dm (*)) - as2 (if),

where wj2 is the spherical function of US2. The assertion of the theoremfollows from this and the fact that 4>0 is cyclic. ^

Let us now consider the special case 5 = 0. Then (see [ 1 ]) Uo splits intothe direct sum: Uo = T° © T1 © . . . © Tn © . . . , where T° is the unitrepresentation and Tn (n > 1) is the ft-th symmetrized power of T intro-o

duced at the beginning of §3.4. In the notation of § 1 we have Tn = Vp°n,where pJJ is the unit representation of Sn. Thus, Uo = © VPn.

« = o5. Elementary representations of Diff X associated with the Poisson

measure. According to §3, the tensor product £/£ = Ux <g> V9 ofrepresentations Ux and Vp of Diff X is called an elementary representation.By Theorem 1 of §3, an elementary representation £/£ is irreducible if andonly if p is an irreducible representation of Sn (n = 0, 1, 2, . . . ).

THEOREM 4. Two irreducible elementary representations U^ and U9* ofDiffX, d i m Z > 1, are equivalent if and only if \x = X2 and px and p2

are equivalent representations of Sn.PROOF. We restrict Up

x to Diff(X, m). From §4.4 it follows thatUx = 9 ^ " , where p£ is the unit representation of Sn (the bar denotes

71=0

the restriction to Diff(A", m)). Consequently,

(5) vi^vp® S Fpop°.

From the decomposition (5) it follows easily on the basis of Theorem2 in §2 that U* i U^ when px i p 2 . Hence, a fortiori, U^ i Up*

when Pi ^ p2- It only remains to discuss the representations £/£ and

^ A '

Let (7? f [/{. We denote by A the vacuum vector in the

representation space of Ux. (i = 1, 2) and by F an arbitrary vector in the


representation space of Vp. From (5) it follows easily that under an iso-morphism of the representation spaces of Up and £/? the vectorfx (g> F goes over into the vector fx <g) F\ therefore,

for any \p E Diff X. Hence ux = ux , where ux is the spherical function

for £/x (see §4.3), and therefore, \ = X2.

§5. The ring of elementary representations generated bythe Poisson measure

1. The decomposition of the tensor product Ux ® Ux of representationsof Diff X into irreducible representations. First we prove a general theoremabout representations EXP^T. Let G be an arbitrary group and T a unitaryrepresentation of G in a real space H\ let j3: G -> i / be a 1-cocycle. Thena new unitary representation Us = EXPSj3r can be defined as in [1] and [2],where s is an arbitrary real number. Let H' be the dual space to H and /za measure in any nuclear completion H of H\ defined by the characteristicfunctional:

The operators of the representation Us = EXPSj3 T act in the complexHilbert space Ll(H) according to the formula:

(Us (g) O) (F) = e*s<F. P<s»(D (T* (g) F).

THEOREM 1. If s] + s\ = s[2 + S22, ^ ^ &s ® 6^ - ^s; ® ^ ; -PROOF. We define operators At, t e R, in Ll(H) ® I^ (^ ) by the

formula:

F2) = O(cos t Fx + sin ^ F2, —sin ^ ^ + cos t F2).

From the definition of the Gaussian measure j2 it follows tha t^ f for anyt G R is a unitary operator. Further, from the definition of Us it followseasily that

A? (USl (g) <g> US2 (g)) At = C/Sl cos t+s2 sin * (g) ® U-$l sin t+s2 cos i (g).

Consequent ly , ^ ® f/s2 = ^SlCO3r + sa sinr ® 5L S i sinf + s2 cost f o r a n Yf GR; hence the assertion of the theorem follows immediately.

COROLLARY. USi 0 US2 = Uj(S] + s]) ® ^0-Now let £/x be the representation of Diff X generated by the Poisson

measure with parameter X (see §4), let Vp be the representation of Diff Xdefined in § 1 (p runs over the representations of Sn).


THEOREM 2.

(1) Uu 0 UK2 - © (Uk 0 FP°),n=0

where X = Xx •/• X2, flwc? P° & ^ wmY representation of Sn. Every termin (1) is an irreducible representation of Diff X.

PROOF. Let T be a representation of Diff X in Z^ (X) defined by= 3 JÔO/dJT1*). A 1-cocycle 0: Diff X -> Z^ (X) is given by

(*) = J J/2(x) - 1. We consider the representation Us = EXP^Tof Diff X. By Theorem 1 we have, for any Xj > 0, X2 > 0

(2) ^yr , ® ^Vr2 = ^yxi+x; ® ô-

On the other hand, it was proved in §4.4 that U^x = Ux for any

X > 0, and Uo = © FP n . Consequently, (2) implies that« = o

C/x 0 C/x ^ 0 (t/x 0 Fp»), where X = Xx + X2.2 w = 0

The irreducibility of Ux 0 F n follows from the main theorem of §3.

2. The decomposition of the tensor product of two elementaryrepresentations of Diff X associated with the Poisson measure.

THEOREM 3. C/?» 0 W* = © Up^ ° ^°PnXl K n = 0 ** + ** . '

(For the definition of the operation pj o p 2 , see § 1.)

PROOF. By definition, Ufr = UK 0 Vp^, U** = Ux^ 0 Vp\ Consequently,

Ux\ ® ux] -Wxt ® ^x,)1 ® (^Pl ® F P 2 ) - ^ Theorem 2,

^ ® ^ = ® (^,+Xa 0 K P ^- F u r t h e r ' F P l ® F P 2 - F P l ° P 2 forany

Pi>P2 ( see § 0 - Hence the required result is obtained straightaway.COROLLARY. The set of representations of Diff X that split into the

direct sum of irreducible representations of the form Ux 0 Fp is closedunder the operation of taking the tensor product.

§6. Representations of Diff X associated with infinitelydivisible measures

The group Diff X acts naturally in the space $F (X) of generalizedfunctions on X. Therefore, representations of Diff X can be constructedfor any quasi-invariant measure /I on JF(X). We have already noted earlierthat the configuration space Tx has a natural embedding in jF (X), and,therefore, the representations considered earlier are part of a considerably


wider class of representations. Here we consider a special class of measuresin $F (X) (infinitely decomposable measures), which generate the same stockof representations as the representations Ux <g> Vp studied in §§4 and 5.

1. The measure /xr in the space of generalized functions on X. Let jf{X)be the space of generalized functions on X\ we define an action of Diff Xon &(X) by < \jj*F, />= <F, / ° i//>.

Let us consider a positive definite function xr on R of the form-f-OO

( j (e*a«—l

where r is a non-negative finite measure on R (not necessarily normalized).(Xr(O is the Fourier transform of a certain infinitely divisible measure on R.)Let m be a fixed smooth positive measure on X, ra(X) = °°. A measure/I = piT is given on jF(X) by the characteristic functional

(1) L% (/) = exp ( j In xt (/ (*)) <M*)) =exp ( j j (***'<*> - l)dro(*)<*T (oc)) .X R X

We list some basic properties of M, which follow easily from thisdefinition.

1) If X = Xx U • • • U Xn is a finite partitioning of X andJ^(X) = ^(A^) © . . . © ^r(Ar

w) the corresponding decomposition of / ( I )into a direct sum, then pi =/z1X . . . X jurt, where pt- is the projection of /xonto the subspace jF(^/), / = 1, . . . , « (infinite decomposability).

2) The measure pt = p[T is concentrated on the set Jo (X) of generalizedoo

functions of the form X ak8 , ak = 0, where o: E supp r, and { fe} is afc= l fe

set without accumulation points in X (that is, a configuration in X)\ 5X

denotes the delta-function on X concentrated at x G I 1

3) The measure /x is quasi-invariant under Diff X, and

dm(\l)x)where J^ (x) = .(Since \jj is finite, only finitely many factors

dm(x)3 ^(xk) are distinct from 1.)

Let us note the particular case when r is concentrated at one point,a - 1. Then t f 0 W = (S5X.; {xj G F x } , where Tx is the configuration^ 0 ( X ) -> F x in which each generalized function 25^. is associated with aconfiguration {xt} G F ^ . It is not difficult to verify that the image of nunder this map is the Poisson measure on Fx with parameter X = r(R).

1 The converse is also true: any infinitely decomposable measure in &(X) concentrated on a set of thetype indicated is a measure MT for a certain T; when X = R1, this fact is very well known (see, for example,J.L. Doob, Stochastic processes, Wiley & Sons, New York 1953.


2. The representation of Diff X associated with j2T. A unitary representa-tion of Diff X is given in Ll(^(X)) by the formula

(2)

In the particular case when the measure r on R is concentrated at asingle point and is equal to X at this point, the representation VT so con-structed is equivalent to the representation Ux corresponding to thePoisson measure with parameter X (see the remark above). Here we shallobtain the decomposition of VT into elementary representations.

We denote by ,$FQ (X) the set of all generalized functions of the form2 <^jcdx , ak =£ 0, where {xk} G Tx; it was mentioned above that JfQ (X)

oo

is a subset of full measure in & (X). We introduce the space R°° = n R ,/= l

Rz- = R with measure v = r0 X . . . X r0 X . . . , where r0 is the normalizedmeasure on R: r0 = -r , X = r(R).

Next, let i(y, x) be an admissible indexing (for the definition see §2) inX and consider the sequence of maps ak: Tx -+ X (k = 1, 2, . . . )defined by ak(y) G y, i(y, ak(y)) = k. We define a map

JI:

by

Standard arguments establish the following result.LEMMA 1. The map TT is measurable in both directions; the image of

^Q{X) is a subset of full measure in (Tx X R°°;/xX v)\ the image o//IT

under IT is the product measure JJL X v, where ji is the Poisson measure onFx with parameter X = r(R).

By means of IT the action of Diff X can be carried over from ,iF0 COto I\r X R°°. It is not difficult to see that the action of Diff X onTx X R°° is given by

(3) ip: (Y, a) ^ (ty'^f a0(^> Y))»

where a = (OLX , . . . , an,...), and ao = (cea(1), . . . , a a ( n ) , . . . ) ; hereo(\p, y) is the map Diff X X Tx ^ S^ defined by / (see §2).

Lemma 1 and (3) imply the next result.LEMMA 2. The representation VT of Diff X is equivalent to the

representation acting on L2(TX) <g> Z (R°°) according to the formula

Here JU is the Poisson measure with parameter X = r(R), andv = T0 X . . . X r0 X . . .


Let us now consider the space Z (R°°) = Z,J (R)® . . . ®Lj (R)®. . .with a given unitary representation T of S^ : (T(O)<P)(OL) = y(ao). We splitZ,£(R°°) into invariant subspaces.

We fix an orthonormal basis eQi el9 . . . in L2r (R), where e0 = 1 (that

is, e0 is the function on R everywhere equal to 1). It is known that thevectors e^® . . . ® e, ® . . . , where 2/^ < °° (so that all the indices ik

apart from finitely many are zero) form a basis in Z^(R). Now S^ permutesthese vectors. We divide the set A of basis vectors into orbits under Sx.

Let us consider all possible collections of natural numbers of the form

(5) (nl9 . . ., nh; il9 . . . , ih),

where nx > . . . > nk, ip ¥= iq when p ¥= q, and ip > ip+1 if np = np+1;k = 0, 1, . . . (k = 0 corresponds to the empty set). With each collection (5) weassociate a basis vector in A:

(6) e^ ® . . . ® eii ® . .. ® eife <8). . . ® eih <g> eQ ® e0 ® . . . .

We denote by A1^''"'l* the orbit of 5^ in A generated by (6) and by

i / ! 1"""? the subspace of I2(R°°) spanned by the vectors of ^4 '1 ' "" '* .

It is easy to establish that the orbits A1^'' " ' lhn are pairwise distinct and

that their union is the whole of A. Hence it follows that Z,£(R°°) splits intoa direct sum of invariant subspaces

the sum is taken over all collections (nx, . . . , nk\ i\, . . . , ik) of the type(5). Note that representations of S^ in H^1'""1* and in Hl

n1""'1* are

equivalent.The decomposition (7) we have just obtained leads to the following

result.LEMMA 3. The space L2(TX) X Z^(R~) decomposes into the direct

sum of VT-invariant subspaces:(8) LI (Tx) ® Ll (R00) = 2 e K (Tx) ® V ; ; ; ; ^ .

The representations of Diff X in I j ( r ^ ) ® K\\'.'.'.'X a n d i n

x ® Hln1'"'1* are equivalent.

We denote by KTWl"-"rtfe the restriction of FT to ZÔV ® K\\'.\'.\nk'

LEMMA 4. v"l""*nk = (/xp"i°'"°P"fe w/zere X = r(R), and p^ is the unit

representation of Sn.The assertion of the lemma is easily established if we use the realization

of elementary representations introduced in §3.3.Lemmas 3 and 4 give the next result.


THEOREM 1. The representation VT of Diff X in L2^ (JF(X)) defined by(2) splits into a discrete direct sum of irreducible elementary representationsof the form UP

K, where X = r(R).REMARK 1. If r is concentrated at two points on R, and if the measures

of these points are \x and X2, respectively, then, as is easy to show,V~ = UXi (g> JJX . In this way we obtain from Lemmas 3 and 4, in parti-cular, the decomposition of the tensor product Ux 0 Ux into irreduciblerepresentations. This was obtained by another method in §5.

REMARK 2. The representation VT can be treated as a continual tensorproduct of Poisson representations Ux\ Theorem 1 then gives a decompositionof the continual tensor product of Ux into irreducible representations.

REMARK 3. The representation so constructed is a cyclic subrepresent-ation in EXP^ T, where T is the representation of Diff X in the real spaceL2{X X R, m X r) and j3 is the 1-cocycle: [j3(i//)](x, a) = J^2(x) - 1, see§7. If r = 5^ , x0 =£ 0, then we obtain the Gaussian form for the Poissonrepresentation Ux (see §4.4).

3. Criteria for representations VT of Diff X to be equivalent. By Lemma

3, the multiplicity with which y"itmm''nk occurs in VT depends only on the

numbers nl9 . . . , nk and on the dimension of L2 (R). Therefore, Lemmas3 and 4 also imply the following result.THEOREM 2. Let T and r" be two non-negative finite measures on R suchthat

1) r'(R) = r"(R);2) the supports of r' and T" are either both infinite or consistent of the

same finite number of points.Then the representations VT> and VT» of Diff X are equivalent.By Theorem 2, each representation VT is given, up to equivalence, by a

pair of numbers: the parameter X = r(R) of the Poisson measure(0 < X < oo) and the index h, which is equal to n if r is concentrated onn points, and is °o if supp r is infinite. It is convenient, therefore, to denotethese representations by Vx h (instead of the previous notation VT).

4. The tensor product of representations VT = Vxh. THEOREM 3.v\lthx ® Vx2,h2 - Vxi+x2,h^h2'^

thus> the set of representations VT = Vxh

is closed under the operation of tensor multiplication.PROOF. We have Vx.h. = FT., where rt is any non-negative finite

measure on R scuh that r^R) = Xz- and |supprz| = hf if ht < «>, andsupp Tf is any infinite set if ht - °° (i = 1, 2). The measures T1 and r2

can always be chosen so that supp TX C\ supp r2 = 0. Let us considerr = T! + r2 on R. Obviously, VT = Vx +x h +h . Therefore, it is sufficientfor us to check that VT ^ VTi ® VT\

We denote by 1il9 /z2) a nd M the measures on & (X) corresponding to

T1? r2 , and r on R, respectively, and by LT (/), LT (/), and LT(f) their


characteristic functionals. Since supp Tt O supp r2 = <j>, we haveLT(f) = LTi(f) LTi(f). Hence

2£ & (X)) ~ L*~ (& (X)) ® L i ( ^ (X)),

and so VT = VT ® VT , as required.

§7. Representations of the cross product 3 = C°°(X)*Diff X

1. Definition of & and the construction of representations. Let usintroduce the (additive) group C°°(X) of all real finite functions / on X ofclass C°°. Now Diff X acts on C°°(X) as a group of automorphisms/ -+ f o i//"1. In this way we can define the cross producty = C°°(X)-Diff X of C°°(X) with the multiplication:(A, *iXf2, * 2 ) = (A + / 2 ° r 1 , * i* 2 ) -

Let pi be an arbitrary quasi-invariant (under Diff X) measure in the spacejf(X) of generalized functions on X. We consider Ll(,lF (X)) and associatewith each element (f, \jj) & the following operator V(f, \jj) in

(1)

It is easy to check that the V(f, \jj) are unitary and form a representationof 3. This representation of # is cyclic with respect to C°°(X) (the constantis a cyclic vector). It is irreducible if and only if the measure ju on $F (X)is ergodic with respect to Diff X.

2. Representations associated with infinitely decomposable measures. Fromnow on we restrict ourselves to the measure jut on :f {X) introduced in §6,that is, measures with characteristic functionals of the form

Lx (/) = exp ( \ f (eia^>— 1) dm {

where m is a smooth positive measure on X, m(X) = «>, and r is a non-negative finite measure on R. The representation of 3 corresponding tothis measure is now denoted by VT (the measure m on X is assumed tobe fixed).

It is not difficult to prove that these measures are ergodic with respectto Diff X\ consequently, the representations VT of & =C°°(Z)*Diff X areirreducible.

LEMMA I. If dim X > 1, then Ji is ergodic with respect to the sub-group Diff (X, m) C Diff X of diffeomorphisms preserving m.

The proof goes as for Poisson measures (see §4).COROLLARY 1. The restriction of VT to the subgroup C°°(X)-Diff(X, m)

is irreducible.


COROLLARY 2. The only vectors in Ll(JF(X)) that are invariant underDiff (X, m) are the constants.

Let 4>0 be the function in L~(JF(X)) that is identically equal to 1. Thefollowing function on IS is called the spherical function of VT:

*c(/,*) = (^(/,^)O0, O0>,

where the brackets denote the scalar product in Ll (,f(X)). Since <t>0 is acyclic vector in Ll(Jf (X)), the representation VT is uniquely determined byuT(f, n M

By a simple calculation we obtain

(2) ux (/, *) = exp ( j j (J^2 Or) ete**> - 1) dm (x) dx (a)) ,

rf/nOIT*)where 3 Ax) ^ — — - .

Obviously, if r t = r2, then MT = wT . Since 4>0 is defined invariantly inZ,£(jF(X)), we have the following result.

LEMMA 2. / / T J r2 , ^ ^ representations VT and VT of& = C°°(Z)- Diff(X) (dim X > 1) are inequivalent.

3. The Gaussian form of the representations VT. Let us consider thec o m p l e x H i l b e r t s p a c e H o f f u n c t i o n s o n l X R w i t h t h e n o r m

||cp||2= J j |q>(*. *)\*dm(z)dx{a).

A unitary representation T of § is given in H by

(T(f, *) cp) (a:, a) = cto/(x)j^(a;)(pW-ia;f a ) t

We define map |3: ^ ->• H by

It is easy to verify that for any glr g2 £ ^ we have:^ 1 ^ 2 ) = 0tei) + ^ 1 ) ^ 2 ) , so that P is a 1-cocycle.

Let us construct from T and the 1-cocycle |3 a new representationFT = EXP^T of S (see [1], [2]). We denote the dual space of H by H\The standard Gaussian measure on the completion H' of H' is the measureH with the characteristic functional

The representation FT of is given on L^(Hr) by

(V, (g) O) (F) = eJ Re <F« «ff»<D (r* (g) F),


where the operator T*(g) is given by (T*(g)F, /> = <F, T(g)f).

THEOREM. Let T be a measure on R such that f elOitdr{a) is a realfunction of t. Then the restriction of the representationVT 0fS =C°°(Xy Diff X to the cyclic subspace SS c /,£(#') generated bythe vector £1 = 1 is equivalent to the representation of VT defined in

PROOF. From the definition of VT it follows that

(Vx (/, \|?) Q, Q) = exp ( j j (cos (a/ (x)) j j / 2 (x) -1) dm (x) d% (a))

(here the diamond brackets denote the inner product in L^(H')). Conse-quently, from the hypothesis of the theorem,

(Vx.(f,

where uT is the spherical function of Vr (see §7.2). Hence the assertionof the theorem follows immediately.

REMARK 1. A representation of C°°{X)* Diff X was constructed in [15]by means of the N/V limit. This representation coincides with that con-structed here for the Poisson measure (the connection with the Poissonmeasure was apparently not noticed), and the transition to a Fock modelin [15] is equivalent to the realization of this representation as EXP^T(see above).

We emphasize that a representation of the cross product can be con-structed for any measure in .ff (X) that is quasi-invariant under Diff X.However, only those that are constructed from an infinitely decomposablemeasure have the structure EXP^T, because it is only in this case thatthere is a vacuum vector.

REMARK 2. Instead of C°°(X) we can consider an arbitrary group ofsmooth functions C°°(X, G) = Gx on X with values in a Lie group G andthe cross product C°°(T, G)- Diff X.

If a unitary representation IT of G is given on a space H, then therepresentation T of this cross product acts naturally in the space

SB = \ ®Hxdm(x), Hx = H. This is irreducible if IT is irreducible. If

p: C°°(X, G)-Diff X-+3B is a non-trivial cocycle (see [1]), then inEXP SB we get a representation EXP^T of C°°(X, G)- Diff X.

APPENDIX 1On the methods of defining measures on the

configuration space Tx

1. Let Xx C . . . C Xn C . . . be a sequence of open submanifolds in X withcompact closures such that X = U Xn. The projections pk: Tx -* Bx

n k


are given by putting pk 7 = 7 0 1 ^ . Let M be a measure on Tx and[ik - pkn its projection on Bx(k = 1, 2, . . . )• Then the measures txk

are mutually compatible, that is, for any k > 1 we have P\k[ik = M/>

where plk\ Bx -> Bx is the natural projection. A well known theorem ofKolmogorov about the extension of measures enables us to establish theconverse: if {\ik} is any compatible sequence of measures on {BXfe}, thenthere is a unique measure on Tx such that pk\x = fxk(k = 1, 2, . . . ).

We can abandon the compatibility conditions and consider sequences ofmeasures nk on Bx for which lim Pikixk - M(/) exists (in the weak sense)

for all /. In this case the measures JU(/) on B x are compatible and definea measure /i on Fx.

We also recall that Fx is naturally embedded in the space of generalizedfunctions (7 -• 2 5X), therefore, the methods for defining measures in

X<Ey

linear spaces are applicable here (by means of the characteristic functionaland so on); see, for example, [5].

2. A fundamentally different method describing measure on Fx hasreceived attention in statistical physics [7]. It generalizes the method ofspecifying Markov measures (by transition probabilities). It consists ingiving conditional measures on By (or their densities with respect toPoisson measure) as functions on Txx Y f°r aH compact domains Y C X bymeans of a single function (the potential) on B^. The question of existenceand uniqueness of the measure on Tx with a given system of conditionalmeasures is, as a rule, very difficult. Curiously enough, in this case the conditionfor a measure ju on Tx to be quasi-invariant under Diff X can be formulatedvery simply: all the conditional measures on By, where Y is any open set withcompact closure, must be equivalent to a quasi-invariant (under Diff Y)measure on BY • By now there are many such measures known in statisticalphysics that are not equivalent to the Poisson measure (Gibbs measures).

A measure on Tx can also be given with the help of so-calledcorrelation functions on B^; a correlation function defines uniquely aninitial measure on Fx (see, for example, [12]).

3. Let us introduce yet another method of defining measures on Tx. Wesay thatâ normalized Borel measure JJL on X°° (see §0.3) is admissible if:

1) fJL(X°°) = 1, that is, X°° is a subset of full measure in X°°;2) /! is quasi-invariant under Diff X.If /x is an admissible measure on X°°, then its projection JU = pix on Tx

(that is, ju(C) = ix(p~lC) for any measurable set C C F j ) is a quasi-invariantmeasure on Fx. This method of defining a measure on Tx is of limitedinterest, however, it is convenient for constructing various examples.

It is easy to show that an admissible measure M on X°° is ergodic if andonly if it is regular (regularity means that it satisfies the 0 - 1 law). Inparticular, any admissible product measure fx = mx X . . . X mn X . . . is


ergodic. The following lemma is analogous to the Borel-Cantelli lemma.LEMMA. The product measure y =ml X . . . X mn X . . .on X°° is

admissible if and only if f mt(Y) < <» for any compact set Y C X.

EXAMPLE OF AN ADMISSIBLE MEASURE. Let X = Rn. We denoteby ma the Gaussian measure with centre a t f l G R and with unitcorrelation matrix: dma(x) = (7-n)-~nl2e~u~a^ 2 dx, where dx is the Lebesguemeasure on R". By a direct computation it is not difficult to check thatju = maX . . .X ma X . . . is admissible if, for example,\\an\\>c log n, n = 1, 2, . . ..

If \\an\\ -• °° sufficiently quickly, then it is easy to verify that /i is con-centrated on the set As = n (sTx)o, where s: Tx -• X°° is the CrOSS-section corresponding to a certain admissible indexing / in Tx. However, juis not invariant under left translations in X°°\ (Tx\ = xi+l. Hence itsprojection /x = pfx is a non-saturated measure in Tx (see Proposition 7 in§2.3).

Another example refers to the group Diff X, where X is a compactmanifold. In this case, let X°° denote the^set of all sequences in X thatconverge in X. It is easy to verify that X^/S00 = Tx is the union of allcountable subsets of X with a unique limit point (one for each subset).Let x0 6 J , let p be a continuous metric in X, and let {mn} be a sequence

of smooth measures in X such that lim \ p(x, xo)dmn(x) = 0. It is clear

that the product measure mx° = II mn is concentrated on X°°. We intro-n

duce the measure in = f m*° <ix0 ~ t n e mixing of the measures mx° in

Z°°. This in projects onto a measure m on F^ that is quasi-invariant andergodic under Diff X. Another more complicated example of a measureon the countable subsets of a compact manifold that is quasi-invariant andergodic under Diff X was constructed earlier in [8].

APPENDIX 2

iS^-cocyles and Fermi representations

According to the standard definition, a 1-cocycle on Diff X with valuesin the group S^(YX) of measurable maps Yx -> S^ is a mapa: Diff X -* ^ ( P ^ ) satisfying the following condition:

(l) <y(%, yM%, îlv) = or(% 2» v)-

Two 1-cocycles ox and o2 are said to be cohomologous if there exists ameasurable map o0: Tx -• 5^ such that

Represenrations of the group of diffeomorph isms 105

(2) a2(o|), y) = <ro(Y)ai(i>, vWiV^)-

Let / be a measurable indexing in F^, ak = Tx -+ X (k = 1 , 2 , . . . ) ^the sequence of measurable maps defined by i (see §2.2); and 5: Tx -> Z°°the cross-section of the fibration1 X°° -> Tx defined by i\

Further, let As = II

We say that a measurable indexing / is correct if for any 7, 7 ' E F^the conditions I7 n 7T| = | 7 ' n # | and 7 n (X \ 7Q = 7 ' O (X \ JO for acertain compact set K C X imply that ak(y) = 0^(7') for all indices kexcept finitely many.2 A cross-section s: Fx -> X00 defined for a correctindexing / is also called correct. If / is correct, then the set A^ is invariantunder Diff X. _

To each correct cross-section s: Tx -> X°° there corresponds a 1-cocycleos defined by the following relation (see §2.2):

REMARK. There are examples of cocycles that are not generated bycorrect cross-sections.

Cocycles os generated by correct cross-sections s are also called correct.We give, without proof, some properties of correct cocycles os.1) The cross-section s is uniquely determined by the correct cocycle os

corresponding to it.2) Any two correct cocycles os and as are cohomologous as cocycles

with values in S^F^) , that is,

(3) crS2 (of, 7) = oQ(y)a8i (i|>, Y K 1 ^ " 1 ^

where a0 is a measurable map F x -> S°°.3) No correct 1-cocycle is cohomologous to the trivial cocycle.4) Let os be a correct cocycle, o0: Tx ^ S°° a measurable map, and o

a 1-cocycle defined by

afo, 7) = tfo(Y)tfs(^ 7) ffo1^"1?)-

For a to be correct it is necessary and sufficient thato0(y)<Jol(\lj-ly) e S^ for any \p G Diff X and 7 e Tx.

5) Two correct cocycles os and as are cohomologous as cocycles withvalues in S^ (Tx) if and only if the cross-sections Sx and s2 are cofinal,that is, s2 = ao° 5X where o0 G ^ ^ ( r ^ ) .

1 Note that, generally speaking, the fibration X°° -* Vx has no continuous cross-sections. It can beshown that this fibration has no continuous quotient fibrations with fibre Z2.

The condition of correctness is weaker than the condition of admissibility introduced in § 2.2


For each 1-cocycle a: Diff X -+ 5^(1^) we define a corresponding1-cocycle aa\ Diff X -* Z2{VX) by

(4) aa(f, y) = sign a(ip, v)

(sign o is the parity of a E ^ ) .It is obvious that when ox and a2 are cohomologous, then the correspond-

cocycles aa and aa are cohomologous. However, there exist cocycles ox

and o2 (even correct ones) that are not cohomologous, although thecorresponding cocycles aa and aa are.

EXAMPLE. The correct cocycles Oi(\p9 y) and o2(\jj, 7) = o0o(\p, 7)ao1,where a0 ^ S^. Since sign (OQOOZ1) = sign a, we have aa = aa . However,the cocycles ox and o2 are themselves not cohomologous (see property 4).

A SUFFICIENT CONDITION FOR COCYCLES TO BE COHOMOLOGOUS.Let a2(i//, 7) = ao(7)o ri(^,7)ô1(^"17)- If cr0(7) = croao(7), where00 ^•S'oo(r^) and a0 is an arbitrary element of S°°, then aa ~ aa^.

Note that each Z2-cocycle a: defines a Z2-covering of Tx with a givenaction of Diff X on it. The elements of this covering are pairs(e, 7), 7 e r x , 8 = ± 1; and Diff X acts by ^(e, 7) = (ea(ij/, 7), ty~ly).

Z2-cocycles of the form (4), where a is a correct cocycle, are calledcorrect Z2-cocycles, and the Z2-coverings of Tx defined by them are alsocalled correct.

LEMMA. Correct Z2-cocycles are non-trivial. ^Let / be a correct indexing in Tx, s: Tx -> X°° the cross-section defined

by /, ix a quasi-invariant measure in Tx, and T a unitary representation ofS^ acting on a Hilbert space H. We associate with the triple (/, jtz, T) aunitary representation V of Diff X in the space 3$ C L^(AS, H) offunctions / : As ->- H such that

for every a G S^ ; the representation operators are defined by

(5)

where p is the projection As ->• F^.ALTERNATIVE DEFINITION: 7 is given in the space Ll(Tx, H) of

functions f: Tx -* H for which | |/ | |2 = f 11/(7) 11^*(7) <°°- the operators

F(\//)have the form

(6)

where a5 is the 1-cocycle generated by a correct cross-section s: F^ -> X°°.It is obvious that the representations of Diff X so defined are equivalent.Note that V= Ind^ i f f x(7o TT), where (7 is the subgroup of all diffeo-


morphisms \p £ Diff X for which ^7 = 7, and TT is the projectionGy -* Gy\ G° = S^. (G° is the subgroup leaving every point x G 7 fixed).

Two 1-cocycles a and a' are said to be equivalent with respect to ameasure pt if a(\//, 7) = o'(\p, 7) mod 0 with respect to /z for any\p £ Diff X. Note that the right-hand side of (6) does not change if the1-cocycle is replaced by any equivalent cocycle. Therefore, the 1-cocycleso need only be defined up to an equivalence.

The properties 1) - 5) of 1-cocycles can be reformulated withoutdifficulty for equivalence classes of 1-cocycles. Moreover, 2) can be mademore precise as follows: the map a0 in (3) is uniquely determined mod 0if n is ergodic.

Let us consider the particular case when T = Ind^°°x sn (p X / ) , where pis a unitary representation of Sn and / is the unit representation of S"OS" is the group of finite permutations of n + 1, n + 2, . . . ) . Comparingthe definition of V with that of the elementary representations Up (see §3)it is easy to check that V = £/£. Hence, in particular, in this example allthe correct cocycles lead to equivalent representations.1

Quite a different example is the Fermi representation. Let us consider aZ2-cocycle <x(\p, 7) (see above) and define a representation of Diff X inL2(TX) by

This is called a Fermi representation of Diff X.When the Z2-cocycle is generated by a correct cross-section

s: Tx -* X°°, that is, when a(\p, 7) = sign OS(\JJ9 7), where os:Diff X -+ S^ (Vx) is a correct cocycle, then there is another convenientrealization of V: the representation is given in SB by a function fix) onA s c r such that

1) f(xa) = sign of(x) for any a G S^ (an "odd" function).

2) il/H2= J \f(sy) |2 î(y) < 00.

The representation operators are defined by (5). In this case it can beshown by the same arguments as in §3 that // JJL is ergodic and the cross-section s: Tx -+ X00 is generated by an admissible indexing i (in the senseof §2.2), then the Fermi representation V(\jj) is irreducible.

REMARK 1. Apparently, there exist non-equivalent Fermi representationsof Diff X constructed from the same measure n on Tx (for the con-struction of a Fermi representation by means of an N/V limit, see [19]and [20]).

REMARK 2. The group Diff X has factor representations of type II -it is sufficient to take a representation of S^ of type II in H (for example,1 There is a more general fact: if a representation T of 5» can be extended to a representation of 5°°,then the representation V of Diff X corresponding to T is uniquely determined, up to equivalence, by Tand a measure ju on Tx..


the regular representation) and to construct a representation of Diff X inLl{Vx, H) from it.

APPENDIX 3

Representations of Diff X associated with measures in thetangent bundle of the space of infinite configurations

The representations of Diff X discussed above are of "zero order", thatis, they do not depend on the differentials of the diffeomorphisms. In thiscontext let us note that these representations can be extended torepresentations of the group of measurable finite transformations of amanifold X with a quasi-invariant measure.

However, one can construct representations of Diff X of positive order(by a representation of order k we mean one depending essentially on theA:-jet of the diffeomorphisms). A number of representations of Diff X inspaces of finite functional dimension were defined in [9]; in the terminologyof this paper these representations are connected with the space Bx offinite configurations. But there are also representations of positive orderconnected with the space Tx of infinite configurations. Let us take as anexample a representation of order 1 connected with this space.

Let (i be a measure in Tx that is quasi-invariant under the action ofDiff X. We consider the "tangent bundle" TTX over Tx, that is a fibrebundle over Tx, where the fibre over 7 = {xt} is the direct product

II Tx. I of the tangent spaces at the points xt £ 7.i= 1 i _ _ _ _

The space JTX can be regarded as the factor space TX^/S00, where TX°°

is the subset of (TX)°° = U TXt, Xt = X, consisting of points1=1

{(xi9 uf); u,- G Tx.Xt, {xt} e X^yjhe topology in TTX and the a-algebraof Borel sets are induced from TX°°.

Let XT be a normalized measure in TX.X that is equivalent to the00 00

Lebesgue measure, and let X7 = II XT be the product measure in II TY X./= 1 l / = 1 xi

In this way a measure X is introduced in TYX such that its projection00

onto Tx is M and that the conditional measures in TyFx =11 Tx. X areX7. The action of Diff X on (TTX, X) is defined by i//(% a) = (1//7, d\jja),

00

where a = II ax, ax G TXX and d\l/ is the natural action on TTX. The

measure X is quasi-invariant under this action. This leads to a unitaryrepresentation of Diff X in L^(TTX). A proof that this is irreducible forergodic measures n can be modelled on §3. The parameters of therepresentations just constructed are the measure JJL in Tx and the measures

Represen tations of the group of diffeomorphisms 109

X? in r ^ I VIt is easy to see how to construct representations of order 1 analogous

to the elementary representations. We do not say much about representationsof higher order, because difficulties in describing them arise even in the caseof a finite number of particles (see [9]).

REMARK. The representations of Diff X listed in this appendix can beextended to representations of the cross-product C°°(Xy Diff X.

References

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[2] A. M. Vershik, I. M. Gel'fand and M. I. Graev, Irreducible representations of the groupGx and cohomology, Functsional. Anal, i Prilozhen. 8:2 (1974), 67-69. MR 50 # 530.= Functional Anal. Appl. 8 (1974), 151-153.

[3] D. B. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodicdiffeomorphisms, Trudy Moskov. Mat. Obshch. 23 (1970), 3-36.

[4] A. Weil, L'integration dans les groupes topologiques et ses applications, Actual. Sci.Ind. 869, Hermann & Cie, Paris 1940. MR 3 # 198.Translation: Integrirovanie v topologicheskikh gruppakh i ego prilozheniya, Izdat. Inost.Lit., Moscow 1950.

[5] I. M. Gel'fand and N.Ya. Vilenkin, Obobshchennye funktsii, vypA. Nekotoryeprimeneniya garmonicheskogo analiza. Oskashchennye giVbertovy prostranstvaGos. Izdat. Fiz.-Mat. Lit., Moscow 1961. MR 26 #4173.Translation: Generalized functions, vol.4, Some applications of harmonic analysis,Equipped Hilbert spaces, Academic Press, New York-London 1964.

[6] A. M. Vershik, Description of invariant measures for the actions of some infinite-dimensional groups, Dokl. Akad. Nauk SSSR 218 (1974), 749-752.= Soviet Math. Dokl. 15 (1974), 1396-1400.

[7] R. L. Dobrushin, R. A. Minlos and Yu. M. Sukhov,Prilozhenie k knige Ryuelya:Statisticheskaya mekhanika (Supplement to Ruelle's book Statistical mechanics).Mir, Moscow 1971.

[8] R. S. Ismagilov, Unitary representations of the group of diffeomorphisms of the circle,Funktsional Anal, i Prilozhen. 5:3 (1971), 45-53.= Functional. Anal. Appl. 5 (1971), 209-216.

[9] A. A. Kirillov, Unitary representations of the group of diffeomorphisms and some ofits subgroups, Preprint IPM, No.82 (1974).

[10] A. A. Kirillov, Dynamical systems, factors, and group representations, Uspekhi Mat.Nauk 22:5 (1967), 67-80.= Russian Math. Surveys 22:5 (1967), 63-75.

[11] V. A. Rokhlin, On the fundamental ideal of measure theory, Mat. Sb. 25 (1949),107-150. MR 11 #18

[12] D. Ruelle, Statistical mechanics. Rigorous results, W. A. Benjamin Inc., Amsterdam1969.Translation: Statisticheskaya mekhanika. Strogie rezul'taty, Mir, Moscow 1971.

1 1 0 4. M. Vershik, I. M. Gel'fand andM. I. Graev

[13] S. V. Fomin, On measures invariant under a certain group of transformations, Izv.Akad. Nauk SSSR Ser. Mat. 14 (1950), 261-274. MR 12 #33.

[14] G. Goldin, Non-relativistic current algebras as unitary representations of groups,J. Mathematical Phys. 12 (1971), 462-488. MR 44 # 1330.

[15] G. Goldin, K. J. Grodnik, R. Powers and D. Sharp, Non-relativistic current algebra inthe A/K limit, J. Mathematical Phys. 15 (1974), 88-100.

[16] A. Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Math.261, Springer-Verlag, Berlin-Heidelberg-New York 1972.

[17] J. Kerstan, K. Mattes and J. Mecke, Unbegrenzt teilbare Punktprozesse, Berlin 1974.[18] D. Knutson, X-rings and the representation theory of the symmetric group, Lecture

Notes in Math. 308, Springer-Verlag, Berlin-Heidelberg-New York, 1973.[19] R. Menikoff, The hamiltonian and generating functional for a non-relativistic local

current algebra, J. Mathematical Phys. 15 (1974), 1138-1152. MR 49 # 10285.[20] R. Menikoff, Generating functional determining representations of a non-relativistic

local current algebra in the iV/K limit, J. Mathematical Phys. 15 (1974), 1394-1408.[21] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965),

286-294. MR 32 #409[22] R. S. Ismagilov, Unitary representations of the group of diffeomorphisms of the space

R", n>2, Funktsional. Anal, i Prilozhen. 9:2 (1975), 71-72.= Functional Anal. Appl. 9 (1975), 144-145.

Received by the Editors, 15 May 1975

Translated by A. West

AN INTRODUCTION TO THE PAPER'SCHUBERT CELLS AND COHOMOLOGY

OF THE SPACES GIFGraeme Segal

It is well known that a generic invertible matrix can be factorized as theproduct of an upper triangular and a lower triangular matrix. A more precisestatement is that any invertible n X n matrix g can be Written in the formb1wb2, where bx and b2 are upper triangular and w is a permutation matrix.Here w is uniquely determined by g, though bx and b2 are not. The matrices gfor which w is the order-reversing permutation i*-*n —i + 1 form a dense opensetinGLn(C).

This double-coset decomposition GLW (C) = U BwB, where B denotes thew

upper triangular matrices and w runs through the permutation matrices, has ananalogue for any connected affine algebraic group G. The role of B is played bya Borel subgroup (i.e. a maximal soluble subgroup), and the role of thepermutation matrices by the Weyl group W = N(H)/H, where H s (Cx); is amaximal algebraic torus in G and N(H) is its normalizes The decomposition isnowadays called the Bruhat decomposition; but Gelfand had earlier recognizedits importance in his work on the representations of the classical groups.

It is best to think of the decomposition as the decomposition of the homo-geneous space X = G/B into the orbits of the left action of B. The space X playsa central role in representation theory. It turns out that it is a complexprojective algebraic variety, and that the orbits of B are algebraic affine spacesCm of various dimensions, the "Bruhat cells". The closures of the cells arealgebraic subvarieties which in general have singularities. It is important thatthe maximal compact subgroup K of G acts transitively on X, so that X has analternative description as K/T, where T = K O B is a maximal torus of K.

If G = Ghn (C) then X is the flag manifold: a flag in Cn is an increasingsequence of subspaces

F=(F1 CF2 C...CFn - C " )

with dim(Fk) = k. For GLn (C) acts transitively on the set of all flags, and Bis the isotropy group of the standard flag C C C 2 C . . . C Cn . In this casethe cells are indexed by permutations w of {1, 2, . .> ,, n}, and we can take as arepresentative point in the cell Xw the flag Fw such that F™ is spanned by

{euKi)>euKi)> ' • ->ew{k)}> w h e r e {*i> • • •>*/!> is the standard basis of C". Z ^can be defined by "Schubert conditions": it consists precisely of the flags Fsuch that dimCFfc ^Cm) = vkm, where

vkm - card {/: i < k, w(i) < m },

The dimension of Xw is the length l(w) of w, defined by

l(w)= £ | M;(0-/I.

i n

112 Schubert cells and cohomology of the spaces G/P

Alternatively l(w) is the number of pairs (/, /) such that / < / but w(i) > w(j). Infact if Nw is the subgroup of B consisting of matrices (ai7-) with diagonal ele-ments aH = 1 and such that <z/y- = 0 unless i < / and w(i) > w(j) then the mapNw ~* Xw given by g*-+g'Fw is an isomorphism of algebraic spaces.

An analogous discussion can be carried out for the other classical groups. IfC" has a non-degenerate bilinear form < , >, and G is the group of auto-morphisms of Cn which preserve the form, then G/B can be identified with theset of flags F such that F^ =Fn_k for k= 1, . . . ,«.

Returning to the general case, we have a topological cell decomposition ofX into cells {Xw} wG w which are all of even dimension. The homology groupsof X are therefore free abelian with the classes [Xw ] of the cells as a naturalbasis.

On the other hand there is a completely different way of describing thecohomology ring of X. For every algebraic homomorphism X: B -* Cx, orequivalently for every character X of the compact torus 71, there is a holo-morphic line bundle Ex on Xx Associating to X the first Chern class cx = cx (Ex)of Ex gives an isomorphism T-* H2(X\ Z). (Jis the lattice of weights of G: inthe paper it is called 1} | . ) The classes cx generate the cohomology ring of Xmultiplicatively over the rationals, and H*(X; Q) = R/J, where R is the poly-nomial algebra over Q generated by the cx, and / is the ideal generated by thehomogeneous W-invariant polynomials of positive degree. (When G = GLW(C)there are n obvious line bundles Ex, . . .,En on X: the fibre of Ef at a flag F isFi/F^ j . The classes xt = c(Ej) span H2 (X; Z). The elementary symmetricfunctions in the xt vanish because they are the Chern classes of Ex © . . .©£„ ,which is a trivial bundle.)

It is natural to ask for the relation between these descriptions of thehomology and the cohomology. In other words if p is a homogeneouspolynomial of degree k in the cK, and w is an element of W length k, what isthe value (p, [Xw] > of p on the cell Xwl One can also ask how to express thecohomology class Poincare dual to a cell Xw as a polynomial in the Chernclasses.

To answer these questions it is enough in principle to determine the cap-product cx n[Xw]e H2k_ 2 (X; Z) for each X G f and each w e W of length k.The paper uses a simple and very attractive geometrical argument to do this.One begins by observing that by linearity it is enough to consider weights Xwhich are in the interior of the positive Weyl chamber. In that case X can beembedded as a projective algebraic variety in P(VX), the projective space of theirreducible representation Vx of G with highest weight X, as the orbit under Gof the highest weight vector fx. (Vx is the dual of the space of holomorphicsections of Ex.) The cohomology class cx = cx(Ex) is then the class dual to theintersection X O IT, where II is a hyperplane inP(VX); and the cap-productwith cx can be interpreted as the geometric operation of intersection with II.This is amenable to calculation because of the following properties of theembedding X^P(VX):

Schubert cells and cohomology of the spaces G/P 113

(i) the centre of the cell Xw maps to the point of P(VX) represented by theweightj/ector/^ G Vx of weight wX,

(ii) Xw is precisely Jthe intersection ofZ with a linear subspace of P(VX), and(iii) the boundary Xw ~XW of Xw is Xw D Uw, where 11^ is the hyperplane

perpendicular of fw.Now let us recall that the Weyljgroup W - which we are regarding as a group

of automorphisms of the lattice T — is generated by the reflections oy in thehyperplanes of T perpendicular to the roots 7 of G. If w G W has length k itturns out that the (k - 1 )-dimensional cells in the boundary of Xw are preciselythe Xwa such that l(woy) = k — 1. Thus the cap-product cx E [Xw ] is

necessarily of the form 2 n [Xwa ], where n is a positive integer. To deter-7 T

mine ny one must calculate the order to which the linear form (fw,), whenregarded as a function on Xw, vanishes on the cell XUXJ . That is easy to dobecause the formula

where E_y is the standard element of & in the (- 7) root-space, defines a holo-morphic curve in Xw which passes through the centre fwa = woy fe of Xwa

when t = 0, and is transversal to Xwa . We calculate

(fW9 woy exp(tE_y)fe )=(oyfe,exp(tE_y)fe ) = 0(A),

where ny = < X, Hy ), Hy being the co-root associated to 7, i.e. the element ofthe dual lattice to T characterized by the property

for all x £ T.The formula

gives us the pairing between homology and cohomology in the form

(cXicX2...cXnAXw])=X(\1,Hyi)..A\k,Hyi),

where the sum is over all strings 7 j , . . ., yk of positive roots such that

I shall not describe here the elegant algebraic formulations the authors derivefrom this.

It ought, however, to be mentioned that the methods apply equally well notonly to the space G/B, but to G/P for every parabolic subgroup P of G. Themost obvious case of this is the Grassmannian Gr^ n of /^-dimensional subspacesof C", which is GLW (C)/P, where P is the appropriate group of echelon matrices.

114 Schubert cells and cohomology of the spaces G/P

(Iri terms of compact groups Gr^ n = Un/Uk. X Un_k.) The analogue of Gr . „for the orthogonal groups is the Grassmannian of isotropic ^-dimensional sub-spaces of Cn for some non-degenerate quadratic form on C" : this space can beidentified with On/Uk X On^2k- When k = 1 it is a complex projective quadrichypersurface.

SCHUBERT CELLS AND COHOMOLOGY OFTHE SPACES G/P

I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand

We study the homological properties of the factor space G/P, where G is a complex semi-simple Lie group and P a parabolic subgroup of G. To this end we compare two descriptionsof the cohomology of such spaces. One of these makes use of the partition of G/P intocells (Schubert cells), while the other consists in identifying the cohomology of G/P withcertain polynomials on the Lie algebra of the Cartan subgroup// of G. The results obtainedare used to describe the algebraic action of the Weyl group W of G on the cohomologyof G/P.

Contents

Introduction 115§ 1. Notation, preliminaries, and statement of the main results . . .117§ 2. The ordering on the Weyl group and the mutual disposition

of the Schubert cells 120§3. Discussion of the ring of polynomials on f) 124§4. Schubert cells 133§5. Generalizations and supplements 136

References 139

Introduction

Let G be a linear semisimple algebraic group over the field C of complexnumbers and assume that G is connected and simply-connected. Let B bea Borel subgroup of G and X = G/B the fundamental projective space of G.

The study of the topology of X occurs, explicitly or otherwise, in alarge number of different situations. Among these are the representationtheory of semisimple complex and real groups, integral geometry and anumber of problems in algebraic topology and algebraic geometry, in whichanalogous spaces figure as important and useful examples. The study ofthe homological properties of G/P can be carried out by two well-knownmethods. The first of these methods is due to A. Borel [ 1 ] and involvesthe identification of the cohomology ring of X with the quotient ring ofthe ring of polynomials on the Lie algebra I) of the Cartan subgroup

115

116 /. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand

H c G by the ideal generated by the ^-invariant polynomials (where W isthe Weyl group of G). An account of the second method, which goes backto the classical work of Schubert, is in Borel's note [2] (see also [3]); itis based on the calculation of the homology with the aid of the partitionof X into cells (the so-called Schubert cells). Sometimes one of theseapproaches turns out to be more convenient and sometimes the other, sonaturally we try to establish a connection between them. Namely, we mustknow how to compute the correspondence between the polynomialsfiguring in Borel's model of the cohomology and the Schubert cells.Furthermore, it is an interesting problem to find in the quotient ring ofthe polynomial ring a symmetrical basis dual to the Schubert cells. Theseproblems are solved in this article. The techniques developed for thispurpose are applied to two other problems. The first of these is thecalculation of the action of the Weyl group on the homology of X in abasis of Schuberts cells, which turns out to be very useful in the study ofthe representations of the Chevalley groups.

We also study the action of W on X. This action is not algebraic (itdepends on the choice of a compact subgroup of G). The correspondingaction of W on the homology of X can, however, be specified in algebraicterms. For this purpose we use the trajectories of G in X X X, and weconstruct explicitly the correspondences on X (that is, cycles in X X X)that specify the action of W on H^ (X, Z). The study of such correspon-dences forms the basis of many problems in integral geometry.

At the end of the article, we generalize our results, to the case when Bis replaced by an arbitrary parabolic subgroup P c G. When G = GL(n)and G/P is the Grassmann variety, analogous results are to be found in [4].

B. Kostant has previously found other formulae for a basis ofH*(X, Z), X = G/B9 dual to the Schubert cells. We would like to expressour deep appreciation to him for drawing our attention to this series ofproblems and for making his own results known to us.

The main results of this article have already been announced in [13].We give a brief account of the structure of this article. At the

beginning of § 1 we introduce our notation and state the known resultson the homology of X = G/B that are used repeatedly in the paper. Therest of § 1 is devoted to a statement of our main results.

In § 2 we introduce an ordering on the Weyl group W of G that arisesnaturally in connection with the geometry of X, and we investigate itsproperties.

§ 3 is concerned with the ring R of polynomials on the Lie algebra f)of the Cartan subgroup He G. In this section we introduce the functionalDw on R and the elements Pw in R and discuss their properties.

In § 4 we prove that the elements Dw introduced in § 3 correspond tothe Schubert cells of X.


§ 5 contains generalizations and applications of the results obtained, inparticular, to the case of manifolds X(P) = G/P, where P is an arbitraryparabolic subgroup of G. We also study in § 5 the correspondences on Xand in particular, we describe explicitly those correspondences that specifythe action of the Weyl group W on the cohomology of X. Finally, in thissection some of our results are put in the form in which they were earlierobtained by B. Kostant, and we also interpret some of them in terms ofdifferential forms on X.

§1. Notation, preliminaries, and statement of the main results

We introduce the notation that is used throughout the article.G is a complex semisimple Lie group, which is assumed to be connected

and simply-connected;B is a fixed Borel subgroup of G;X - GIB is a fundamental projective space of G\N is the unipotent radical of B\H is a fixed maximal torus of G, H c B;<$ is the Lie algebra of G; rj and %l are the subalgebras of <$ correspond-

ing to H and N;Ij* is the space dual to t);A C £)* is the root system of t) in @ ;

A+ is the set of positive roots, that is, the set of roots of I) in 9i,A_ = -A+, £ C A+ is the system of simple roots;

W is the Weyl group of G; if y £ A, then aY : I)*-> f)* is an elementof W, a reflection in the hyperplane orthogonal to y. For each element1

w 6 W = Norm(//)/77, the same letter is used to denote a representativeof w in Norm (H) C G.

l(w) is the length of an element w G W relative to the set of generators{aa, a 6 2} of W, that is, the least number of factors in the decomposition

(1) w = oaioa2...oai, a* 6 2.

A decomposition (1), with / = l(w), is called reduced; s G W is theunique element of maximal length, r = l(s);

N_ - sNs~1 is the subgroup of G "opposite" to N.For any w G W we put Nw = w N_wl n N.HOMOLOGY AND COHOMOLOGY OF THE SPACE X. We give at this

point two descriptions of the homological structure of X. The first ofthese (Proposition 1.2) makes use of the decomposition of X into cells,while the second (Proposition 1.3) involves the realization of two-dimen-sional cohomology classes as the Chern classes of one-dimensional bundles.

We recall (see [5]) that ^ = wALw^niVis a unipotent subgroup of

Norm H is the normalizer of H in G.

118 IN. Bernstein, I. M. GeVfand, S. I. GeVfand

G of (complex) dimension l(w).1.1. PROPOSITION (see [5]), Let o 6 X be the image of B in X. The

locally closed subvarieties Xw = Nwo C X, w E W, yield a decompo-sition of X into N-orbits. The natural mapping Nw -* Xw (n H* nwo) isan isomorphism of algebraic varieties.

Let Xw be the closure1 of Xw in X, [Xw] £ H2i±w) (XW,Z) thefundamental cycle of the complex_ algebraic variety Xw andsw £ H2t(w) (X,Z) the image of [Xw ] under the mapping induced by theembedding Xw C-+ X.

1.2. PROPOSITION (see [2]). The elements sw form a free basis of#* (*,Z).

We now turn to the other approach to the description of the cohomo-logy of X. For this purpose we introduce in Ij the root system{Hy, y £ A} dual to A. (This means that oy\ = x ~ X (Hy)y for allX 6 £) *, 7 6 A). We denote by !)Q CZ ^ the vector space over Q spannedby the Hy. We also set f)£ = {x 6 ^ * I x (^7) 6 Z for all 7 6 A} and

Let R = S i(^) be the algebra of polynomial functions on f)Q withrational coefficients. We extend the natural action of W on ^* to R. Wedenote by / the subring of PV-invariant elements in R and set/+ = { / e / | / ( 0 ) - o } , J = I+R.

We construct a homomorphism a: /?-> M*(X9 Q) in the following way.First let x 6 *)z- Since G is simply-connected, there is a character0 € Mor (7/, C*) such that 6 (exp /z) = exp xW, h G t). We extend 0to a character of B by setting 0(n) = 1 for « 6 N. Since G->-Ar is aprincipal fibre space with structure group B, this 0 defines a one-dimensional vector bundle Ex on X We set c*i(x) = cx, wherecx G iPiX, Z) is the first Chern class of Ex. Then ax is a homomorph-ism of tjz into ^ ( Z , Z), which extends naturally to a homomorphismof rings a: R -> #*(X, Q).

Note that W acts on the homology and cohomology of X. Namely,let K c G be a maximal compact subgroup such that T = K. n / / is amaximal torus in A". Then the natural mapping K/T-+X is a homeomorph-ism (see [ 1 ]). Now W acts on the homology and cohomology of X inthe same way as on K/T.

1.3. PROPOSITION ([1], [8]). (i) The homomorphism a commuteswith the action of W on R and H* (X, Q).

(ii) Ker a = J, and the natural mapping a: R/J -> H* (X, Q) is anisomorphism.

In the remainder of this section we state the main results of this article.The integration formula. We have given two methods of describing the

As Xw is a locally closed variety, its closure in the Zariski topology is the same as in the ordinarytopology.


cohomological structure of X. One of the basic aims of this article is toestablish a connection between these two approaches. By this we under-stand the following. Each Schubert cell sw £ #* (X, Z) gives rise to alinear functional Dw on R according to the formula

Dw(f) = <*w, a (/))

(where < , > is the natural pairing of homology and cohomology). Weindicate an explicit form for Dw.

For each root 7 6 A, we define an operator AY : R -* R by the formula

f

(that is, Ayf(h) = [/(h) - f(o7h)]/y(h) for all h 6 \ )• Then we have thefollowing proposition.

PROPOSITKDN. Let w = attj . . . oav at £ S. If/(">) < /, thenA a i • • • A a i = 0 . / / l{w) - /, r/ze/^ /7ze operator A a ^ . . . A a i dependsonly on w and not on the representation of w in the formw = aai . . . o a i ; we put A w = A a i . . . A a y

This proposition is proved in §3 (Theorem 3.4).The functional Dw is easily described in terms of the Aw : we define for

each w E W another functional Dw on R by the formula Dw f = Aw /(0).The following theorem is proved in §4 (Theorem 4.1).

THEOREM .Dw = Dw for all w G W.We can give another more explicit description of Dw (and thus of Dw).

To do this, we write u)x ^ w2, M>I, w2 £ W, y £ A+, to express.the factthat ifi = oyw2 and / (w;2) ~ I (^1) + 1.

THEOREM. Let w 6 W, /(w) = /.(i) / / / E /? w <2 homogeneous polynomial of degree k ^

^ ( / ) = 0.(ii) If xi, - - -, Xi €M ,then 4,(xi • • - -Xi) = S X i ( ^ )

where the sum is taken over all chains of the form

w0

ry y y- > W i -Z . . . 4

(see Theorem 3.12 (i), (v)).The next theorem describes the basis of H*(X, Q) dual to the basis

{sw I w 6 W7} of H*(X, Z). We identify the ring R = R/J with H*(X, Q)by means of the isomorphism a of Proposition 1.3. Let {Pw \ w 6 W} bethe basis of R dual to the basis {sw \ w 6 ^F} of ^(A' , Z). To specify Pw,we note that the operators Aw: R -+ R preserve the ideal J C R (lemma3.3 (v)), and so the operators Aw : R -* R are well-defined.

THEOREM, (i) Ler s £ W be the element of maximal length, r = /(s)Ps = pr/r! (mod /) = IWI"1 [] 7 (mod / ) , {where P 6 %

VGA+

120 /• M Bernstein, I. M. Gel'fand, S. I. Gel'fand

the sum of the positive rootsjind \W\ is the order of W)(ii) If w e W, then Pw = AW.UP8 (see Theorem 3.15, Corollary 3.16,

Theorem 3.14(i)).Another expression for the Pw has been obtained earlier by B. Kostant

(see Theorem 5.9).The following theorem gives a couple of important properties of the Pw

THEOREM (i). Let X € %, w£W. Then x-Pw - 2 w x W pw

(see Theorem 3.14 (ii)).(ii) Let $ : H^Xf Q) -> H*(X, Q) be the Poincare duality. Then

& (sw) = a (Pw8) (see Corollary 3.19).THE ACTION OF THE WEYL GROUP. The action of W on H* (X, Q)

can easily be described using the isomorphism a: R/J -+ H*(X9 Q), but weare interested in the problem of describing the action of W on the basis{sw) of H^X, Q).

THEOREM. Let a 6 2 , w 6 W. Then oasw = - sw if l(woa) = l(w) - 1

and oasw == — sw + 2J w'a(Hy) sw., if l(woa) = /(w) + 1 (seev

Wf—>WG

Theorem 3.12 (iv)).In § 5 we consider some applications of the results obtained. To avoid

overburdening the presentation, we do not make precise statements atthis point. We merely mention that Theorem 5.5 appears important to us,in which a number of results is generalized to the case of the varietiesX{P) = G/P (P being an arbitrary parabolic subgroup of G), and alsoTheorem 5.7, in which we investigate certain correspondences on X.

§2. The ordering on the Weyl group and the mutualdisposition of the Schubert cells

2.1 DEFINITION (i) Let wx, w2 6 W, y 6 A+. Then wx ^ w2 indicatesthe fact that OyWr = w2 and l(w2) = l(u)\) + 1.

(ii) We put w < w'- if there is a chain

w = Wi-> w2 -*• . . . - > • wk = w'.

It is helpful to picture W in the form of a directed graph with edges drawnin accordance with Definition 2.1 (i).

Here are some properties of this ordering.2.2 LEMMA. Let w = o a j . . . oai be the reduced decomposition of an

element w 6 W. We put yt = att i . . . a a . _ i ( a , ) . Then the roots

7 i » • • ., Ifi are distinct and the set {Yi> • • •» Y*} coincides with A+ n n>A_.

This lemma is proved in [6] .2.3 COROLLARY, (i) Let w = oai . . . o^ be the reduced decompo-

sition and let y G A+ be a root such that w~1y G A-. Then for some i( 2 ) OyOa,i . . . cr a . = a a i . . . Gaii.


(ii) Letw 6 PF, Y £ &+-Then I (w) < / (oyw)Jf and only if w^y £ A+.P R O O F ( i) F r o m L e m m a 2 .2 w e d e d u c e t h a t y = a t t j . . . o^.^ (a f ) for

some z, a n d ( 2 ) fo l lows .(ii) If w^y £ A_, t h e n b y ( 2 ) oyw = o0Ci . . . aa._t aa.+i . . . oap t h a t is

l(oyw) < l{w). I n t e r c h a n g i n g w a n d oyw, w e see t h a t if w~xy g A + , t h e nl(oyw).

2.4 LEMMA. Letwu w2 6 W, a 6 2, y 6 A+,and y ± a. Let y = o^y. If

(Ta^2 .

Conversely, (3) follows from (4).PROOF. Since a £ 2 and y =£ a, we have 7' = aa7 6 A+. It is there-

fore sufficient to show that l(oaw2) > l(w2) = l(wi). This follows fromCorollary 2.3, because aau;2 = oy'Wx and (ffa^)"1?' = w cTaV' = 2*7 6: A_by (3). The second assertion of the lemma is proved similarly.

2.5 LEMMA. Let w, w' 6 W, a 6 2 and assume that w < w . Thena) either oaw < w or oaw < oaw\b) either w < oaw' or oaw < oaw'.PROOF a) Let

We proceed by induction on A;. If aaiy < w or aaw = w2, the assertion isobvious. Let w < o^w, oaw ¥^ w2 • Then aaw; < oaw2 by Lemma 2.4. Weobtain a) by applying the inductive hypothesis to the pair (w2, w').

b) is proved in a similar fashion.2.6. COROLLARY. Let a 6 2 , ^ J!> u?;, w2 _ > M£. If one of the

elements Wi, wi z'5 smaller (in the sense of the above ordering) than oneof w2, w2, then Wi < w2 < w2 and Wi < w\ < w2.

The property in Lemma 2.5 characterizes the ordering <. More precisely,we have the following proposition:

2.7 PROPOSITION. Suppose that we are given a partial ordering w H won W with the following properties:

a) / / a f 2 , w 6 W with l(oaw) = l(w) + 1, then w -\ oaw.b) If w H wf, a £ 2 , then either oaw -) w' or oaw —| oaw'.Then w -\ w' if and only if w < w'.

122 /• N. Bernstein, I. M. Gel'fand, S. I. Gel'fand

PROOF. Let s be the element of maximal length in W. It follows froma) that e —j w -\ s tor all w 6 W.

I. We prove that w < w implies that w -\ if'. We proceed by inductionon /(it/). If l{w) = 0, then w = e, w = e and so w H H/. Let /(u/) > 0and let a 6 2 be a root such that l(oaw') = l{w) - 1. Then by Lemma2.5 a), either aaw < oaw' or w < oaw'.

(i) w < oaif' => if -\ oaw' (by the inductive hypothesis), =» if H u^'.(using a)).

(ii) O^M; < aau;' => aau; - | aait>' (by the inductive hypothesis), => eitherw -\ oaw' or w -\ w' (applying b) to the pair (oaw, o^w')), => w -\ w'.

II. We now show that w - | w implies that w < wr. We proceed bybackward induction on l(w). If t(w) = r = /(s), then w; = 5 , u;' = s, and sou; < w . Let /(u;) < r and let a be an element of 2 such thatl(o7w) = /(w) + 1. By b) either oaw -) u;' or oait; H aait>'.

(i) aaw; H M;' =*• oaw < u;' (by the inductive hypothesis) => if < w'.(ii) aaii> H oaw' => aait; < aait>' => if < H/ (by Corollary 2.6).

Proposition 2.7 is now proved.2.8 PROPOSITION. Let w 6 W arid let w = aax . . . oai be the reduced

decomposition of w.a) / / 1 < I'I < h < • • • < k < / ^ « ^

(5) i*f = a B i i . . . a o V

fên u;' < w.b) / / a;' < w, then w can be represented in the form (5) for some

indexing set{ij).c) / / w -* if, ^ e « there is a unique index /, 1 < / < /, 5wc/z /zar

(6) a?' = att l . . . â ._ laa .+ i . . . a « r

PROOF. Let us prove c). Let if' - if. Then by Lemma 2.2 there is atleast one index / for which (6) holds. Now suppose that (6) holds for twoindices /, /, i < j . Then oa.+i • • • %• = Oat • • • °aHl • T n u s

5

aa. . . . oa. = aa.+i . . . a ^ , , which contradicts the assumption that thedecomposition if = aaj . . . oai is reduced.

b) follows at once from c) if we take into account the fact that thedecomposition (6) is reduced. We now prove a) by induction on /. Wetreat two cases separately.

(i) it > 1. Then by the inductive hypothesis w' < oaj . . . oav that is,w < oai if < if.

(ii) ix = 1. Then, by the inductive hypothesis,°ax w' = oa. . . . oa. < oaiw = oa2 . . . oav By Corollary 2.6, if' < if.

Proposition 2.8 yields an alternative definition of the ordering on W(see [7J). The geometrical interpretation of this ordering is very interesting


and useful in what follows.2.9 THEOREM. Let V be a finite-dimensional representation of a Lie

algebra (& with dominant weight X. Assume that all the weights w\w 6 W, are distinct and select for each w a non-zero vector fw £ V ofweight w\. Then

w'^w<=>fw, £U («R) fw

{where U (91) is the enveloping algebra of the Lie algebra %l).PROOF. For each root y £ A we fix a root vector £V6@ in such a

way that [Ey, E.y] - Hy. Denote by 2fv the subalgebra of @, generated byEy, E-y, and Hy. 21Y is isomorphic to the Lie algebra sl2(C). Letw' - > w and let V be the smallest 2IY -invariant subspace of V containing

2.10 LEMMA. Let n = w'\(Hy) G Z, n > 0. The elements{ElyU- \i --- 0, 1, . . ., n)form a basis of V. Put f= E^fw>. ThenE_~f = 0, Etf = c%. (cf ^ 0) and fw = cf (c =£ 0).

PROOF. By Lemma 2.2, w'~l y e A+, hence Eyfw> =~cEyw'fe = cw'Ew>-iyfe = 0, that is, fw> is a vector of dominant weightrelative to $ v . All the assertions of the lemma, except the last, followfrom standard facts about the representations of the algebra Wv ^ sl2 (C).Furthermore, / and fw are two non-zero vectors of weight w\ in V, andsince the multiplicity of w\ in V is equal to 1, these vectors are propor-tional. The lemma is now proved.

To prove Theorem 2.9 we introduce a partial ordering on W by puttingw -\ w if fw> G U(W)fw. Since all the weights w\ are distinct, therelation -\ is indeed an ordering; we show that it satisfies conditions a)and b) of Proposition 2.7.

a) Let a 6 2 and l(oaw) = l(w) + 1. Then w ^ oaw, and by Lemma2.10, fw e U (31) foaW, that is, w H oaw.

b) Let w -\ w . We choose an a G 2 such that w ^ oaw. Replacingw by oaw , if necessary, we may assume that oaw' -> it;'. We prove thataau; - | w\ that is, faawÛ(%l)fw>. It follows from Lemma 2.10 thatE-afw ~ 0 and fOaW = cEn_afw. Let $« be the subalgebra of @ generatedby 31, f) and5Ta. Since w H w', fw£UW)fw> and so /OaU, = ci??a/«; = XL'*where XÛ(â). Any element X of £/ ($a) can be represented in the

formx= 3\YiYi+YE-a, wheret=i

Therefore, /aau> = 2 ^^7™- = S cî/u,' 6 # (s^) /«- and Theorem 2.9 isproved.

We use Theorem 2.9 to describe the mutual disposition of theSchubert cells.

2.1J. THEOREM (Steinberg [7]_). Let w 6 W, Xw c X a Schubert cell,and Xw its closure. Then Xw> c Xw if and only if w' < w.


To prove this theorem, we give a geometric description of the variety

xw.Let V be a finite-dimensional representation of G with regular dominant

weight X (that is, all the weights w\ distinct). As above, we choose foreach w E W a non-zero vector /„, 6 F of weight w\. We consider thespace P(V) of lines in V; if / 6 K, / i= 0, then we denote by [f] £ A*0a line passing through /. Since X is regular, the stabilizer of the point[fe] £ P(V) under the natural action of G on P(V) is B. The G-orbit ofL/g] in P(V) is therefore naturally isomorphic to X = G/B. In what follows,we regard X as a subvariety of P(V).

For each u> 6 W we denote by <£„, the linear function on V given by<t>w(fw) = U 0u;CO = 0 if / 6 V is a vector of weight distinct from w\.

2.12 LEMMA. Let ft V and [f] 6 X. Then

PROOF. We may assume that f - gfe for some g € G.Let [/] C û;> that is, g 6 NwB. Then / = C! exp (Y)wfe for some

y 6 » , hence / 6 U(W)U and 0^(0 ^ 0.On the other hand, it is clear that for each / 6 V there is at most one

w 6 W such that /£#($») /w and <t>w(f) * 0. The Lemma now followsfrom the fact thatX = U Xw.

We now prove Theorerri 2.11a) Let Xw> c Xw. Then [fw>] G Xw, and by Lemma 2.12, fw. 6 ^(^)/«,.

So w < w;, by Theorem 2.9.b) To prove the converse it is sufficient to consider the case w ^ w.

Let n = w\(Hy) £,Z. Just as in the proof of Theorem 2.9, a) we can showthat n > 0, JEJ/W = c/w' and Ef1 fw = 0.

Therefore fHm r " exp (r^7) 4 =% fw', that is, [/„/] 6 ^ • Hence,

XW' C J w .

§3. Discussion of the ring of polynomials on Ij

In this section we study the rings R and R. For each w 6 W we definean element /^ 6 ^ and a functional -Dw on R and investigate their proper-ties. In the next section we shall show that the Dw correspond to Schubertcells, and that the Pw yield a basis, dual to the Schubert cell basis, for thecohomology of X.

3.1 DEFINITION, (i) R = © Rt is the graded ring of polynomialfixations on 1)Q with rational coefficients. W acts on R according to therule wf(h) = Aw* h).

(ii) / is the subring of W-invariant elements in R,


(iii) J is the ideal of R generated by 1+.(iv) R = R/J.3.2 DEFINITION. Let y £ A. We specify an operator Ay on R by the

rule

Ayf lies in R, since f — oyf = 0 on the hyperplane y = 0 in 1)Q-The simplest properties of the Ay are described in the following lemma.3.3 LEMMA.(i) A_y = -Ay, A$ = 0.(ii) wAyw~x = Awy.(iii) oyAy = - Ayoy= Ay, oy = —yAy + 1 = Ayy — 1.

(iv) Ayf = 0 o ayf = f(v) AyJ C / .(vi) Let x 6 £)Q- Then the commutator of Ay with the operator of

multiplication by X has the form [Ay, xl = x(Hy)oyPROOF, (i) — (iv) are clear. To prove (v), let f - fif2, where

/i 6 /+, / 2 € i?.. It is then clear that Ayf = fx.Ayf2 6 J As to (vi), sinceoyX = X - xCtfyh, we have

y *>The following proper ty of the Ay is fundamental in what follows.3.4 THEOREM. Let a h . . ., a, G S , and put w = a a j . . . a a j ;

a) Ifl(w) < /, then A(Oii tt|) = 0.b) / / l(w) = /, then A{Qti a / ) depends only on w and not on the

set « ! , . . . , az. In this case we put Aw = A^^ a / ) .The proof is by induction on /, the result being obvious when / - 1.For the proof of a), we may assume by the inductive hypothesis that

/(atti . . . oail) = / — l5 consequently l(oai . . . o^ô^^ = 1 — 2.Then o^ aa.+1 . . . a a w = aaH1 . . . oail oai for some / ( we have appliedCorollary 2.3 to the case w = o a H . . . atti , 7 = aj). We show thatA = 0

Since / - / < / , the inductive hypothesis shows thatAatAai+l . . . Aal_i = Aa.+i . . . A^Â^, and so by lemma 3.3 (i)

To prove b), we introduce auxiliary operators Bi0li a / ) , by setting

We put wt = oai . . . oa.. Then in view of Lemma 3.3 (ii, iii) we have


(where A% stands for wAyw'1).3.5 LEMMA. Let %et)q. The commutator of Bi0li ai)with the

operator of multiplication by x is given by the following formula:1

i

(8) [#«»., «.), xl =

PROOF. We have

By Lemma 3.3 (ii, vi), [A™*1, x l = x(wi+1Hai) ow.+lOL..

Since ow.+ia. = w^wf* , we have

r , = x (î+i^«4) <2 • • • 4 ^ 1 + ^ r 1 ^ • • Aat.

We want to move the term wi+1Wil to the left. To do this we notethat for ;' < /

Therefore,

^ ; 4 ; ; A,,By (7), applied to the sequence or roots (al5 . . ., 6th . . ., az), we have

Tt = X ( ^ I + I ^ B J ) ^ i + ^ i 1 ^ £ a j ) f

a n d L e m m a 3 . 5 is p r o v e d .I f / ( a ^ . . . a a . . . . a a / ) < / - 1, t h e n 7J = 0 b y t h e i n d u c t i v e

h y p o t h e s i s . I f / ( a t t i . . . a a . . . . a a { ) = / - 1, t h e n , p u t t i n gw' = oUi . . . a a . . . . oai a n d 7 = a t t j . . . a a . . 1 ( a f ) , w e s e e f r o m L e m m a2 . 2 t h a t M ; ' ^ u;, a n d a l s o

X (wi+iHa.) = w'x (w'wi+lHai) = W'X (âj • • • Oa^Hat) =and

A indicates that the corresponding term must be omitted.


Using Proposition 2.8 c) and the inductive hypothesis, (8) can berewritten in the following form:

[£(ai a,), Xl = 2 *>'% (Hy) if* A*.w' —> w

The right-hand side of this formula does not depend on the represen-tation of w in the form of a product oUi . . . oav The proof of theorem3.4 is thus completed by the following obvious lemma.

3.6. LEMMA. Let B be an operator in R such that B(\) = 0 and[B, X] = 0 for all X 6 %. Then B = 0.

3.7. COROLLARY. The operators Aw satisfy the following commutatorrelation:

v

We put St = Rf (where Rt c R is the space of homogeneous poly-nomials of degree /) and S =@St. We denote by ( , ) the natural pairingS X R -> Q. Then W acts naturally on S.

3.8 DEFINITION, (i) For any X6^Q we let x* denote the transform-ation of S adjoint to the operator of multiplication by \ in jR-

(ii) We denote by Fy: S -• S the linear transformation adjoint toAy: R -* R.

The next lemma gives an explicit description of the Fy.3.9 LEMMA. Let 7 6 A. For any D 6 S there is a D £ S such that

7* 0) = D. If D is any such operator, then D - oyD = Fy(D), (in par-ticular, the left-hand side of this equation does not depend on the choiceof S).

PROOF. The existence of D follows from the fact that multiplication by7 is a monomorphism of R. Furthermore, for any / 6 R we have

(D-OyD, /) = (S, /-crY/) = (5, V-Y) = (Y'(5), ^v/) = (A ^v/),hence D - oyD = Fy.

REMARK. It is often convenient to interpret S as a ring of differ-ential operators on f) with constant rational coefficients. Then the pairing( , ) is given by the formula (Z), /) = (Df) (0), D £S, f £R. Also, it is easyto check that x*(^) = W, xL where X 6 ^Q and D 6 S are regarded asoperators on R.

Theorem 3.4 and Corollary 3.7 can be restated in terms of the operatorsFy

3.10 THEOREM. Let au . . ., a, 6 2 , w = ao . . . aa.(i) / / /(w) < /, then Fai . . . Fai = 0.(ii) / / l(w) = I, then Fttl . . . Fai depends only on w and not on

a,, . . ., a,. In this case the transformation Fa; . . . Fa_ ,-5 denoted byFw. {Note that Fw = A*).

128 /. N. Bernstein, L M. Gel'fand, S. I. GeVfand

(iii) l%*,Fww]= S w'x(Hy)Fw>w.v

w' —> w

3.11. DEFINITION. We set Dw = Fw (1).As we shall show in §4, the functionals Dw correspond to the Schubert

cells in H+ (X, Q) in the sense that (Dw,f) = <sw, <*(/)> for all f € R.The properties of the Dw are listed in the following theorem.3.12. THEOREM, (i) Dw 6 SUw).(ii) Let we W, a 6 2 . Then

[0 if l(woa) = l(w) - 1,

(iii) Let%e QQ- r/ze«x* (A*) = S i

Vw'—> w

(iv) Let a 6 2 . Tfterc

[ - A , , ifâîy = < —Dw+ S ^ ' a (^v) ^ ' if Z (w;aa) = Z (w) + 1.

(v) £er M; g W, I (w) = Z, %i> . . ., %z G^Q. T/z^n

(A* > Xi5 • • •, Xi) = 2 Xi(^7i) . . . xi(Hi), where the summation extendsover all chains

PROOF, (i) and (ii) follow from the definition of Dw and Theorem3.10 (i).

(iii) X* ( A J = X*Fww(\) = [x*, Fww] (1) (since x* (1) = 0), and (iii)follows from Theorem 3.10 (iii).

It follows from Lemma 3.3 (iii) that oa = a*Fa - 1. Thus, (iv) followsfrom (ii) and (iii).

(v) We put Dw = Dw -i . Then the 5^ satisfy the relation

(9) X*(A.)= Sv

xu' —>Since (£>, x/) = (x*(^)» /)> (v) is a consequence of (9) by induction on /.

Let SB be the subspace of S orthogonal to the ideal / c R. It followsfrom Lemma 3.3 (vi) thatch is invariant with respect to all the Fym It isalso clear that 1 6 SS. Thus, Dw 6 SS for all w 6 AT.

3.13. THEOREM. 77ze functionals Dw, w £ W, form a basis for SB.PROOF, a) We first prove that the Dw are linearly independent. Let

s 6 W be the element of maximal length and r = l(s). Then, by Theorem3.12 (v), Ds(p

r) > 0 and so Ds * 0. Now let 2 cwDw = 0 and let w beone of the elements of maximal length for which cw =£ 0. Put / = l(w).


There is a sequence c , . . ., ar.j f° r which woai . . . a a r / = s. LetF = Far.t • • • Fal • ^ follows from Theorem 3.10 that F£>~ = Ds and

FZ^ = 0 if /(it;) > /, w # w. Therefore F( $>„,/)„,) = c~D, * 0.b) We now show that the Dw span c$£. It is sufficient to prove that if

/ € /* and (/>„,,/) = 0 for all w £ W, then / 6 / . We may assume that /is a homogeneous element of degree k. For & = 0 the assertion is clear.

Now let k > 0 and assume that the result is true for all polynomials /of degree less than k. Then for all a 6 2 and w 6 W,(Dw, Aaf) = (FaDw, f) = 0, by Theorem 3.10 (i) and (ii). By the inductivehypothesis, Aaf 6 J, that is, / — oaf = aAaf 6 / . Hence for all w 6 W,f=wf (mod / ) . Thus, | W I"1 2 M?/ = / (mod / ) . Since the left-hand side

belongs to /+, we see that / 6 J. Theorem 3.13 is now proved.The form ( , ) gives rise to a non-degenerate pairing between R = R/J

and(§#. Let {Pw} be the basis of R dual to {Dw}. The following propertiesof the Pw are immediate consequences of Theorem 3.12.

3.14 THEOREM, (i) Let w 6 W, a 6 2 . Then0 if l(waa) = /(M;) + 1,Fwaa if l(W0a) = 1(W) - 1.

_ (

(ii)XPa,= 2J wx{Hy)Pw> for

(iii) Let a 6 2.

Pw if

Pw- S wa{Uy)Pw> if

From (i) it is clear that all the Pw can be expressed in terms of thePs. More precisely, let w = oai . . . ooip l(w) - r — I. Then

To find an explicit form for the Pw it therefore suffices to compute thePs 6 R.

3.15 THEOREM. P s = | W I"1 f] Y (mod/).

PROOF. We divide the proof into a number of steps. We fix an elementh £ f) such that all the wh, w (: W, are distinct.

1. We first prove that there is a polynomial Q (: R of degree r suchthat

(10) Q(sh) = 1, £>(">&) - 0 for M; ^ 5.

For each u; G IV we choose in R a homogeneous polynomial Pw of degree


l(w) whose image in R = R/J is Pw. Since {Pw} is a basis of R, any polynomialfe R can be written in the form / = 2 Pwfw. where j£,e/ (this is easily provedby induction on the degree of/) . Now let Q' 6 R be an arbitrary poly-nomial satisfying (10) and let Q' = 2 ? ^ w , gw£I* It is clear that<? = 2 io ( ) û; meets our requirements.

2. Let Q be the image of Q in R, and let Q = 2 rr^P^ be the represen-

tation of Q in terms of the basis {Pw} of R. We now prove that

cs = (—1)' U (Y(A))-».Y6A+ _ _

To prove this we consider ASQ. On the one hand ASQ = c8, by Theorem3.13 (i); on the other hand, A$Q is a constant, since 2 is a polynomial ofdegree r. Hence, ASQ = cs.

We now calculate ASQ, Let s = oai . . . oar be the reduced decompo-sition. We put wt = oa. . . . oa (in particular, w0 = e), yt = wfl a x ,Q, = Aa.+i . . . AarQ. '

LEMMA. Qt is a polynomial of degree i,

and Qi(wh) = 0 if w p wt.PROOF. We prove the lemma by backward induction on /. For / = r

we have wr = s, Qr = Q, and the assertion of the lemma follows from thedefinition of Q.

We now assume the lemma proved for Qh i > 0. In the first place, itis clear that (2,-_, = AaiQt is a polynomial of degree i — 1.

Furthermore,Qi (wh) — Qt (a„ .wh)

Cu, (wh) = AaiQi (wh) = a ^ ' ^ •

If w - wiA , then w < wt, oa.w = wt a n ^

OLi(u>i-ih) = (wf.\at) (h) = - (wr\oLi) (h) = - yt(h). Therefore, using theinductive hypothesis, we have

But if u; ^ M;,-.! , Corollary 2.6 implies that u; ^ w;,- and aa.u; ^ wt. So(2i_! (w;^) = 0, and the lemma is proved.

Note that by Lemma 2.2, as / goes from 1 to r, yt ranges over all thepositive roots exactly once. Therefore

( 0YGA+

3. Consider the polynomial Alt (Q) = 2 ( - \)Kw)wQ; Alt (Q) is skew-

symmetric, that is, oa Alt(Q)= -Alt(Q) for all y G A.Therefore Alt(Q) is divisible


(in R) by [( 7. Since the degrees of Alt(2) and [I 7 are equal (to r),V£A+ V£A+

Alt(g) = X [I y- Furthermore, A\t(Q) (h) = ( - l)r , so thatYGA+

4. We put Alt(fi) = 2 ( - l)/(u;) ^ 5 - By Theorem 3.14 (iii),Alt(P.) = S ( - \)liw) wPs = \W\ P.. Therefore Alt(G) = cs IH/I Ps + termsof smaller degree. Since Alt(C) is a homogeneous polynomial of degree r,we have

(12) Alt (Q) = cs\W\Ps.

By comparing (11) and (12) we find that

P* = \W\-* 0 v(niod^).VGA+

The theorem is now proved.3.16 COROLLARY. Let p be half the sum of the positive roots. Then

P$ = pr/r\ (mod / ) .PROOF. For each % £ I)* we consider the formal power series exp x on

lj given by

Then we have (see [ 9 ] )

2(-l)'<w>exp(i£;p)= J[[exp| -exp(- | ) ] .

Comparing the terms of degree r we see that

VGA+

If pr(mod J) = \PS, X e C, then (wpr) (mod / ) = \wPs = X(- l)Z(u;)Ps.

Thus, S. ^ ( — 1)/(U3) (^p)r == XPS (mod/). The result now follows from

Theorem 3.15.To conclude this section we prove some results on products of the

Pw in R.3.17. THEOREM, (i) Let a 6 2 , M; 6 TF. r/iew

PoaPw= 2 %a(Hw-iy)PW>,w —> w'

where Xa 6 z is r/ze fundamental dominant weight corresponding to theroot a (that is, Xa (fy) = 0 for a ± p 6 2, x<* (^4) = 0-

(ii) Let wl9 w2 6 W, Kwx) + /(w2) = r. Then P^ PW2 = 0 for


W2 * WXS9 PWlPWl.= Psj_ v

(iii) Let w £ W, f £ R . Then f Pw = j£wCw'Pw>.

(iv) / / wt ^ w2sy then PWlPW2= 0.PROOF, (i) By Theorem 3.12 (v), POQI = Xa (mod / ) . Therefore (i)

follows from Theorem 3.14 (ii).(ii) The proof goes by backward induction on l(w2). If l(w2) = r, then

w2 = s, wx = e and PWI = 1.To deal with the general case we find the following simple lemma useful,

which is an easy consequence of the definition of the Ay.3.18 LEMMA. Let 7 £ A, / , g £ R. Then Ay(Ayf-g) = Ayf-Ayg.Thus, let w2 £ W9 l(w2) = I < r, and choose a 6 2 so that

^2 ~* 0 ^ 2 - We consider two cases separately.

A) Wi ^ oaWi. We observe that the following equation holds for anyw e• w

(13) Z(i 5) = r — l(w).

Since in our case l(oaw2) = / + 1 and l(oawi) - r — I -\- 1, we see thatoawis i= oaw2, and so wxs ¥= w2. On the other hand, PWi = AaPOaw2 and^ ! = AaPoî by Theorem 3.14 (i). Therefore, an application of Lemma3.18 shows that

Since Ko^Wi) + l(w2) = / • - / + 1 + / > r, we have P a ^ i ^ = °-Hence PWlPW2 = 0 as well.

B) OaWi ^ Wi. In this case, P0aU; i = ^Pu ; ! an<* Pw2 = AaPoaWl, by

Theorem 3.14 (i). Again applying Lemma 3.18, we have

Since the Pw form a basis of /?, any element / of degree r in R has theform / = XPS, X 6 C. Furthermore, ^ a P s = i>aaS =£ 0. But

deg PWIPW2 = deg ^ u ; , - âaiy2 = r- Therefore (14) is equivalent to

Applying the inductive hypothesis to the pair (oaWi, oaw2), we obtainpart (ii) of the theorem.

(iii) is an immediate consequence of Theorem 3.14 (ii).(iv) follows from (ii) and (iii).We define the operator <fT>: R-+ Sf of Poincare duality by the formula

(&f)(g) = Ds(fg), /, g £5", &f 6 $£.3.19. COROLLARY. ®PW = Dws.


§4. Schubert cells

We prove in this section that the functional Dw, w 6 W introduced in§ 3 correspond to Schubert cells sw, w 6 W.

Let sw 6 H*(X, Q) be a Schubert cell. It gives rise to a linear functionalon H*(X, Q), which, by means of the homomorphism a: R -* H*(X, Q)(see Theorem 1.3), can be regarded as a linear functional on R. Thisfunctional takes the value 0 on all homogeneous components Pk withk J= l(w), and thus determines an element Dw £ SKw).

4.1. THEOREM. Dw = Dw (cf. Definition 3.11).This theorem is a natural consequence of the next two propositions.PROPOSITION '[.be = 1, and for any X€*)z

(15)V

—> w

PROPOSITION 2. Suppose that for each w 6 W we are given anelement Dw 6 Sl(w), with De = 1, for which (15) holds for any% 6 fyz •Then Dw = Dw.

Proposition 2 follows at once from Theorem 3.12 (iii) by inductionon l(w).

We turn now to the proof of Proposition 1.We recall (see [10]) that for any topological space Y there is a bilinear

mapping

Hl(Y, Q ) x / / ; - ( r , Q ) - ^ IIM (F, Q)

(the cap-product). It satisfies the condition:

(16) 1. (cny.z) = (yiC>z)

for all y£Hs(Y, Q), z£H]-l(Y, Q), cf,IP (Y, Q).2. Let / : Yx - F2 be a continuous mapping.. Then

(17) U(f*cny) -cnUy

for all y 6 Hj (7i, Q). ^ 6 //* (V2 Q)-By virtue of (17) we have for anyx6nz , 1£R

(X* 0w), f) = 0 W , if) = (^ , a! (X) a (/)) - ( ^ n a4 (x), a (/)).

Therefore (15) is equivalent to the following geometrical fact.PROPOSITION 3. For all x€fyz

(18) * a ; n a 1 ( x ) = I wf%(Hy)sw,

We restrict the fibering Ex to Xw C X and let cx 6 H2 (Xw, Q) be thefirst Chern class of Ex. By (17) and the definition of the homomorphismal:

:bfc->H2(X, Q), it is sufficient to prove that

134 /• N- Bernstein, I. M. Gel'fand, S. I. Gel'fand

(19) swncx= 2V

in H2liwy-2(XW, Q).

To prove (19), we use the following simple lemma, which can be verifiedby standard arguments involving relative Poincar6 duality.

4.2 LEMMA. Let Y be a compact complex analytic space of dimensionn, such that the codimension of the space of singularities of Y is greaterthan 1. Let E be an analytic linear fibering on Y, and c 6 fP(Y, Q) thefirst Chern class of E. Let M be a non-zero analytic section of E and^mtYi = div n the divisor of n. Then [Y] n c = S ^ t ^ l £ H2n-2(Y, Q),where [ Y] and [ Yi are the fundamental classes of Y and Y(.

Let w 6 W, and let Xw c X be the corresponding Schubert cell. FromLemma 4.2 and Theorem 2.11 it is clear that to prove Proposition 3 it issufficient to verify the following facts.

4.3. PROPOSITION. Let w -> w. Then Xw is non-singular at pointsx € Xw>.

4.4. PROPOSITION. There 4s a section ix of the fibering Ex over Xw

such that

vw' —>

w'%(Hy)Xw

To verify these facts we use the geometrical description of Schubertcells given in 2.9. We consider a finite-dimensional representation of G ona space V with regular dominant weight X, and we realize X as a subvarietyof F(V). For each w 6 IV we fix a vector /„, 6 V of weight wX.

PROOF OF PROPOSITION 4.3. For a root y 6 A+ we construct athree-dimensional subalgebra WY cz @ (as in the proof of Theorem 2.9).Let i: SL2(C) -> G be the homomorphism corresponding to the embedding

K a b \1 (/a 0

H ;r = ^L0 a LJ) i\[) a 1

f/1 OM / 0 1\and NL = \[ . I \ a nd the element a = I . ^ 1. We may assume thatc H, i(B') c B.

Let F be the smallest 2Iv-invariant subspace of V containing fw>. It isclear that V is invariant under i(SL2 (C)), and that the stabilizer of the line[fw>] is B'. This determines a mapping 5: SLaiQlB'- -> ^ . The spaceSL2(C)/B' is naturally identified with the projective line P1. Let o, °° 6 P1

be the images of e, o 6 iSTZ/2(C).We define a mapping J: Nw> X P1 -» I by the rule

(a:, z) >—• o:-6(z).4.5. LEMMA. The mapping J /za5 r/ze following properties:(i) EW,' X {o}) = Xw.f E(^w. X (P1 \ o)) c: Xw.


(ii) The restriction of £ to (Nw> x P1 \ °°)) is an isomorphism onto a

certain open subset of X^.Proposition 4.3 clearly follows from this lemma.

PROOF OF LEMMA 4.5. The first assertion of (i) follows at once fromthe definition of Xw>. Since the cell Xw is invariant under N, the proof ofthe second assertion of (i) is reduced to showing that 5(z) '€ Xw forz € P1 \ o. Let h 6 SL2(C) be an inverse image of z. Then h can bewritten in the form h = bxob2, where br, b2 6 B'. It is clear thati{bi)fw' = dfw' and i(o)fw> = c2fw , where cx, c2 are constants. ThereforeKh)fw> = c1c2i(bl)fw, that is, 6(z) 6 Xw.

To prove (ii), we consider the mappingw'-iol-.Nv. X ( P x \ 00) -+X.

The space P ^ 0 0 is naturally isomorphic to the one-parameter subgroupNL c SL2(Q.

The mapping f: A^/X AL -• X is given by the rulel(n, rii) = ni(ni) [fw>], n g iV^, wt 6 N\

Thus,i^'"1 ° I (w, wO = ( M ; ' " 1 ^ ' ) (I//"1* (w,) u;') [/,].

We now observe that w'~lNw*w c AL (by definition of A^'). andw~li(N')w 6 AL (since u;'"1 7 6 A+). Furthermore, the intersection of thetangent spaces to these subgroups consists only of 0, because Nw> C N,i(N'_) C 7V_. The mapping AL -> X {n h> n[ / e ] ) is an isomorphism onto anopen subset of X. Therefore (ii) follows from the next simple lemma,which is proved in [5 ] , for example.

4.6. LEMMA. Let Nx and N2 be two closed algebraic subgroups of aunipotent group N whose tangent spaces at the unit element intersect onlyin 0. Then the product mapping N{ X N2 -* N gives an isomorphism ofN1 X N2 with a closed subvariety of N.

This completes the proof of Proposition 4.3.PROOF OF PROPOSITION 4.4. Any element of \)£ has the form

X = X - X', where X, X' are regular dominant weights. In this case,E% — Ex <g) Erf, and it is therefore sufficient to find a section /i with therequired properties in the case x = X.

We consider the space P(V), where V is a representation of G withdominant weight X. Let 7?vbe the linear fibering on P(V) consisting ofpairs (P, 0), where 0 is a linear functional on the line P C V. ThenE\ ~ i*(vv), where /: X -> P(V) is the embedding described in §2.

The linear functional <pw on V (see the proof of Theorem 2.11) yields asection of the bundle 7?. We shall prove that the restriction of ju to thissection on Xw is a section of the fibering E^ having the requisite properties.

By Lemma 2.12, n(x) =£ 0 for all x 6 Xw. The support of the divisordiv id is therefore contained in XW\XW = U Xw>.

136 /• N. Bernstein, I. M. Gel'fand, S. I. Gel'fard

Since Xv is an irreducible variety, we see that div ju = 2J ayXw', wherew —*• w

ay 6 Z, ay > 0. It remains to show that ay = w'x(Hy).In view of Lemma 4.5 (i) and (ii), the coefficient ay is equal to the

multiplicity of zero of-the section 5*Gu) of the fibering S*(EX) on P1 atthe point o, that is, the multiplicity of zero of the function*K0 = 0u/((exp tE_y)fw') for t = 0. It follows from Lemma 2.10 that\l/(t) = ctn, hence ay = n = w'\(Hy). This completes the proof of Proposi-tion 4.4 and with it of Theorem 4.1.

§5. Generalizations and supplements

1. Degenerate flag varieties. We extend the results of the previous sec-tions to spaces X(P) = G/P, where P is an arbitrary parabolic subgroup ofG. For this purpose we recall some facts about the structure of parabolicsubgroups P c G (see [7]).

Let 0 be some subset of 2, and A0 the subset of A+ consisting oflinear combinations of elements of 0. Let G0 be the subgroup of Ggenerated by H together with the subgroups Ny = {exp tEy\ I £ C} for7 6 A0 u - A0, and let NQ be the subgroup of TV generated by the Ny

for 7 £ A+\A0. Then G0 is a reductive group normalizing NQ, andP& = GQN& is a parabolic subgroup of G containing B.

It is well known (see [7], for example) that every parabolic subgroupP c G is conjugate in G to one of the subgroups PQ. We assume in whatfollows that P = P@, where 0 is a fixed subset of 2. Let WQ be the Weylgroup of G0. It is the subgroup of W generated by the reflections oa,a 6 0.

We describe the decomposition of X{P) into orbits under the action of B.5.1. PROPOSITION, (i) X(P) = JJW Bwo, where o 6 X(P) is the image

of P in G/P(ii) The orbits Bwxo and Bw2o are identical if WiW2

l 6 W© and other-wise are disjoint.

(iii) Let WQ be the set of w 6 W such that w® C A+. Then each cosetof W/W@ contains exactly one element of W@. Furthermore, the elementw 6 WQ is characterized by the fact that its length is less than that ofany other element in the coset wWQ.

(iv) / / w E W®, then the mapping Nw ->• X(P) (n -» nwo) is an iso-morphism of Nw with the sub variety Bwo c X(P).

PROOF, (i)—(ii) follow easily from the Bruhat decomposition for G andG0. The proof of (iii) can be found in [7], for example, and (iv) followsat once from (iii) and Proposition 1.1.

Let w 6 W0, XW(P) = Bwo, let XW(P) be the closure of XW(P) andIXW(P)] 6 H2liw)(Xw(P), Z) its fundamental class. Letsw(P) € H2liw)(X(P), Z) be the image of [XW(P)] under the mapping

Schubert cells and cohomology of the spaces GfP 137

induced by the embedding Xw (P)C-+X(P). The next proposition is ananalogue of Proposition 1.2.

5.2. PROPOSITION ([2]) . The elements sw(P), w 6 W&, form a freebasis in H*(X(P), Z).

5.3. COROLLARY. Let ocP: X -+ X(P) be the natural mapping. Then(«p)*sw = 0 if w £ W&, (ctp)^ = sw(P) if w e W&.

5.4. COROLLARY. (aP)*: H^X, Z) -* H*(X(P), Z) w fl« epimorphism,

and (aP)*: H*(X(P), Z) -» / /* (jr, Z) w a monomorphism.

5.5. THEOREM, (i) Im(aP)* c i/*(X, Z) = R coincides with the set ofWQ-invariant elements of R.

(ii) Pw e Im(aP)* /or u, € W& and {(ccp)*-1?^^ is the basis inH*(X(P), Z) dual to the basis {sw(P)}w£Wi@ in H*(X(P), Z).

PROOF. Let w 6 R/&. Since (Pw, ^ > = 0 for Wi £ H£, />„, is orthog-onal to Ker(aP)^, that is, Pw 6 Im(aP)*. Now (ii) follows from the factthat <(aP)*PWi sw'(P)) = <PW, v > for w, w 6 W&. To prove (i), it issufficient to verify that the Pw, w 6 W©, form a basis for the space ofW& -invariant elements of R. We observe that an element / 6 R isW0 -invariant if and only if ^4 a / = 0 for all a 6 0 . Since w 6 W® if andonly if l(woa) = l(w) + 1 for all a £ 0 , (i) follows from Theorem 3.14(i).

2. CORRESPONDENCES. Let Y be a non-singular oriented manifold.An arbitrary element z 6 //* (Y X y, Z) is called a correspondence on y.Any such element z gives rise to an operator z+: H*(Y, Z) -• H%(Y, Z),according to

n 2), c G ^ ( y , Z ) ,

where 7r X , TT2 : ^ X Y -+ Y are the projections onto the first and secondcomponents, and cP is the Poincare duality operator. We also define anoperator z*: H*(Y, Z) -> H*{Y, Z) by setting z*&) - ^[(Jti)*((jx2)*(6) fl 2)g g / f* (y , Z). It is clear that z* and z* are adjoint operators.

Let z be assigned to a (possibly singular) submanifold Z c y X y, insuch a way that z is the image of the fundamental cycle [Z] 6 H*(Z, Z)under the mapping induced by the embedding Z c__^ y X y. Then

z*(c) = (p2)* (IZ] fl (P i )*^ ) ,where P i , p2 • Z -> y are the restrictions of ni , n2 to Z.

If, in this situation, pl\ Z -• y is a fibering and c is given by a sub-manifold C C y, then the cycle

[zi n ( P I ) * ^

is given by the submanifold Pi1 (C) c Z.We want to study correspondences in the case Y = X = G/B.5.6. DEFINITION. Let w £ W. We put Zw = {(gwo, go)} C X X X

and denote by zw the correspondence zw = [Zw ] C H*(X X X, Z).5.7. THEOREM. (zw)* = Fw .PROOF. We calculate (zw)# ( v ) .


Since the variety Zw is G-invariant and G acts transitively on X, themapping px: Zw -* X is a fibering. Thus,

It is easily verified that p\x (Xw>) = n? (Xw>) n Z ^ . W e putY = TTI1 (*„,') n Zw c JT X X. Then

(20) Y = {(»M?'O, rcu/fcwo) | rc £ JV, 6 6 #}.

Since the dimension of the fibre of px : Zw -+ X is equal to 2l(w), we seethat dim Y = 2l(w) + 2/(u/). It is clear from (20) that

9i(Y) = {nw'bwo \n £N, b £ B} = Bw'Bwo.

It is well known (see [6] . Ch. IV, §2.1 Lemma 1) that

Bw'Bwo = Bw'wo U ( (J Bwto).I (wi)<l(w)+l(w')

Thus, two cases can arise.a) l(w'w) < l(w) + l(w). In this case, dim p2(7) < 2/(u/) + 2l(w), and

so (zw)#(v) = (Pa)*m = 0.b) /(u;fu;) = /(a;') + /(u;). In this case, p2(Y) = Xw>w + X\ where

dim Z' < dim Xw>w = 2/(u;') + 2l(w). Thus, (p 2 ) jy ] = [Zw>w], that is,(zu;)*(5u;') = sw'w • Comparing the formulae obtained with 3.12 (ii), we seethat (zw )* = Fw .

5.8. COROLLARY. zu; = 25u;/s 0 5u;'u;> where the summation extendsover those w 6 W for which l(w'w) = l(w) + /(«;').

In § 1 we have defined an action of W on H#(X, Z). This definitiondepended on the choice of a compact subgroup K. Using Theorem 5.7 wecan find explicitly the correspondences giving this action.

In fact, it follows from Lemma 3.3 (iii) that oa = a*Fa - 1 for anya E 2. The transformation Fa is given by the correspondence ZOQL. Theoperator a* can also be given by a correspondence: if Ua = 2^(7, is adivisor in X giving the cycle < (a) 6 H2r-2(Xi Z) (for example,J7a = 2 a(/^)Xafl), then the cycle £/a = 2c,6^, where

f/f = {(x, x) \x£Ut}cz X X X, determines the correspondence that givesthe operator a*. The operator oa in H*(X, Z) is therefore given by thecorrespondence Ua*ZOa - 1 (where * denotes the product of correspon-dences, as in [11]). Using the geometrical realization of the product ofcorrespondences (see [11]), we can explicitly determine the correspondenceSa that gives the transformation 1 + oa in H*(X, Z), namely, Sa = 2qUiwhere Ut = {(z, y) 6 X X X \ x 6 Uh ~x~ry 6 P{a}} • I n t h i s expression,%, y E G are arbitrary representatives of x, y, and P{ay is the parabolicsubgroup corresponding to the root a.

3. B. Kostant has described the Pw in another way. We state his result.


Let h e % be an element such that <x(h) > 0 for all a e X. LetJh ={fe R \ f{wh) = 0 for all w e W} be an ideal of R.

5.9. THEOREM, (i) Let w e W, l(w) = I. There is a polynomial Qw G Rof degree I such that

(21) Qw(wh) = 1, Qw(w'h) = 0 if l(w')^l(w), w'^w.The Qw are uniquely determined by (21) to within elements of Jh. (ii) LetQ% be the form of highest degree in the polynomial Qw. The image of Q%

in R is equal to II

The proof is analogous to that of Theorem 3.15.4. We choose a maximal compact subgroup K cG such that K n B c H

(see § 1). The cohomology of X can be described by means of the^-invariant closed differential forms on X. For let x e §z» and let Ex be thecorresponding one-dimensional complex G-fibering on X. Let £5X be the2-form on X which is the curvature form of the connection associated withthe ^-invariant metric on Ex (see [12]). Then the class of the form

cox — cox is cx G IP(X, Z). The mapping x "* <>x ex*ends to a mapping

6: R -* £2*y(X), where £2*,, is the space of differential forms of even degreeon X. One can prove the following theorem, which is a refinement ofProposition 1.3 (ii) and Theorem 3.17.

5.10. THEOREM (i) Ker 6 = J, that is, d induces a homomorphism ofrings 0: R -* Sl*v(X). (ii) Let wx, w2 e PV, ^ 4 u;2s- 77*e« r/ze restrictionof the form W(PWj ) to XWi is equal to 0. (iii) Let wx, w2 G W, Wi 4 w2s.Then B{PWx) 0 ( P J 3 ) = 0. *

References

[1] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenesdes groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207;MR 14 #490.Translation: in 'Rassloennye prostranstva\ Inost. lit., Moscow 1958

[2] A. Borel, Kahlerian coset spaces of semisimple Lie groups, Proc. Nat. Acad, Sci.U.S.A. 40 (1954), 1147-1151; MR 17 # 1108.

[3] B. Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2)77 (1963), 72-144; MR 26 # 266.

[4] G. Horrocks, On the relations of S-functions to Schubert varieties, Proc. LondonMath. Soc. (3) 7 (1957), 265-280; MR 19 #459.

[5] A. Borel, Linear algebraic groups, Benjamin, New York, 1969; MR #4273.Translation: Lineinye algebraicheskie gruppy, 'Mir', Moscow 1972.

[6] N. Bourbaki, Groupes et algebras de Lie, Ch. 1—6, Elements de mathematique, 26, 34,36, Hermann & Cie, Paris, 1960-72.Translation: Gruppyialgebry Li, 'Mir', Moscow 1972.

[7] R. Steinberg, Lectures on Chevalley groups, Yale University Press, New Haven,Conn. 1967.

[8] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc.

140 /. N. Bernstein, I. M. Gel'fand, S. I. Gel"fand

Sympos. Pure Math.; Vol. Ill, 7-38, Amer. Math. Soc, Providence, R. I., 1961;MR 25 #2617.= Matematika 6: 2 (1962), 3-39.

[9] P. Cartier, On H. Weyl's character formula, Bull. Amer. Math. Soc. 67 (1961),228-230; MR 26 #3828.= Matematika 6: 5 (1962), 139-141.

[10] E. H. Spanier, Algebraic topology, McGraw-Hill, New York 1966; MR 35 # 1007Translation: Algebraicheskaya topologiya, 'Mir', Moscow 1971.

[11] Yu.I. Manin, Correspondences, motives and monoidal transformations, Matem.Sb. 77 (1968), 475-507; MR 41 # 3482= Math. USSR-Sb. 6 (1968), 439-470.

[12] S. S. Chern, Complex manifolds, Instituto de Fisca e Matematica, Recife 1959;MR 22 #1920.Translation: Kompleksnye mnogoobraziya, Inost. Lit., Moscow 1961.

[13] I. N. Bernstein, I. M. Gel'fand and S. I. Gel'fand, Schubert cells and the cohomologyof flag spaces, Funkts. analiz 7: 1 (1973), 64-65.

Received by the Editors 13 March 1973

Translated by D. Johnson

FOUR PAPERS ON PROBLEMSIN LINEAR ALGEBRA

Claus Michael Ringel

This volume contains four papers on problems in linear algebra. They formpart of a general investigation which was started with the famous paper [Q] onthe four subspace problem. The r subspace problem asks for the determinationof the possible positions of r subspaces in a vector space, or, equivalently, ofthe indecomposable representations of the following oriented graph

(*)

with r + 1 vertices. For r > 5, this problem seems to be rather hard to attack,however one may try to obtain at least partial results dealing with special kindsof representations. Also, the r subspace problem can be used as a test problemfor more elaborate problems in linear algebra. This seems to be the case forsome of the investigations published in this volume, they have been generalizedrecently to the case of arbitrary oriented graphs [M, S].

Three of the four papers deal with the r subspace problem. (We shouldremark that there is a rather large overlap of [F] and [I, II]. However, themain argument of [F], the proof given in section 7, is not repeated in [I, II],whereas [I, II] give the details for the complete irreducibility of the repres-entations pt j which only was announced in [F]. We also recommend thesurvey given by Dlab [8].) Given r subspaces Ex, . . .,Er of a finite-dimensionalvector space V, we obtain a lattice homomorphism p from the free modularlattice Dr with r generators ex, . . ., er into the lattice L(V) of all subspaces ofV given by p(ef) = Et. Such a lattice homomorphism is called a representationofDr. In [F], Gelfand and Ponomarev introduce a set of indecomposablerepresentations ptj with 0 < t < r and / E N , which we will call the prepro-jective representations (in [F], the representations pt j with 1 < t <r are calledrepresentations of the first kind, those of the form po x representations of thesecond kind; in [I, II] there may arise some confusion: pt t is denoted byp]j, whereas the symbol ptl used in [I, II] stands for the same type of repres-entation but with a shift of the indices, see Proposition 8.2 in [II]). For theconstruction of the preprojective representations, we refer to section 1.4 of

141

142 Claus Michael Ringel

[F]: one first defines a finite set At(r, I) (which later we will identify with aset of paths in some oriented graph), considers the vector space with basis theset At(r, /), and also a subspace Zt(r, I) generated by certain sums of the canon-ical base elements of At(r, I). The residue classes of the canonical base elementsof At(r, /) in Vtl = At(r, l)/Zt(r, /) will be denoted by £a (with ocGAt(r, /)).Now, the representation pt z is given by the vector space Vtj together with acertain r tuple of subspaces of Vt /, all being generated by some of the gener-ators £a. Note that this implies that ptl is defined over the prime field k0 ofk. (Gelfand and Ponomarev usually assume that the characteristic oik is zero,thus k0 = Q. However, all results and proofs remain valid in general.)

The main result concerning these representations pt / asserts that in casedim Vt i > 2, the representation pt / is completely irreducible. This means thatthe image of Dr under the lattice homomorphism ptl\ Dr -* L(Vtj) is the setof all subspaces of Vt / defined over the prime field k0, thus pt j(D

r) is a pro-jective geometry over k0. The first essential step in the proof of this result isto show that the subspaces k%a are of the form p(ea) for some ea GDr. (In[F], this is only announced, but it is an immediate consequence of theorem8.1 in [II].)

The second step is to show that any subspace of Vtj which is defined overthe prime field, lies in the lattice of subspaces generated by the k%a provideddim Vtl>2. Combining both assertions, we conclude that ptj is completelyirreducible unless dim Vt j < 2. The proof of the second step occupies section9 of [II]. Here, one considers the following situation: there is given a setR = {£a | a } of non-zero vectors of a vector space V (= Vtj), with the follow-ing properties:

(1) R generates V(2) R is indecomposable (there is no proper direct decomposition

V= V © V" withR = (R H V') U(RD V")), and(3) R is defined over the prime field (there exists a basis of V such that any

£a 6 R is a linear combination of the base vectors with coefficients in the primefield fr0).Then it is shown that the lattice of subspaces of V generated by the one-dimensional subspaces k%a, is isomorphic to the lattice of subspaces of frg > withn = dim V.

Perhaps we should add that the representations p: Dr -* L(V) with V beinggenerated by the one-dimensional subspaces of the form p(a), aGDr, seem tobe of special interest. In this case, the one-dimensional subspaces of the formp(a), a^Dr determine completely p(Dr). (Namely, let b €Dr, and U the sub-space generated by all one-dimensional subspaces of the form p(x), xsatisfying p(x) C p(b), and choose xx, . . ., xs such that

p(b) C£/ep(jc1)0...ep(jcJ) = tf©p(2; x/).Thus,p(Z?)=C/©(p(i: xt)/ = 1 i = 1

Assume, U is a proper subspace of p(b). Then there exists t < s with

Four papers on problems in linear algebra 143

t- 1 tp( 2 xA n p(b) = 0, whereas p( 2 *.•) n p(Z>) is non-zero, and therefore one-

/ = l i = l

dimensional. This however implies tha t p ( 2 J C ) PI p (^ ) = p(b 2 xz-) isi-\ / = l

contained in t/, a contradiction. Thus p(b) = t/.). For r > 4, there always areindecomposable representations which do not have this property.

In the case r = 4, we may give the complete list of all lattices of the formp(D4), where p is an indecomposable representation. Besides the projectivegeometries over any prime field, and of arbitrary finite dimension =£ 1, and the

lattice , we obtain all the lattices S{n, 4) introduced by Day,

Herrmann and Wille in [6]. Let us just copy S(14, 4) and note that anyinterval [cn, cm ] is again of the form S(n - m, 4).

(In fact, in case either p: D4 -* L(V) or its dual is preprojective and dim V> 2,we have seen above that p(D4) is the full projective geometry over the primefield. If neither p nor its dual is preprojective, p is said to be regular. If p is


regular and non-homogeneous, say of regular length n (see [9]), thenp(D4) « S(n, 4), whereas for p homogeneous, we have

Gelfand and Ponomarev use the representations ptj ofDr in order to getsome insight into the structure of Dr. The existence of a free modular latticewith a given set of generators is easily established, however the mere existenceresult does not say anything about the internal structure ofDr. In fact, it hasbeen shown by Freese [14] that for r > 5, the word problem in Dr isunsolvable. The free modular lattice/)3 in 3 generators elf e2, e3 was firstdescribed by Dedekind [7], it looks as follows:

We have shaded two parts of D3, both being Boolean lattices with 23 elements.For r > 4, Gelfand and Ponomarev have constructed two countable families ofBoolean sublattices B+(l) and B~(l) with 2r elements, where / G N, and suchthat

and

. . . B\l . . . <B\2)

called the lower and the upper cubicles, respectively. Let B = U B (/), and


B*= U B\l)./ E N

The elements of these cubicles have an important property: they are perfect.This notion has been introduced by Gelfand and Ponomarev in [F] for thefollowing property: a is said to be perfect if p(a)is either O or V for anyindecomposable representation p: Dr -> L{V). This means that for any repres-entation, the image of a is a direct summand. For any perfect element a, letNk(a) be the set of all indecomposable representations p: Dr -* L(V), with Fafinite dimension vector space over the field k and which satisfy p(a) = 0. It isshown in [F] that fora G5+ , the setNk(a) is finite and contains only prepro-jective representations. Dually, for <z E 2? ~, the set Nk (a) contains all but afinite number of indecomposable representations, and all indecomposablerepresentations not in Nk(a) are preinjective (the representations dual to pre-projective ones are called preinjective).

In dealing with perfect elements it seems to be convenient to work modulolinear equivalence. Two elements a, a £Z)r are said to be linear equivalentprovided p(a) = p(a') for any representation p: Dr -+ L(V). Of course, anyelement linear equivalent to a perfect element is also perfect. Up to linearequivalence, one hasB~ <B+ and Gelfand and Ponomarev have conjecturedthat, up to linear equivalence, all perfect elements belong to B ~ U B+. However,this has to be modified. Herrmann [19] has pointed out that there areadditional perfect elements arising from the different characteristics of fields.For example, for any prime number/?, and m>2, there is some perfect ele-ment dpm ED r such thatNk(dpm ) contains all representations ptj with Km,and, in case the characteristic of A: is p, then, in addition, the representationpom, and nothing else. Thus, it is even more convenient to work in the freep-linear lattice Dr

p, the quotient of Dr modulo p-linear equivalence where p iseither zero or a prime. Here, two elements, a, a' EZ)r are said to be p-linearequivalent provided p(a) = p(a') for any representation p in a vector spaceover a field of characteristic p.

The modified conjecture now asserts that any perfect element isp-linearly equivalent to an element in B ~ U B+. This indeed is true, as we wantto show. Thus, assume there exists a perfect element aEDr which is notp-linear equivalent to an element of B~ U B+. Gelfand and Ponomarev haveshown that then N(a) = Nk{a) contains all preprojective representations andno preinjective representation. In a joint paper [10] with Dlab, we have shownthat for r > 5, the set N(a) either contains only the preprojective repres-entations or else all but the preinjective representations. The elements x ED r

are given by lattice polynomials in the variables ex, . . ., er. Of course, there willbe many different lattice polynomials which define the same element x. Alattice polynomial with minimal number of occurrences of variables defining xwill be called a reduced expression of x and this number of variables in areduced expression will be called the complexity c(x) of x. Now, letp: Dr -> L(V) be a representation, U a one-dimensional subspace of V, and

146 Oaus Michael Ringel

p'\Dr -+ L(V/U) the induced representation, with p'(e() = (p(et) + U)/U for thegenerators et, 1 < / <r . We claim that for x E D r , w e have

dim p'(x) < c(x) — 1 + dim p(x).

[For the proof, we consider instead of p the representation p": Dr -> Z(K)with p"(x) the full inverse image of p'(x) under the projection V V/U, thusdim p"(x) = 1 + dim p'(x), for* eDr. Also note that p(x) C p"(x) for all*.By induction on c(x), we show the formula

dim p"(x) — dim p(x) < c(x).

Since dim U = 1, this clearly is true for x = et, with p"(ei) = p(et) + £/. Nowassume the formula being valid both for x1 and x2. For x = xx + x2 withc(x) = c(xi) + c(x2), we have

dimp"0c) = dim p"{xx + x 2 ) < d i m p ( x ! + x2) + c(xx) + c{x2)

= dim p(x) + c(x).

Similarly, for x = xtx2 with c(x) = c{xx) + c(x2), we have

dim p"(x) = dim p"(x1x2) = dim p"(Xj) + dim p"(x2) - dim p'^Xj

<dim p(Xi) + cCî) + dim p(x2) + c(x2) - dim p(xx + x2)

= dim p(x1x2) + c(x!) + c(x2) = dim p(x)

This finishes the proof. ]It is now sufficient to find a preprojective representation p: Dr ^ L(V)

with dim V > c(a) and a one-dimensional subspace U of F such that theinduced representation p in F/£/has no preprojective direct summand.Namely, our considerations above imply that dim p\a) < c(a) - 1 < dim V/U,due to the fact that p(a) = 0, and therefore there exists at least one indecom-posable representation o in N(a) which is not preprojective. As a consequence,in case r > 5, we know that N{a) contains all but the preinjective repres-entations. By duality, we similarly show that N(a) contains only the pre-projective representations, thus we obtain a contradiction. So, let us constructa suitable preprojective representation with the properties mentioned above. Infact, instead of considering representations of Dr, we will work inside theabelian category of representations of the oriented graph (*). We denote byPt,i = (yt,h Pt,i(ei)> • - •> Pt,i(er^ *ne g raPn representation corresponding toptl. Take any homomorphism <p: P0A ->PQ 2 such that R = Cok \p is regular(that is, has no non-zero preprojective or preinjective direct summand. Forexample, there always exists such a <p with R being the direct sum of twoindecomposable representations of dimension types (1; 1,1, 0, . . ., 0) and(r - 3; 0, 0, 1, 1, . . ., 1).) Now apply $"' for i E N. We obtain exactsequences


thus, the inclusion

ft =$-'(*) o . . . o $-ifo>) o p. PQl ^POJ+2

has regular cokernel (extensions of regular representations being regular, again).We now only have to choose i such that dim Voj+2 > c(a). This finishes theproof in case r > 5. (For r = 4, we again take <p: Po x -» PQ 2 with Cok y? beingthe direct sum of two representations of dimension types (1; 1, 1, 0, 0) and(1,0, 0, 1, 1), and form ft. The indecomposable summands of Cok ft all belongto one component C of the Auslander—Reiten quiver, thus we conclude asabove that C C N(a). By duality, one similarly shows that there are repres-entations in C which do not belong to N(a), so again we obtain a contradiction.Note that in case r = 4, the conjecture has been solved before by Herrmann[19].)

We consider now the general problem of representations of an oriented graph(F, A). We do not recall the definition of the category Z,(F, A) ofrepresentations of (F, A) over some fixed field k, nor the typical examples, butjust refer to the first two pages of [BGP]. We only note that L(T, A) can alsobe considered as the category of modules over the path algebra k(T, A), see[ 17], and A:(F, A) is a finite-dimensional A>algebra if and only if (F, A) doesnot have oriented cycles. In [ 15], Gabriel had shown that (F, A) has onlyfinitely many indecomposable representations if and only if F is the disjointunion of graphs of the form An, Dn, E6, En and Es (they are depicted on thethird page of [BGP]). It turned out that in case F is of the form An, Dn, E6,E1 or Es, the indecomposable representations of (F, A), with A an arbitraryorientation, are in one-to-one correspondence to the positive roots of F. It isthe aim of the paper [BGP] to give a direct proof of this fact. It introducesappropriate functors which produce all indecomposable representations fromthe simple ones in the same way as the canonical generators of the Weyl groupproduce all positive roots from the simple ones. We later will come back tothese functors and their various generalizations.

Given a finite graph F, let Er be the Q-vector space of functions Fo -* Q, anelement of Er being written as a tuple x = (xa) indexed by the elementsa G F 0 . For /3 E Fo , we denote its characteristic function by 0 (thus j3a = 0 fora =£ j8, and j ^ = 1). Any representation V of (F, A) gives rise to an elementdim V in Ev, its dimension type. For any orientation A of F, and any ]3_E Fo ,there is a unique simple representation L$ of dimension type dim L^ = j3. Incase there are no oriented cycles in (F, A), we obtain in this way all simplerepresentations of (F, A), thus, in this case, ET may be identified with therational Grothendieck group G0(F, A) ® Q (here, G0(F, A) is the factor group

zof the free abelian group with basis the set of all representations of (F, A)


modulo all exact sequences) with dim being the canonical map (sending arepresentation to the corresponding residue class). On Ev, there is defined aquadratic form B. In fact, for any orientation A of F, we may consider the(non-symmetric) bilinear form BA on Ev given by

BA(x,y) = 2 xOiya- 2 xa0)yM/ e r0 /er ,

and B is the corresponding quadratic form B(x) = BA (x, x). Note that B ispositive definite if and only if F is the disjoint union of graphs of the formAn, Dn, E6, En and E8, and in these cases, the root system for F is bydefinition just the set of solutions of the equation B(x) = 1.

For k algebraically closed and B being positive definite we will outline adirect proof that dim: L(T, A) -* Er induces a bijection between the indecom-posable representations of (F, A) and the positive roots. There is the followingalgebraic-geometric interpretation of B due to Tits [ 15]: The representationsof (F, A) of dimension type x may be considered as the algebraic variety

ra*(F, A) =

and there is an obvious action on it by the algebraic group

Gx = n GL(a, k)/Ac*er0

with A being the multiplicative group of k diagonally embedded as group ofscalars. Clearly

B(x) = dim Gx + 1 - dim mx(T, A).

Using this interpretation, Gabriel has shown in [16] that it only remains toprove that the endomorphism ring of any indecomposable representation is k.So assume Fis indecomposable, and that there are non-zero nilpotent endo-morphisms. Then V contains a subrepresentation (/with End(£/) = k andExt1 (U, U)¥=0. [Namely, let 0 =£ \p be an endomorphism with image S of

r

smallest possible length, thus <p2 = 0, and let W = © Wf be the kernel of <£,/ = l

with all Wt indecomposable. Now S C W, thus the projection of S into someWj must be non-zero. Since S was an image of a non-zero endomorphism ofsmallest length, we see that S embeds into this Wt. We may assume / = 1. Thusthere is an inclusion t: S -> Wx. If Wx has non-zero nilpotent endomorphisms, weuse induction. Otherwise End(H/

1) = k. Also, ExtHî, ^ i ) ^ 0> since on the onehand Ext1^, Wx) =£ 0 due to the exact sequence

i = 1


and, on the other hand, the inclusion i gives rise to a surjection Ext1 (t, W).Here we use that L(T, A) is a hereditary category]. The bilinear form BA hasthe following homological interpretation [25]:

£A(dim V, dim V) = dim* Hom(F, V) - dim^ Ext^K, F'),

for all representations V, V. Consequently, the existence of a representationU satisfying End(U) = k, Ext1 (U, U) ¥= 0 would imply that

B(dim U) = BA (dim £/, dim U) < 0,

contrary to the assumption that 5 is positive definite. This finishes the proof.For any finite connected graph F without loops, Kac [21, 22] gave a purely

combinatorial definition of its root system A. Note that A is a subset of ET

containing the canonical base vectors 0, for 0 E Fo , and being stable under theWeyl group W, the group generated by the reflections a^ along j3 with respectto B. The set A can also be interpreted in terms of root spaces of certain(usually infinite dimensional) Lie algebras [21]. Denote by A+ the set of rootswith only non-negative coordinates with respect to the canonical basis. ThenA is the union of A+ and A_ = - A+. In case F is of type An, Dn, E6, E1 orE8, the root system is finite and coincides with the set of solutions ofB(x) = 1. Otherwise the root system is infinite and will contain besides certainsolutions of B(x) = 1 also some solutions of B(x) < 0. The elements x of theroot system which satisfy B(x) = 1 are called real roots, they are precisely theelements of the W-orbits of the canonical base elements. The remainingelements of the root system are called imaginary roots, and Kac has deter-mined a fundamental domain for this set, the fundamental chamber.

Now, one has the following results (at least if k is either finite oralgebraically closed): For any finite graph F without loops, and anyorientation A, the set of dimension types of indecomposable modules isprecisely the set A+ of positive roots. For any positive real root x, there existsprecisely one indecomposable representation V of (F, A) with dim V = x. Forany positive imaginary root x, the maximal dimension JJLX of an irreduciblecomponent in the set of isomorphism classes of indecomposablerepresentations of dimensions is precisely 1 ~B(x, x). (Note that the subsetof indecomposable representations in mx (F, A) is constructive, andGx -invariant, thus we can decompose it as a finite disjoint union ofGx -invariant subsets each of which admits a geometric quotient. By definition,JJLX is the maximum of the dimensions of these quotients.) In particular, we seethat the number of indecomposable representations (or of the maximaldimension of families of indecomposable representations) of (F, A) does notdepend on the orientation A. For F of the form An, Dn, E6, E7 or E8, this isGabriel's theorem (of course, there are no imaginary roots). For F of the formAn, Dn, E6, En, or E8, the so called tame cases, these results have been shownby Donovan—Freislich [13] and Nazarova [23], see also [9]; in fact, in these


cases one obtains a full classification of all indecomposable representations;also, it is possible in these cases to describe completely the rational invariantsof the action of Gx on mx (F, A), for any dimension type x, see [27]. Ofcourse, the oriented graphs of finite or tame representation type are ratherspecial ones. It has been known since some time that the remaining (F, A) arewild: there always is a full exact subcategory of Z,(F, A) which is equivalentto the category Mk{X, Y > of k (X, Y >-modules (k (X, Y) being the polynomialring in two non-commuting indeterminates). In this situation, the results aboveare due to Kac [21, 22]. Note that this solves all the conjectures ofBernstein—Gelfand—Ponomarev formulated in [BGP]. However, there remainmany open questions concerning wild graphs (F, A). One does not expect toobtain a complete classification of the indecomposable representations of sucha graph, but one would like to have some more knowledge about certainclasses of representations. For example, there does not yet exist a combina-torial description of the set of those roots which are dimension types ofrepresentations V with End(F) = k.

We have mentioned above that the root system A of F is stable under theWeyl group W and that any W-orbit of A contains either one of the basevectors j3 (with j3 £ Fo) or an element of the fundamental chamber. One there-fore tries to find operations which associate to an indecomposable repres-entation V of (F, A) with A an orientation, and a Weyl group element w E Wa new indecomposable representation of (F, A'), where A' is a possiblydifferent orientation of F. By now, several such operations are known (see[BGP, 21, 28]), the first one being the reflection functors Ff, F* introducedby Bernstein, Gelfand and Ponomarev in [BGP]. Here, for the definition ofFp, the vertex 0 is supposed to be a sink, thus the simple representation Lpwith dimension vector dim Lp = J3 is projective. This concept has beengeneralized by Auslander, Platzeck and Reiten [ 1 ] dealing with any finitedimensional algebra A (or even an artin algebra) with a simple projectivemodule L. For this, we need the Auslander—Reiten translates T,T~1 . Recall

pthat TXA is defined for any A -module XA : let Px -* Po -> XA -* 0 be a minimalprojective resolution of XA, then Tr XA is by definition the cokernel of themap Hom(p, AA ) and TX = D Tr X, r ~* X = Tr D X, with D the usual dualitywith respect to the base field k. So assume L is a simple projective A -module,let P be the direct sum of one copy of each of the indecomposable projectivemodules different from L, and B = End(P © r"1 L). The functor considered byAuslander, Platzeck and Reiten is F= Hom^ (P © r"1 L, —) from the categoryMA of A -modules to MB . The functor induces an equivalence of the full sub-category T of MA of all modules which do not have L as a direct summandand a certain full subcategory of MB. Note that P © r"1 L is a tilting module inthe sense of [ 18], except in the trivial case of L being, in addition, injective.(A tilting module TA is defined by the following three properties:


(1) proj. dim. TA < 1, (2) there exists an exact sequence0 -*AA -+T' ^T" -• 0, with T', T" being direct sums of direct summands ofTA , and (3) Ext1 (TA, TA ) = 0. Now, if LA is simple projective and notinjective, the middle term Y of the Auslander—Reiten sequence

starting with L is projective. This sequence shows, on the one hand, thatproj. dim. T~1L = 1. On the other hand, it also gives an exact sequence of theform needed in (2). Finally, Ext^ (P © T~1L, P © r"1 L) «^ D Hom(P © T~1L, L) = 0, since any non-zero homomorphism from a moduleto L is a split epimorphism.)

A certain composition of the reflection functors F£ (or Fp, respectively) isof particular interest, the Coxeter functor <l>+ (or 3>~). An explicit calculationfor the r-subspace situation is given in [F], in the special case of the 4-subspaceproblem it had been defined before in [Q]. The Coxeter functors are endo-functors of L(T, A), they are only defined in case (F, A) does not have orientedcycles (non-oriented cycles are allowed, see [9]). Note that the assignment ofan orientation A without oriented cycles is equivalent to the choice of a partialordering of Fo (let a < j3 iff there exists an oriented path a. - a0 ôcl *- . . .. . . «- oim - ($), and also to the choice of a Coxeter transformation: this is aWeyl group element of the form c = oa . . . oa with a1, . . ., o^ being the

elements of Fo in some fixed ordering (take an ordering of F which refines thegiven partial ordering). So assume from now on that (F, A) is a connectedoriented graph without oriented cycles, and let c be the corresponding Coxeterelement. The Coxeter functors 3>+ and <£~ defined in [BGP] have the follow-ing properties: if V is an indecomposable representation of (F, A), then eitherV is projective and then <0+(F) = 0, or else V is not projective, and then <£+(F)again is indecomposable, $"^>+(F) ^ Fand dim ^>+(F) = c dim V. Thus theCoxeter functor <l>+ realizes the action of the Coxeter transformation on theset of all representations without non-zero projective direct summands. Theusefulness of the Coxeter functors seems to have its origin in their relation tothe Auslander-Reiten translation r. Namely, Gabriel ([17], Prop. 5.3, seealso [ 1,5]) has shown that r can be identified with C+ ° T, where T is thefunctor which maps the representation (V, f) to (F, - / ) . In particular, for Fbeing a tree, we can identify r with C+ itself.

In order to explain the value of the Auslander—Reiten translation r (andtherefore of the Coxeter functors), we have to recall the definition of theAuslander—Reiten quiver of a finite dimensional algebra yl. Its vertices are theisomorphism classes [X] of the indecomposable A -modules X, and, if X, Yare indecomposable modules, then there is an arrow [X] -* [ Y] iff there existsan irreducible map X -> Y (a map/is said to be irreducible provided it isneither a split monomorphism nor a split epimorphism, and for any factori-zation f = f" © / ' , we have tha t / ' is a split monomorphism o r / " is a split

152 Clam Michael Ringel

epimorphism [2]). Now, the Auslander—Reiten quiver is a translation quiverwith respect to r: if X is indecomposable and not projective, then there existsan irreducible map Y ->• X iff there exists an irreducible map TX -> Y.

For the finite dimensional hereditary algebras A, the structure of theAuslander—Reiten quiver is known. We will recall this result in the specialcase of A = k(F, A). First, we need some notation. Define Z(F, A) as follows:

its vertices are the elements of Fo X Z, and for any arrowl o « o ] 9 there

(a, z) (a*,z)are arrows (/, z) * (/, z) and (/, z) • (/, z + 1), for all z E Z, see[24] and also [ 17, 29]. Note that in case F is a tree, Z(F, A) does not dependon the orientation A and just may be denoted by ZF. If / C Z, let /(F, A) bethe full subgraph of all vertices (z, z) with / G /. In particular, we will have toconsider N(F, A) and N"(F, A), where N = { 1, 2, 3, . . .} andN ~= { — 1, -2 , - 3 , . . .}. Also, denote by A^ the following infinite graph

The result is as follows: in case F is of the form An, Dn, E6, En or Es, theAuslander-Reiten quiver of fc(F, A) is a finite full connected subquiver of ZF.(In case Dn with n = 0(2), the Auslander-Reiten quiver of k(F, A) is[ l , r c - l ] (F, A), in case of En o r£ 8 , i t i s [1,9] (F, A) or [1,15] (F, A),respectively, in the remaining cases, it is slightly more difficult to describe,see [17, 29]). In all other cases, the Auslander—Reiten quiver of A:(F, A) hasinfinitely many components, all but two being quotients of ZA^ (see [26]),the remaining two being of the form N(F, A) and N "(F, A). The component ofthe form N(F, A) contains the indecomposable projective modules: in fact,the indecomposable projective module Pf corresponding to the vertex / G Fo

appears as indexed by (/, 1), and the module indexed by (/, z), z EN, is just<£~z + 1 (P.)f this component is called the preprojective component. Similarly,the component of the form N"(F, A) is called the preinjective component, itcontains the indecomposable injective module Jt corresponding to / E Fo asindexed by (i, —1), and the module indexed by (/, -z) , z G N, is just ^>+z-1 (J.).

Let us consider in more detail a preprojective component &, and the modulesbelonging to:^; they will be called preprojective modules. In case F is of typeAn, Dn, E6, E-j, or E8, we let .^denote the full Auslander—Reiten quiver; inany case, we note that an indecomposable representation of (F, A) is said tobe preprojective iff it is of the form "ZP, with P indecomposable projectiveand z > 0. (A general theory of preprojective modules has been developed byAuslander and Smal0, see [3]). For an indecomposable preprojective repres-entation X, there are only finitely many indecomposable modules Y suchthat Hom(y, A") =£ 0, all of them are preprojective again, and any non-invertible homomorphism Y -> X is a sum of compositions of irreducible maps.In particular, if X, Y are indecomposable and preprojective and

Four papers on problems in linear algebra 15 3

Hom(X, Y) =£ 0, then there is an oriented path [X] - • . . . - » [Y] m0>. In fact,the complete categorical structure of the full subcategory of preprojectivemodules can be read off from the combinatorial description of âs atranslation quiver: the category of all preprojective modules is equivalent to thequotient category <} 0* of the path category of ^modulo the so called meshrelations (see [4, 24, 17]). Note that the category ()0> allows to reconstructall the modules in 0*. Namely, any module XA is isomorphic toH o m ^ ^ , XA ), thus, if AA = 0 P?i9 then XA can be identified with

© Hom(Pz., X)nf, and Hom(P/f X) can be calculated inside 0-^, since both

Pj, X are preprojective.Starting from the preprojective component 0 of A;(F, A), one may define a

(usually infinite-dimensional) algebra II as follows: Take the direct sum of allhomomorphism spaces Hom(/\ 1), it, /)) in <> 0* and define the product of tworesidue classes w, w' of paths w: (/, 1 ) - * . . . - * (f, /) and10': (/', 1)-* . . . -* (* ' , / ' ) as follows: in case t=j',letwwf be the residue classof the composed path r"/+1(">')° w: (/, 1)"*. • • "• (*', / + / ' - l),andOotherwise. There is a purely combinatorial description of II in terms of (F, A)due to Gelfand and Ponomarev, see [R]. Let F be obtained from (F, A) byadding to each arrow a: i -•/ an additional arrow «*:/-> /. We clearly canidentify II with the factor algebra of the path algebra kT modulo the idealgenerated by the element £ era* + Z a**a. Note that this description is

independent of the choice of the orientation A. Also, we see from bothdescriptions that II contains as a subalgebra &(F, A), thus we may consider II asa right k(F, A)-module, and the first description now shows that the A:(F, A)-module n ^ ( r A ) decomposes as the direct sum of all preprojective representationsof (F, A) each occurring with multiplicity one, and therefore is called the prepro-jective algebra of F. (For the proper generalisation to the case of a species, werefer to [11]. We also should note the slight deviation of the preprojectivealgebra from the model algebra defined in [M], which reduces to the algebra Ar

given in [I, II] in the case of the r-subspace situation. Namely, here the constantpaths have square zero, whereas they are idempotents in n. Now, in II the sum ofthe constant paths is the identity element. In order also to have an identity ele-ment, Gelfand and Ponomarev add to the direct sum of all preprojective modulesan additional one-dimensional space ke. There is a change of definition proposedin [S], using the constant paths as idempotents as in II, but adding again anadditional identity element.) Since II is the direct sum of the preprojective repres-entations of (F, A), it follows that II is finite dimensionaHf and only if F is of theform^4w, Dn, E6, Elf or Es. In [12], the tame cases An, Dn, E6, E7 and Es havebeen characterized by the fact that the Gelfand-Kirillov dimension of II is 1,whereas it is °° for the wild cases.


Let us return to the special case of the r subspace graph (*), with r > 4. Thedescription above gives that the preprojective component îs of the form

(0,1) (0,2) (0,3)

If we denote the arrows in the following way:

then the mesh relations are as follows: at af = 0 for all /, and 2 ai at = 0.

Thus, if we want to determine the total space of the representation labelled(t, /), we have to calculate Hom((0, 1), {t, /)) inside the category 0 ^ , and thisamounts to the calculation of all possible paths from (0, 1) to (t, /), taking thisas the basis of a vector space and factoring out the mesh relations. However,taking from the beginning into account the relations <xf af = 0, we just as wellmay work with the vector space generated by the set At{r, I) and factoring outthe remaining mesh relations. This shows that we obtain as total space thevector space Vtj. Similarly, the r different subspaces of the representationlabelled {t, I) are given by the various Hom((y, 1), (t, /)), 1 < / < r , againcalculated in 0 ^ , and therefore coincide with the subspaces ptj(e;-). In thisway, we obtain directly the description of the preprojective representations ofDr given by Gelfand and Ponomarev (and a direct proof of Proposition 8.2 in[F]).

Finally, let us note in which way the preprojective component of Dr deter-mines the lattice B + of perfect elements belonging to the upper cubicles. Forany perfect element a, we have denoted by N(a) the set of indecomposablerepresentation p satisfying p(a) = 0. We claim that for a £B+, the set N(a) is a


finite, predecessor closed subset of :/(an element x is said to be a predecessorof y in case there is an oriented path x -+ . . . ->y. For the proof, we first notethat clearly N(a) H;/is predecessor closed, since for indecomposable repres-entations p, p with Hom(p, p) J= 0, and a perfect, p GN(a) implies p GN(a).Since not all of:/is contained in N(a), it obviously follows that N(a) Hi/isfinite. However, any complete slice of </generates all representations outsideof,/, thus taking a complete slice of .ôutside of N(a) H^/, we easily see thatno indecomposable representation outside of & can belong to N(a), thusN(a) C./.) Thus TV determines a map from B + to the set of all finite,predecessor closed subsets of &. This map is bijective and order-reversing, thusB+ is anti-isomorphic to the lattice of finite, predecessor closed subsets of.:/.

References

[Q] Gelfand, Ponomarev: Problems in linear algebra and classification of quadruples ina finite dimensional vector space. Coll. Math. Soc. Bolyai 5, Tihany (1970),163-237.

[BGP] Bernstein, Gelfand, Ponomarev: Coxeter functors and Gabriel's theorem. UspekhiMat. Nauk 28 (1973), Russian Math. Surveys 28 (1973), 17-32, also in thisvolume.

[F] Gelfand, Ponomarev: Free modular lattices and their representations. UspekhiMath. Nauk 29 (1974), 3-58. Russian Math. Surveys 29 (1974), 1-56, also in thisvolume.

[I] Gelfand, Ponomarev: Lattices, representations and algebras connected with them.I.Uspekhi Math. Nauk 31 (1976), 71-88. Russian Math. Surveys 31 (1976),67—85, also in this volume.

[II] Gelfand, Ponomarev: Lattices, representations and algebras connected with them.II. Uspechi Math. Nauk 32 (1977), 85-106. Russian Math. Surveys 32 (1977),91—114, also in this volume.

[M] Gelfand, Ponomarev: Model algebras and representations of graphs. Funkc. Anal, iPril. 13.3 (1979), 1-12. Funct. Anal. Appl. 13 (1979), 157-166.

[R] Rojter: Gelfand-Ponomarev algebra of a quiver. Abstract, 2nd ICRA (Ottawa1979).

[S] Gelfand, Ponomarev: Representations of graphs. Perfect subrepresentations. Funkc.Anal, i Pril. 14.3 (1980), 14-31. Funct. Anal. Appl. 14 (1980), 177-190.

[1] Auslander, Platzek, Reiten: Coxeter functors without diagrams. Trans. Amer. Math.Soc. 250(1979), 1-46.

[2] Auslander, Reiten: Representation theory of artin algebras III, IV, V. Comm.Algebra 3 (1975), 239-294; 5 (1977), 443-518; 5 (1977), 519-554.

[3] Auslander, Smal0: Preprojective modules over artin algebras. J. Algebra (to appear).[4] Bautista: Irreducible maps and the radical of a category. Preprint.[5] Brenner, Butler: The equivalence of certain functors occurring in the representation

theory of artin algebras and species. J. London Math. Soc. 14 (1976), 183-187.[6] Day, Herrmann, Wille: On modular lattices with four generators. Algebra

Universal 3 (1972), 317-323.


[7] Dedeking: Ober die von drei Moduln erzeugte Dualgruppe. Math. Ann. 53 (1900),371-403.

[8] Dlab: Structure des treillis lineaires libres. Seminaire Dubreil. Springer LNM 795(1980), 10-34.

[9] Dlab, Ringel: Indecomposable representations of graphs and algebras. Mem. Amer.Math. Soc. 173(1976).

[10] Dlab, Ringel: Perfect elements in the free modular lattices. Math. Ann. 247 (1980),95-100.

[11] Dlab, Ringel: The preprojective algebra of a modulated graph. Springer LNM 832(1980), 216-131.

[12] Dlab, Ringel: Eigenvalues of Coxeter transformations and the Gelfand-Kirillovdimension of the preprojective algebra. Proc. Amer. Math. Soc. (to appear).

[13] Donovan, Freislich: The representation theory of finite graphs and associatedalgebras. Carleton Lecture Notes 5 (1973).

[14] Freese: Free modular lattices. Trans. AMS 261 (1980), 81-91.[15] Gabriel: Unzerlegbare Darstellungen I. Manuscripta Math. 6 (1972), 71-103.[16] Gabriel: Indecomposable representations II. Symposia Math. Inst. Naz. Alta Mat.

11 (1973), 81-104.[17] Gabriel: Auslander-Reiten sequences and representation finite algebras. Springer

LNM 831 (1980), 1-71.[18] Happel, Ringel: Tilted algebras. Trans. Amer. Math. Soc. (to appear).[19] Herrmann: Rahmen und erzeugende Quadrupeln in modularen Verbanden. To

appear in Algebra Universalis.[20] Hutchinson: Embedding and unsolvability theorems for modular lattices. Algebra

Universalis 7 (1977), 47-84.[21] Kac: Infinite root systems, representations of graphs and invariant theory. Inv.

Math. 56 (1980), 57-92. part II: preprint.[22] Kac: Some remarks on representations of quivers and infinite root systems.

Springer LNM 832 (1980), 311-327.[23] Nazarova: Representations of quivers of infinite type. Izv. Akad. Nauk. SSSR. Ser.

Mat. 37 (1973), 752-791.[24] Riedtmann: Algebren, Darstellungskocher, Uberlagerungen und Zuriick. Comment.

Math. Helv. 55 (1980), 199-224.[25] Ringel: Representations of A'-species and bimodules. J. Algebra 41 (1976),

269-302.[26] Ringel: Finite dimensional algebras of wild representation type. Math. Z. 161

(1978), 235-255.[27] Ringel: The rational invariants of tame quivers. Inv. Math. 58 (1980), 217-239.[28] Ringel: Reflection functors for hereditary algebras. J. London Math. Soc. (2)

21 (1980), 465-479.[29] Ringel: Tame algebras. Springer LNM 831 (1980), 137-287.

COXETER FUNCTORS ANDGABRIEL'S THEOREM

I. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev

It has recently become clear that a whole range of problems of linearalgebra can be formulated in a uniform way, and in this common formu-lation there arise general effective methods of investigating such problems.It is interesting that these methods turn out to be connected with suchideas as the Coxeter—Weyl group and the Dynkin diagrams.

We explain these connections by means of a very simple problem. Weassume no preliminary knowledge. We do not touch on the connectionsbetween these questions and the theory of group representations or thetheory of infinite—dimensional Lie algebras. For this see [3]—[5].

Let F be a finite connected graph; we denote the set of its vertices byPo and the set of its edges by 1^ (we do not exclude the cases where twovertices are joined by several edges or there are loops joining a vertex toitself). We fix a certain orientation A of the graph F; this means that foreach edge / G Vi we distinguish a starting-point a(l) e r 0 and an end-point«/)e r0.

With each vertex a G Fo we associate a finite-dimensional linear spaceVa over a fixed field K. Furthermore, with each edge / G r 2 we associatea linear mapping / ) : Va0) -+ VpU) (a(l) and <3(/) are the starting-point andend-point of the edge /). We impose no relations on the linear mappings/,. We denote the collection of spaces Va and mappings fx by (V, f).

DEFINITION 1. Let ( r , A) be an oriented graph. We define a categoryX (F, A) in the following way. An object of X(T, A) is any collection(K, f) of spaces Va (a G r 0 ) and mappings // (/ G F j ) . A morphism </?:(V, f) -+ (W, g) is a collection of linear mappings </?a: Va -» Wa (a G F o )such that for any edge / G r 2 the following diagram

Kar«i

ri commutative, that is, v>^(i)/( ~ 8i<Pau)-

157

158 /. N. Bernstein, I. M. GeVfand, and V. A. Ponomarev

Many problems of linear algebra can be formulated in these terms. Forexample, the question of the canonical form of a linear transformation/: V -» V is connected with the diagram

The classification of a pair of linear mappings fx : Vx -> V2 andf2 : Vy -> F2 leads to the graph

A very interesting problem is that of the classification of quadruples ofsubspaces in a linear space, which corresponds to the graph

This last problem contains several problems of linear algebra.1

Let (F, A) be an oriented graph. The direct sum of the objects (V, f)and (U, g) in g(T, A) is the object (W, h), where Wa = Va ® Uat

hi =fi®gi(*e r0, / e r , ) .We call a non-zero object (V, f) € X (r, A) indecomposable if it cannot

be represented as the direct sum of two non-zero objects. The simplestindecomposable objects are the irreducible objects La (a £ r 0 ) , whosestructure is as follows: (La)y = 0 for y =£ a, (La)a = K, fx = 0 for all / e IV

It is clear that each object (F, f) of X (r, A) isisomorphic to the directsum of finitely many indecomposable objects.2

In many cases indecomposable objects can be classified.3

In his article [1] Gabriel raised and solved the following problem: tofind all graphs (r, A) for which there exist only finitely many non-isomor-phic indecomposable objects (V, f) e X (r, A). He made the following

Let us explain how the problem of the canonical form of a linear operator /: V -* V reduces to that ofa quadruple of subspaces. Consider the space W = V 0 V and in it the graph of/, that is, the subspaceE4 of pairs (£,/£), where J6K. The mapping/is described by a quadruple of subspaces in W, namelyEx = V 0 0, E2 = 0 © V, E3 = {(£, *) I * e V)(E3 is the diagonal) and E4 = {(£,/£) | % e V}- thegraph of/. Two mappings/and/' are equivalent if and only if the quadruples corresponding to themare isomorphic. In fact, Ex and E2 define "coordinate planes" in W, E3 establishes an identificationbetween them, and then EA gives the mapping.It can be shown that such a decomposition is unique to within isomorphism (see [6], Chap. II, 14,the Krull-Schmidt theorem).We believe that a study of cases in which an explicit classification is impossible is by no means withoutinterest. However, we should find it difficult to formulate precisely what is meant in this case by a"study" of objects to within isomorphism. Suggestions that are natural at first sight (to consider thesubdivision of the space of objects into trajectories, to investigate versal families, to distinguish "stable"objects, and so on) are not, in our view, at all definitive.

Coxeter Functors and Gabriel's Theorem 159

surprising observation. For the existence of finitely many indecomposableobjects in X (F, A) it is necessary and sufficient that F should be one ofthe following graphs:

(n vertices,n> 1)

(n vertices,4)

£7

(this fact does not depend on the orientation A).The surprising fact hereis that these graphs coincide exactly with the Dynkin diagrams for thesimple Lie groups.1

However, this is not all. As Gabriel established, the indecomposableobjects of X (F, A) correspond naturally to the positive roots, constructedaccording to the Dynkin diagram F.

In this paper we try to remove to some extent the "mystique" of thiscorrespondence. Whereas in Gabriel's article the connection with the Dynkindiagrams and the roots is established a posteriori, we give a proof ofGabriel's theorem based on exploiting the technique of roots and the Weylgroup. We do not assume the reader to be familiar with these ideas, and wegive a complete account of the necessary facts.

An essential role is played in our proof by the functors defined below,which we call Coxeter functors (the name arises from the connection ofthese functors with the Coxeter transformations in the Weyl group). Forthe particular case of a quadruple of subspaces these functors were intro-duced in [2] (where they were denoted by 4>+ and 4>~). Essentially, ourpaper is a synthesis of Gabriel's idea on the connection between the cate-gories %{T, A) with the Dynkin diagrams and the ideas of the first partof [2], where with the help of the functors 3>+ and <f>~ the "simple"indecomposable objects are separated from the more "complicated" ones.


We hope that this technique is useful not only for the solution ofGabriel's problem and the classification of quadruples of subspaces, butalso for the solution of many other problems (possibly, not only problemsof linear algebra).

Some arguments on Gabriel's problem, similar to those used in thisarticle, have recently been expressed by Roiter. We should also like todraw the reader's attention to the articles of Roiter, Nazarova, Kleiner,Drozd and others (see [3] and the literature cited there), in which veryeffective algorithms are developed for the solution of problems in linearalgebra. In [3], Roiter and Nazarova consider the problem of classifyingrepresentations of ordered sets; their results are similar to those of Gabrielon the representations of graphs.

§ 1. Image functors and Coxeter functors

To study indecomposable objects in the category X (F, A) we consider"image functors", which construct for each object V & X (F, A) somenew object (in another category); here an indecomposable object goeseither into an indecomposable object or into the zero object. We constructsuch a functor for each vertex a at which all the edges have the samedirection (that is, they all go in or all go out). Furthermore, we constructthe "Coxeter functors" 3>+ and 3>~, which take the category X (F, A)into itself.

For each vertex a e Fo we denote by Ta the set of edges containing a.If A is some orientation of the graph F, we denote by oaA the orientationobtained from A by changing the directions of all edges / e F a .

We say that a vertex a is (—)-accessible (with respect to the orientationA) if 0(/) =£ a for all / G F2 (this means that all the edges containing astart there and that there are no loops in F with vertex at a). Similarly wesay that the vertex 0 is (+)-accessible if a(/) =£ 0, for all / e Ti.

DEFINITION 1.1 1) Suppose that the vertex 0 of the graph F is(+)-accessible with respect to the orientation A. From an object (F, f) inX(V, A) we construct a new object (W, g) in X (F,aÂ).

Namely, we put Wy = Vy for y =£ 0.Next we consider all the edges lx, l2, . . ., h that end at 0 (that is, all

k

edges of F^). We denote by Wp the subspace in the direct sum © VaOi)

consisting of the vectors v = (vx, . . . , vk) (here vt e VocO.)) for which

fi.(Pi) + . . . + fik (ufc) = 0. In other words, if we denote by h the

k

mapping h: © Va^t) ->- V$ defined by the formula


h(uu v2, . . ., vh) = Ux (Vi) + . . . + fih(vh), then Wp = Ker h.We now define the mappings gt. For I $ Fp we put gt = ft. If

I = lj G r^, then gi is defined as the composition of the naturalembedding of Wp in © VaOi) and the projection of this sum onto theterm V^^p = Wa(lj). We note that on all edges / G r^ the orientation hasbeen changed, that is, the resulting object (W, g) belongs to X (r, aÂ).We denote the object (W, g) so constructed by Fp(V, f).

2) Suppose that the vertex a e r 0 is (—)-accessible with respect to theorientation A. From the object (V, f) G X (r, A) we construct a newobject F-(F, /) = (R/, g) e X (r, aaA). Namely, we put

Wy = Vy for 7 = agl = U for Z $ r«

^ = © F^(/.)/Im /z, where {lu . . . , Zft} = r a , and the mapping

h'Va-+ © ^p(^) is defined by the formula h (u) = (fti (y), . . ., /Zfc (L>)).

If / G r a , then the mapping gt: Wp0) -> H a is defined as the compositionh

of the natural embedding of WP0)= VPil) in 0 Vî^ and the projectioni=l

of this direct sum onto Wa.It is easy to verify that F£ (and similarly F~) is a functor from

X (r, A) into X(F, aÂ)(or ^ ( r , aaA), respectively). The followingproperty of these functors is basic for us.

THEOREM 1.1 1) Let (T, A) be an oriented graph and let 0 G r0 bea vertex that is {-^-accessible with respect to A. Let V G X (F, A) be anindecomposable object. Then two cases are possible:

a) V « Lp and F^V = 0 (we reca// that Lp is an irreducible object,defined by the condition (Lp)y = 0 for y * p, (Lfi)p = K, fx = 0 for all/ e r , ) .

b) F*(V) is an indecomposable object, F^F^(V) = V, and the dimensionsof the spaces F^(V)y can be calculated by the formula

(1.1.1) dim F$(V)y = dim Fv for y == p,dimFa(0.

2) / / the vertex a is (-)-accessible with respect to A and ifV G X (r, A) is an indecomposable object, then two cases are possible:

a) V * La, F~(V) = 0.

162 /. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev

b) F~{V) is an indecomposable object, F*F~{V) = V,

(1.1.2) dim F~(V)V = dim Vy for y =/= a,

+ 2 di

PROOF. If the vertex 0 is (+)-accessible with respect to A, then it is(-)-accessible with respect to oÂ, and so the functor F^F^'-X(T, A)-+X{T, A) is defined. For each object V e%(r, A) we constructa morphism fiy: F~^F*{V) -> V in the following way.

If 7 * P, then F~F+(F)7 = Vy, and we put (i* )7 = Id, the identitymapping.

For the definition of (fv)p we note that in the sequence of mappings

> 0 Va(i)--> V3 (see definition 1.1) Ker h = Im /z ; we take forzerP

the natural mapping

It is easy to verify that fy is a morphism. Similarly, for each (—)-accessiblevertex a we construct a morphism pa

v: F -• F^F~(F). Now we state thebasic properties of the functors F~, Fp and the morphisms p°y, /^.

LEMMA 1 . 1 . 1 ) ^ ( ^ 1 © ^ ) = ^ (Vi) 0 â ( 2) • 2) p y is an epimorphismand i$ is a monomorphism. 3) If fv is an isomorphism, then the dimensionsof the spaces Fp(V)y can be calculated from (1.1.1). / / P y is an isomor-phism, then the dimensions of the spaces FZ(Y)ycan be calculated from(1.1.2). 4) The object Ker p^ is concentrated at a {that is, (Ker p<

v)y = 0for 7 ^ a). The object K/Im fv is concentrated at p. 5) / / the object Vhas the form F*W (FJW, respectively), then p% (fv) is an isomorphism.6) The object V is isomorphic to the direct sum of the objects Fp^^yyand F/Im fv {similarly, V « O ^ ( K ) © k e r p%).

PROOF. 1), 2), 3), 4) and 5) can be verified immediately. Let us prove6).

We have to show that F « F^Ff{V) 0 V, where V = F/Im i%. Thenatural projection 1^: Vp -> F'& has a section <p&\ V$ -> F^ ( . F. It is clearthat the morphisms y\ V -+ V and /y: FpFp{V) -• F give a decompositionof F into a direct sum. We can prove similarly that V » F£FZ (V) 0 Ker p*.

We now prove Theorem 1.1. Let F be an indecomposable object of thecategory# (F, A),and 0 a (+>accessible vertex with respect to A. Since


V « Fp F$ (V) © F/Im iy and F is indecomposable, F coincides with one

of the terms.

CASE I). F = F/Im i%. Then F7 = 0 for y ± 0 and, because F is

indecomposable, V ** L&.

CASE II). F = FJFp{V), that is, £ is an isomorphism. Then (1.1.1) is

satisfied by Lemma 1.1. We show that the object W = Fp{V) is indecom-

posable. For suppose that W = Wt © W2.ThenV = F$ (Wx) © F$ {W2)

and so one of the terms (for example, F^{W2)) is 0. By 5) of Lemma 1.1,

the morphism p%\ W -> FpFJ{W) is an isomorphism, but

P0V{W2) c F;F;{W2) = o, that is, w2 = o.So we have shown that the object F${V) is indecomposable. We can

similarly prove 2) of Theorem 1.1.

We say that a sequence of vertices ax, a2, . . . , afe is (+)-accessible with

respect to A if ax is (+)-accessible with respect to A, a2 is (+)-accessible

with respect to a a j A, a3 is (+)-accessible with respect to aa 2at t iA, and so

on. We define a (—)-accessible sequence similarly.COROLLARY 1.1. Let (F , A) be an oriented graph and aXi a2, . . . , afe

a {-^-accessible sequence.1) For any i (1 < i < k\ F ^ • . . . •F« f _ l (L a . ) w ezY/zer 0 or aw

indecomposable object in X (F, A) {here La. £ X {T, oot._i oa._2 . . . oai A))2) Let V e X {r, A) be an indecomposable object, and

F+ F+ • •JF+ (V) = 0afe aft-i " * a i v ;

Tên /or some i

We illustrate the application of the functors Fp and F^ by the followingtheorem.

THEOREM 1.2. LeJ F ^^ <2 grap/z without cycles {in particular, withoutloops), and A, A' two orientations of it.

1) There exists a sequence of vertices ati . . . , afe, {-^-accessible withrespect to A, swc/z //zatf °afeffah t* • • • #cra1A = A ' .

2) Le£ <M, S' be the sets of classes {to within isomorphism) ofindecomposable objects in X (F, A) and X (F, A'), <Jl a o/ft — the set ofclasses of objects F^F^ . . . -F^.^La.) ( K i < f c ) , and<JCr a oM'the set ofclasses of objects FZk-...'FZUi(Lai) (l<i<k). Then the functor

F£k • . . . • F*^ sets up a one-to-one correspondence between QM\QM and

Q/n> \Q/K> .

Where it cannot lead to misunderstanding, we denote by the same symbol La irreducible objects inall categories X(r, A), omitting the indication of the orientation A.


This theorem shows that, knowing the classification of indecomposableobjects for A, we can easily carry it over to A'; in other words, problemsthat can be obtained from one another by reversing some of the arrowsare equivalent in a certain sense.

Examples show that the same is true for graphs with cycles, but we areunable to prove it.

PROOF OF THEOREM 1.2. It is clear that 2) follows at once from 1)and Corollary 1.1. Let us prove 1).

It is sufficient to consider the case when the orientations A and A'differ in only one edge /. The graph V \ I splits into two connectedcomponents. Let r ' be the one that contains the vertex 0(7) (0(7) is takenwith the orientation of A). Let ax, . . ., ak be a numbering of the verticesof r ' such that for any edge /' e r\ the index of the vertex a(l') isgreater than that of j3(/'). (Such a numbering exists because r ' is a graphwithout cycles.) It is easy to see that the sequence of vertices a{, . . ., ak

is the one required (that is, it is (+)-accessible and aafe • . . . • aOiA = A').This proves Theorem 1.2.

It is often convenient to use a certain combination of functors F~ thattakes the category X (r, A) into itself.

DEFINITION 1.2. Let (I\ A) be an oriented graph without orientedcycles. We choose a numbering ax, . . ., an of the vertices of T such thatfor any edge /G Fj the index of the vertex a(/) is greater than that of0(/). We put* + =Kn ' . . . • < < , < * > " = / X ' F:2- . . . ' FZn. We call <D +

and "~ Coxeter functors.LEMMA 1.2. 1) The sequence al5 . . ., an is {^-accessible and

<*„,...,<*! is (-)-accessible.2) The functors $+ and $~ take the categoryX (F, A) into itself. 3) 3>+ and 3>~ do not depend on the freedom ofchoice in numbering the vertices.

The proof of 1) and 2) is obvious. We prove 3) for 3>+. We note firstlythat if two different vertices 71, y2 £ To are not joined by an edge andare (+)-accessible with respect to some orientation, then the functors Fând F+2 commute (that is, F*tFyt = F*tF*a).

Let a2, . . ., otn and ai, . . . a'n be two suitable numberings and letax = a'm . Then the vertices a[, a'2, . . ., amJi are not joined to ax by anedge (if ax and a/ (z < m) are joined by an edge /, then a(/) = a'm - OLX

by virtue of the choice of the numbering of ai, . . . , a'n, but this contradictsthe choice of the numbering of <*!, . . . , an). ThereforeF£ • . . . * F£; = Fa'm_i * • • • * F^F^ -. Carrying out a similar argu-ment with a2, then with a3, and so on, we prove that F^ • . . . • F^

T a A • • • ^ a ; - t o c n • • • ^ , -

The proof is similar for the functor $ .Following [2] we can introduce the following definition.


DEFINITION 1.3. Let ( I \ A) be an oriented graph without orientedcycles. We say that an object V E X ( I \ A) is (+)-(respectively, (-)-)irregular if {<&)kV = 0 ((<$~)kV = 0) for some k. We say that an object V isregular if V * (<£"")* (<D+)feF « (<&)k(<$T)kV for all A:.

NOTE 1. Using the morphisms p0^ and iv introduced in the proof ofTheorem 1.1, we can construct a canonical epimorphism pv: V -» (<J>+)fe ($~)kV

and monomorphism /* : (<!>")* (<l>+)feF -• F. The object F is regular if andonly if for all k these morphisms are isomorphisms.

NOTE 2. If an object F is annihilated by the functor F*s • . . . • F« t

(«!, . . . , OLS is some (+)-accessible sequence), then this object is (+)-irregular.Moreover, the sequence al9 . . . , as can be extended to a h . . . , a s ,a.+1, . . . , am so that F^ • . . . • Fa

+s+i -Fa

+s • . . . • F*t = (S+)s.

THEOREM 1.3. Let (T, A) be an oriented graph without oriented cycles.1) Each indecomposable object V e # ( r , A) w either regular or irregular.2) Le^ <*!, . . . , an Z?e <3 numbering of the vertices of T such that for anyI E Fi the index of a(l) is greater than that of (5(1), PutV{ = K, FZ7' . . . 'F~fH (La.) e X ( r , A), h =F:n-... -Fa

+.+i ( I a . ) e 2 ( r , A )(here ] +(F) = 0 is isomorphic to one of the objects Vt.Similarly, 3>~(F,-) = 0, and if V is indecomposable and 3>~(F) = 0, thenV «» Vt for some i. 3) Each (^-(respectively, (—)-) irregular indecomposableobject V has the form (3>"~)feFi (respectively\ ($+)fei^) for some i, k.

Theorem 1.3 follows immediately from Corollary 1.1.With the help of this theorem it is possible, as was done in [2] for the

classification of quadruples of subspaces, to distinguish "simple" (irregular)objects from more "complicated" (regular) objects; other methods arenecessary for the investigation of regular objects.

§ 2. Graphs, Weyl groups and Coxeter transformations

In this section we define Weyl groups, roots, and Coxeter transformations,and we prove results that are needed subsequently. We mention two differ-ences between our account and the conventional one.

a) We have only Dynkin diagrams with single arrows.b) In the case of graphs with multiple edges we obtain a wider class of

groups than, for example, in [7] .DEFINITION 2.1. Let r be a graph without loops.(1) We denote by %v the linear space over Q consisting of sequences

x = (xa) of rational numbers x a (oc G F o ) .For each 0 E Fo we denote by 0 the vector in %v such that (fi )a - 0

for a * 0 and (0^ = 1.We call a vector x = (xa) integral if xa E Z for all a E Fo.

We call a vector x = (xa) positive (written x > 0) if x ¥= 0 and

166 /. TV. Bernstein, I. M. Gel'fand, and V. A. Ponontarev

xa > 0 for all a G r 0 .2) We denote by B the quadratic form on the space %v defined by the

formula B (x) = 2 xa — 2 xviii)'xV2(i)y where x = (xa), and 71 (/) and y2 (0

are the ends of the edge /. We denote by <, > the corresponding symmetricbilinear form.

3) For each 0 G r0 we denote by op the linear transformation in %v

defined by the formula (o$x\ = xy for 7 =£ 0, {O$X)Q - - Xp + 2 a:v(/>5

where 7(0 is the end-point of the edge / other than 0.We denote by W the semigroup of transformations of £r generated by the

°P $ e r° )•L E M M A 2 . 1 . 1 ) If a , P e r 0 , a ^ j3, r / i e « < a a > = 1 fl«cf 2 ( a j S ) i s

//ze negative of the number of edges joining a and p. 2) Let 0 G Po. 7%e«ffpOO = x - 2 <0, x>0, ag = 1. /« particular, W is a group. 3) T/ze ^rowp W

preserves the integral lattice in %? and preserves the quadratic form B. 4) / /the form B is positive definite {that is, B(x) > 0 for x =£ 0), then thegroup W is finite.

PROOF. 1), 2) and 3) are verified immediately; 4) follows from 3).For the proof of Gabriel's theorem the case where B is positive definite

is interesting.PROPOSITION 2.1. The form B is positive definite for the graphs An,

Dn, E6, Zs7, E* and only for them (see [7], Chap. VI).We give an outline of the proof of this proposition.1. If F contains a subgraph of the form

I f f I f f

2 2 2 2then the form B is not positive definite, because when we complete thenumbers at the vertices in Fig. (*) by zeros, we obtain a vector x 6 ^ r

for which B(x) < 0. Hence, if B is positive definite, then F has the form

(**) Zr Zr-f Zr-2 Zj Z2 Z{

Zj xz Xj x^Xj, a fy y^ % % yf

where p, q, r are non-negative integers.2 For each non-negative integer p we consider the quadratic form in

(p + 1) variables xlf . . ., xp + l

Cp


This form is non-negative definite, and the dimension of its null space is 1.Moreover, any vector x =£ 0 for which Cp{x) = 0 has all its coordinatesnon-zero.

To prove these facts it is sufficient to rewrite Cp{x) in the form

3. We place the numbers xl9 . . ., xp, yl9 . . . , yq, z l5 . . . , zr a at thevertices of r in accordance with Fig. (**). Then

B(xt, y u zt, a) = Cp(xlt ...,xp, a) + Cq(y{, ...,yq, a) +

2{p + l)

Hence it is clear that B is positive definite if and only if

+ + 2(r+l) < l j t h a t ^ + +4. We may suppose that p < q < r. We examine possihle cases.

a) p = 0, q and r arbitrary. A= -|—XJ-+ I > 1> that is, B is

positive definite (series An).b) p = \, q - \, r arbitrary. A > 1 (series Dn),c) p = 1, q = 2, r = 2, 3, 4. ^ > 1 (£*, £7, £8),d) p = 1, <? = 2, r > 5. ^ < 1,

p = 1, q = 3, r > 3. 4 < 1,p > 2,^r > 2 , r > 2. >4 < 1.

Thus i? is positive definite for the graphs An, Dn, E6, Elf Eg and onlyfor them.

DEFINITION 2.2 A vector x E %v is called a roo/1 if for some 0 G r 0 ,w G H / w e have x = w^. The vectors j8 (j3 G r 0 ) are called simple roots. Aroot x is called positive if x > 0 ( see Definition 2.1).

LEMMA 2.2 \) If x is a root, then x is an integral vector and B{x) = 1.2) If x is a root, then (~x) is a root. 3) If x is a root, then either x> 0 or(-JC) > 0.

PROOF. 1) follows from Lemma 2.1; 2) follows from the fact that

oa (<*) = ~<x f ° r a ^ a E Fo •3) is needed only when 5 is positive definite and we prove it only in

this case.We can write the root x in the form attj oaj • . . . • aafej3, where

a l5 . . . , afc, j3 G r 0 . It is therefore sufficient to show that if y > 0 anda e r 0 , then either a j > O o r y = a (and -oay = + a > 0).

Since ILyll = II aII = 1, we have |<a, y)\ < 1. Moreover, 2<a, .y) G Z.Hence 2<a, >> takes one of the five values 2, l,_0, - 1 , - 2 .

a) 2<a, ^> = 2. Then <a, > ' ) = ! , that is, y = a.


b) 2<a, y) < 0. Then oa(y) = y - 2<a, y) a > 0.c) 2<a, y) = 1. Since 2<a, j>> = 2ya - 2 #Y<J> (Y(0 i s the other end-point

Z 6 r a

of the edge /), we have ya > 0, that is, ya > 1. Hence oay = y - a > 0.This proves Lemma 2.2.DEFINITION 2.3. Let F be a graph without loops, and let ai9 . . ., an

be a numbering of its vertices. An element c = aan • . . . • aai (c depends onthe choice of numbering) of the group W is called a Coxeter transformation.

LEMMA 2.3. Suppose that the form B for the graph F is positive definite:1) the transformation c in %v has non non-zero invariant vectors;2) if x 6 %v, x -^ 0, then for some i the vector clx is not positive.PROOF. 1) Suppose that */€$r> y¥=0 and cy = y. Since the trans-

formations oan, <*<*„_,, • • ., 0<*, do not change the coordinate correspondingto ai (that is, for any z6§r (ffa*z)ai = zai for i=^l), we have(oaiy)otl ~ toOa, = yOtl • Hence aai>> = j> Similarly we can prove thatOa3y

= ^, then oasy = ^, and so on.For all a. E r 0 , a ^ = - 2<a, > )a = >>, that is (a, y) = 0. Since the vectors

a(a G r 0 ) form a basis of $r and B is non-degenerate, >> = ().2) Since W is a finite group, for some h we have ch - 1. If all the

vectors x, ex, . . ., c71""1* are positive, then >> = x + ex + . . . + ch~lx isnon-zero. Hence cy = y, which contradicts 1).

§3. Gabriel's theorem

Let (F, A) be an oriented graph. For each object V e X (F, A) weregard the set of dimensions dim Va as a vector in %v and denote it bydim V.

THEOREM 3.1 (Gabriel [1]). 1) If in X{T, A) there are only finitelymany non-isomorphic indecomposable objects, then F coincides with oneof the graphs An, Dn, E6, £7, Es.

2) Let F be a graph of one of the types An, Dn, E6, E7, Es, and Asome orientation of it. Then in X(T, A) there are only finitely many non-isomprphic indecomposable objects. In addition, the mapping V h* dim Vsets up a one-to-one correspondence between classes of isomorphic inde-composable objects and positive roots in %v-

We start with a proof due to Tits of the first part of the theorem.TITS'S PROOF. Consider the objects (V, f) e X(T, A) with a fixed

dimension dim V = m = (ma).If we fix a basis in each of the spaces J&, then the object (F, f) is

completely defined by the set of matrices Ax (/ € Fj), where A{ is thematrix of the mapping fx\ F a ( 0 -> Vp0). In each space Va we change thebasis by means of a non-singular (ma X ma) matrix ga. Then the matricesA i are replaced by the matrices


Let A be the manifold of all sets of matrices Ax (/ G r1) and G thegroup of all sets of non-singular matrices ga (a G r 0 ) . Then G acts on Aaccording to (*); clearly, two objects of X (I\ A) with given dimension mare isomorphic if and only if the sets of matrices {At} corresponding tothem lie in one orbit of G.

If in %(V, A) there are only finitely many indecomposable objects, thenthere are only finitely many non-isomorphic objects of dimension m.Therefore the manifold A splits into a finite number of orbits of G. Itfollows1 that dim A < dim G - 1 (the -1 is explained by the fact that Ghas a 1-dimensional subgroup Go = {g{k)\k £&*}, g(X)a = X'lv T which

acts on A identically). Clearly, dim G = 5 â* dim ,4= 2

Therefore the condition dim A < dim G - 1 can be rewritten in theform2 B(m) > 0 (if m =£ 0). In addition, it is easy to verify thatB((xa)) > B((\xa\)) for all x = (xa) G g r .

So we have shown that if in X(T, A) there are finitely many indecomposableobjects, then the form B in £r is positive definite.

As we have shown in Proposition 2.1, this holds only for the graphs Ani

Dn, E6, En, E8.

We now prove the second part of Gabriel's theorem.LEMMA 3.1. Suppose that (r, A) is an oriented graph, 0 G r0 a (+)-

accessible vertex with respect to A, #rcd F E < (F, A) an indecomposableobject. Then either Fp(V) is an indecomposable object and dimFfi(V) = a^dim F), or V = L0, FfrV) = 0, dim FfcVy ± a^dim V) < 0.A similar statement holds for a (-)-accessible vertex a and the functor F^.

This lemma is a reformulation of Theorem 1.1.COROLLARY 3.1. Suppose that the sequence of vertices otXi . . ., ak is

{^-accessible with respect to A and that V G X (F, A) is an indecomposableobject. Put Vj = Ft/ij^' • • • -KXV, mj = <fa/*aM- • • • -^(dim V)(0 < / < A:). L^/ / be the last index such that mj > 0 for j < i. Then theVj are indecomposable objects for j < /, and V = F2X • . . . • F^Vj. If i < k,then Vi+l = Vi+2 = . . . = Vk = 0ji = La.+i, V^F^-L.-F^LaJ. Similarstatements are true when (+) is replaced* by (-).

We now show that in the case of a graph P of type An, Dn^ £ 6 j £n Or£"8 (that is, B is positive definite), indecomposable objects correspond topositive roots.

a) Let V e X(T, A) be an indecomposable object.

1 This argument is suitable only for an infinite field K. If K = ¥q is a finite field, we must use the fact thatthe number of non-isomorphic objects of dimension m increases no faster than a polynomial in m, andthe number of orbits of G on the manifold A is not less than C-^^" 1 ^ ~ < d i m G~l).

2 We can clearly restrict ourselves to graphs without loops.

170 /• N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev

We choose a numbering aif a2i . . ., <*„ of the vertices of r such thatfor any edge / e Ti the vertex <*(/) has an index greater than that of 0(/).Let c = aa • . . . • aa be the corresponding Coxeter transformation.

By Lemma 2.3, for some k the vector cfe(dim V) e %? is not positive.If we consider the (+)-accessible sequence 0i, 02 , . . . , 0nfe = (<*!,. . ., an,

« ! , . . . , an, . . . , «i, . . . , <*„)(& times),then we have a ^ - . . . • aPi (dim F)= c^(dim V) > 0. From Corollary 3.1 it follows that there is anindex / < kn (depending only on dim F) such thatV = FJt'Fp% • . . . • F^(L0.+il dim F = oPi* . . . • aPi0M). It follows thatdim F is a positive root and F is determined by the vector dim F.

b) Let x be a positive root.By Lemma 23, ckx > 0 for some &. Consider the (+)-accessible sequence

0i, &, • . •, ftife = (ai> • • •, <*n, • • •, <*i, . . ., an) (k times). Then<jpnk* . . . • o&x(x) = cfe(x) > 0. Let / be the last index for whichopppi-/ • • • * opxW > 0- It is obvious from the proof of 3) in Lemma 2.2that ofii* . . . • afii(x) = 0I+1.

It follows that Corollary 3.1 that F = F p , - • • • 'Fh (Lfiui) € # (r, A) is anan indecomposable object and dim V = o^^ . . . • ap.(fii+i) = x.

This concludes the proof of Gabriel's theorem.NOTE 1. When B is positive definite, the set of roots coincides with the

set of integral vectors x e i r for which B(x) = 1 (this is easy to see fromLemma 2.3 and the proof of Lemma 2.2).

NOTE 2. It is interesting to consider categories X(T, A), for which thecanonical form of an object of dimension m depends on fewer than C*\m\2

parameters (here |m| = 2 |ma|, a G r 0 ) . From the proof it is obvious thatfor this it is necessary that B should be non-negative definite.

As in Proposition 2.1 we can show that B is non-negative definite for thegraphs An9 Dn, E6, Ely E8 and Ao, An, Dn, E6, Elf £ 8 , where

(n + 1 vertices, n > 1)

(n + 1 vertices, n > 4)


A A / \ /S ^ .

(the graphs An, Dn, E6, El9 E8 are extensions of the Dynkin diagrams(see [7])).

In a recent article Nazarova has given a classification of indecomposableobjects for these graphs. In addition, she has shown there that such aclassification for the remaining graphs would contain a classification of pairsof non-commuting operators (that is, in a certain sense it is impossible togive such a classification).

§4. Some open questions

Let F be a finite connected graph without loops and A an orientation of it.CONJECTURES. 1) Suppose that x G %? is an integral vector, x > 0,

B(x) > 0 and x is not a root. Then any object V G %(T, A) for whichdim V = x is decomposable.

2) If x is a positive root, then there is exactly one (to within isomorphism)indecomposable object V G X(Y, A), for which dim V - x.

3) If V is an indecomposable object in X(T, A) and i?(dim V) < 0, thenthere are infinitely many non-isomorphic indecomposable objectsV d X(T, A) with dim V = dim V (we suppose that K is an infinite field).

4) If A and A' are two orientations of r and V G X(T, A')[s anindecomposable object, then there is an indecomposable objectV G X{T, A') such that dim V = dim V.

We illustrate this conjecture by the example of the graph (F, A)

ct,

(quadruple of subspaces).F o r each x G %T we p u t p(x) = - 2 < a 0 , x) (if x = (xOi x l 9 x 2 , x 3 y x 4 ) ,

then p ( x ) = Xi + x 2 + x 3 + x 4 - 2 x 0 ) .In [2] all the indecomposable objects in the category X{Y, A) are

described. They are of the following types.1. Irregular indecomposable objects (see the end of §1). Such objects

are in one-to-one correspondence with positive roots x for which p(x) =£ 0.2. Regular indecomposable objects V for which #(dim V) # 0. These

objects are in one-to-one correspondence with positive roots x for whichP(x) = 0.

3) Regular objects V for which i?(dim.F) = 0. In this case dim V hasthe form dim V - (2«, n, n, n, n), p(dim V) = 0. Indecomposable objectswith fixed dimension m = (2n, n, n, n, n) depend on one parameter. Ifm G %v is an integral vector such that m > 0 and B{m) = 0, then it hasthe form m = (2n, n, n, n, n) (n > 0) and there are indecomposableobjects V for which dim V - m.

If / is a linear transformation in ^-dimensional space consisting o f one


Jordan block then the quadruple of subspaces corresponding to it (see theIntroduction) is a quadruple of the third type.

References

[1] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71-103.[2] I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of

quadruples of subspaces in a finite-dimensional vector space, Colloquia MathematicaSocietatis Ianos Bolyai, 5, Hilbert space operators, Tihany (Hungary), 1970,163—237(in English). (For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765= Soviet Math. Doklady 12 (1971), 535-539.)

[3] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in thecollection "Investigations in the theory of representations", Izdat. Nauka, Leningrad1972,5-31.

[4] I. M Gelfand, The cohomology of infinite-dimensional Lie algebras. Actes CongrSsInternat. Math. Nice 1970, vol. 1. (1970), 95-111 (in English).

[5] I. M. Gelfand and V. A. Ponorarev, Indecomposable representations of the Lorentzgroup, Uspekhi Mat. Nauk 23: 2 (1968), 3-60, MR 37 # 5325.= Russian Math. Surveys 23: 2 (1968), 1-58.

[6] C. W. Curtis and I. Reiner, Representation theory of finite groups and associativealgebras, Interscience, New York-London 1962, MR 26 #2519.Translation: Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Izdat.Nauka, Moscow 1969.

[7] N. Bourbaki, Elements de mathematique, XXVI, Groupes et algebres de Lie, Hermann& Co., Paris 1960, MR 24 # A2641.Translation: Gruppy ialgebry Li, Izdat. Mir, Moscow 1972.

Received by the Editors, 18 December 1972.

Translated by E. J. F. Primrose.

Dedicated to the memory ofIvan Georgievich Petrovskii

FREE MODULAR LATTICES ANDTHEIR REPRESENTATIONS

I. M. Gel'fand and V. A. Ponomarev

Let I be a modular lattice, and V a finite-dimensional vector space over a field k. A representationof L in V is a morphism from L into the lattice J£(V) of all subspaces of V. In this paper we studyrepresentations of finitely generated free modular lattices Dr.

An element a of a lattice L is called perfect if for every indecomposable representationp: L -+• %(kn) the subspace p(a) of V = kn is such that p(a) = V or p(a) = 0. We construct andstudy certain important sublattices of Dr, called "cubicles." All elements of the cubicles are perfect.

There are indecomposable representations connected with the cubicles. It will be shown that almostall these representations, except the elementary ones, have the important property of completeirreducibility; here a representation p of L is called completely irreducible if the sublatticep(L) cz J£(kn) is isomorphic to the lattice P(Q, n - 1) of linear submanifolds of projective space overthe field Q of rational numbers.

Contents

§1. Definitions and statement of results 173§2. The category of representations 180§3. Construction and elementary properties of the cubicles B+(l) and

B'(l) 184§4. Representations of the first upper cubicle 190§5. The functors ®+ and <D- 195§6. Proof of the theorem on perfect elements 204§7. The subspaces p(ea) and the maps q>f 213§8. Complete irreducibility of the representations pM 224References 228

§ 1. Definitions and statement of results

1.1. Lattices. A lattice L is a set with two operations: intersection andsum. If a, b G L, we denote their intersection1 by ab and their sum by

The intersection of elements a and b of L is often denoted a n b. We use the notation aft to avoidclumsy formulae. For a + b the notation 0 u & or a V b, is also used fairly frequently.

nWe denote the sum of the elements a,,..., an by V # . and their intersection

a, n j 2 n . . . n an by f] fli- i = = i

173

X74 I- M. Gel'fand and V. A. Ponomarev

a+ b. Each of these operations is commutative and associative. Moreover,for any a, b G L we have the identities

aa = a, a -\- a — a, a(a + b) = a, a -{- ab = a.

An order relation is defined in a lattice L by

(a^ b) <^> (ab = a).

If a, 6, c are arbitrary elements of a lattice L, then it is easy to showthat a(b + c) ^ ab + ac.

A lattice £ is called distributive if for any a, b, c & L

a(b + c) = ab + dc and a + be = (a + b)(a + c).

A lattice L is called modular (or Dedekind) if for any a, b, c £ L suchthat a C fcf

&(a + c) = a + &c.

This relation is called Dedekind's axiom.EXAMPLE 1. Let V be a finite-dimensional vector space over a field

#, and F = kn. The set of all subspaces of V is a lattice in which £F isthe intersection of the subspaces E and F, and £" + F is their sum, that is,(E + J1) == (^ + y I r 6 E, y e F). We denote this lattice by ^ (V) oxX (kn).It is well-known that for n > 1 the lattice <£ (/c77) is modular, but notdistributive.

Let P be the projective space generated by V = kn. Then the latticeX (kn) is isomorphic to the lattice of linear submanifolds of P. We denotethe latter lattice by P(fc, n - 1) or P(fc, F), and call it the projectivegeometry over k.

EXAMPLE 2. Let M be an arbitrary module over a commutative ring A.Then the set of all submodules of M is a modular lattice under theoperations of intersection and sum.

1.2. Basic definitions. Let I be a modular lattice, and F a finite-dimensionalvector space over a field k. A representation of L in F is a morphism fromL into the lattice X(V). Thus, a representation p: L-*X(V) associateswith each element x G X a subspace p(#) ^ V such that for all x, y G I

p(;ry) = p(z)p(y) and p(a: + */) = p Or) + p(y).Let Pi and p2 be representations of a lattice L in spaces Vl and F2,

respectively. We set p(x) = px(^) 0 p2Oz) for every x G X , wherepiW ® p2(#) is the subspace of Vx 0 F2 consisting of all pairs (£, 17) suchthat I G PiW and rj 6 p2( )« ^ is not hard to show that this defines arepresentation 0 in the space V - Vx 0 F2. This representation is called thedirect sum of px and p2 and is denoted by p = px 0 p2-

A representation p is decomposable if it is isomorphic to the direct sumpi 0 p2 of two non-zero representations pj and p2. It is easy to see that arepresentation p in a space F is decomposable if and only if there exist

Free modular lattices and their representations 175

subspaces Ux and U2 such that Ux U2 = 0 and U{ + U2 ~ K, and thatp(a) = f/ip(a) + £/2p(a) f° r every a E L.

DEFINITION. An element a of a modular lattice Z is called perfect iffor every field k and every representation p: L-+X (V) — £ (h*1) the sub-space p(a) gz y has the following property: there is a subspace £/ comple-mentary to p(a) (that is, Un{a) = 0 and £/ + p(o) ~ T7) such that thesubspaces £/ and p(a) define a decomposition of p into the direct sum ofsubrepresentations, that is, p(x) — Up(x) + p(&)p(.z) for every x E Z.

It is easy to check that this definition is equivalent to the following:an element a E L is called perfect if, for every indecomposable representationp\ L-+X (kn) with K = fc", either p(a) = F o r p(d) = 0.

Two elements 0 and & of a modular lattice L are called linearly equivalentif p(a) = p(b) in every representation p:L-*-£ (kn) for any fc and n. Inthis case we write a = b. It can be shown that if L is the free modularlattice with 4 generators, then in L there are unequal, but linearly equivalent,elements.

Such examples are of interest to us in connection with the followingproblem. A modular lattice L is called linear if for any x, y E L and everyrepresentation p: L -> £ (kn) we have x = y if and only if p(^) = p(y).This leads to the following question: can a linear lattice be characterized byadding to the axioms for a modular lattice finitely many identities?

1.3. Cubicles in the lattice Dr. In this paper we study representations ofthe free modular lattice Dr with r generators eu . . . , er.

The key idea in this paper is the construction of an important sublatticeB of Dr whose elements are all perfect, for each integer / > 1 we constructsublattices B+(l) and B~(l), each consisting of 2r perfect elements. We callB+(l) the l-th upper cubicle and B~(l) the l-th tower cubicle.

It is quite simple to define the upper cubicle B+(\). We set

hi (1) = 2 eJ> The sublattice of Dr generated by the elements

fti(l), . . . , hr{\) is then the upper cubicle ^+(1). We shall prove that1) i?+(l) is a Boolean algebra with 2r elements (see §3), and 2) everyelement x E B+(\) is perfect (see §4). The lattice # + ( l ) is, thus, isomorphicto the lattice of vertices of an r-dimensional cube with the natural ordering.The element ht{\) corresponds to the point (1, . . . , 1, 0, 1* . , , , 1) with0 in the z-th place.

Now we construct the lattice B+(l) with / > 1. The elements of thecubicle B+(l) are "constructed" from certain important polynomials1)eti . . .fr which are of independent interest. We proceed to define thesepolynomials.

Let / > 1 and / = {1, . . ., r}. We denote by Air, I) the set whoseelements are sequences of integers a = (il9 , . „ , /,) with iK E / such that

' The elements of Dr are also called lattice polynomials, or simply polynomials.

176 /. M. GeVfandand V. A. Ponomarev

i\ ^ h> h ^ h> - • - > h-i ^ h- *n Particular, A(r, 1) = /. For fixedoc G Air, I) we construct a set F(a) consisting of elements f$ E A(r, I - 1)in the following way:r(a) = {p = fa, . . ., * M ) 6 4(r , Z - 1) | &x $ {flf * ,} , * t £ {j2, * 3 } , . . .

Note that kx ¥= k2i k2 ¥* k3, . . . , kx_2 =£ kx_x, because 0 G ,40, / - 1).With each a G A(r, I) we now associate an element ea G D ' by the follow-ing rule. Let / = 1 and o: = (/j). We set ea = et . For 1=2 and a = (/j, i2)with /j ^ /2

w e s e t

In general, for arbitrary / and a G A(r, I) we set by induction(1.1) ea = e{ i =ek 2 ep.

Now we introduce the elements ht(l), from which we construct the /-thcubicle, just as B+(l) was constructed from the elements ht{\). We denoteby At(r, 1) the subset of Air, I) consisting of all a = (ii9 . . . , //_!, f) whoselast index is fixed and equal to Z\ We set

(1.2) M0 S

(1.3) ht(l ^

The sublattice of Z)r generated by /*!(/), . . . , /?r(/) is called the l-th uppercubicle B+(l).

It is fairly elementary to prove (see §3) that for every / > 1 the sub-lattice B+il) is a Boolean algebra. It is vastly more complicated to provethat the number of elements in B+(l) is equal to 2r and that every elementx G £+(/) is perfect (§§4-7) . We shall also prove (§3) that the elements ofany cubicles B+(l) and B+im) can be ordered in the following way: for everyxt G B+(l) and every ym G B+(m), if / < m, then xx D xm . It follows fromthis that the collection of elements of all of the cubicles B+il) is also alattice, which we denote by B*.

We denote by B~(l) the sublattice of If dual1) to B+il). The sublatticeB~{1) is called the l-th lower cubicle. Apparently, the following is true.

) A lattice polynomial^*, , . . . , jcr) in the variables x , , . . . , xr is dual to a polynomial/(x,,. . . , xr) ifg is obtained f rom/by changing the operation of intersection into addition and addition into inter-section. For example, the polynomials x, (x2 + x3 + . . . + JCP) and x, + X2JC3 . . . xr are dual to eachother. Let a, and a2 be elements of Dr, and a-x = /,.(<?,,. . . , er) for i = 1, 2 be lattice polynomials. Theelementsal anda2 are dual if/2 =£, ( e , , . . . , er), whereg x (e l t . . . , er) is the polynomial dual to/ , ( e , , . . . , e ). We say that £"(/) is dual to B+(l) if B'(l) is the set of all elements dual to the elements of


PROPOSITION. Let x+ G B+(1) and y~ G B~{m). Then y~ C x+ for everyI and m.

We are able to prove this proposition only up to linear equivalence, thatis, for every representation p: Dr -> %(kn) and any x+ G B+(l) andy~ G B~(m) we have p(y~) s p(x+).

MAIN THEOREM. The elements of the sublattices B+(l) and B~{1) areperfect.

CONJECTURE. Let a be a perfect element in Dr. Then there exists anI > 1 such that either a G B+(l) or a G B~(l).

REMARK 1. The elements ea = eii... ^ used to construct the perfectelements in B+(l) are of considerable independent interest. Below, in §1.4,we construct completely irreducible representationspt, i (t = 1, . . ., r; Z = 1, 2, . . .). For these representations the imagesPt, i(ea) of the elements ea G D r are one-dimensional subspaces of Vtl.These pt, u with £ 6 {1» • • • > r}, are called completely irreducible repres-entations of the first kind.

REMARK 2. We shall also construct completely irreducible representationspOtl. of the second kind. In these representations the elementsea, oc G At(r, /), are replaced by elements ft i0 GZ) r , where iv G / andiv =£ zy+1. The elements ft / 0 are "constructed" in the following manner.The set F(a) = T(il, . . . , / M , 0) C ^(r , / - 1) consists in this case of allelements 0 = {ku . . . , A:^!) G ^(r , / - 1) such that kx £ {h, i2},k2f{i2, iH}, . . ., /c/_2 ^{ i / - 2 , ^/-i}» ^;-i $={£M}- We note that all kx in]3 are different from zero and therefore e& is defined by (1.1). We set

fa=:fi1...il_iQ--=eii ]V. gp. For example fi1o = €ii 2J e .

By analogy to the elements er(/), t G / , we set /o(0 = 2 / a » wherea

a = (/1} . . . , / ^ j , 0), /y G /, and the summation is over all such a.We shall prove later that / 0 ( 0 is linearly equivalent to the smallest element

of B+(l - 1), that is, fo(l) ~ f] ht(l — 1). Apparently, in Dr we haver l

/o(O= .D î(^ — 1). for every / > 2. However, we can prove this only

for / = 2.1.4. Representations of the first and second kind. We denote by p(L)

the sublattice of %{kn) consisting of all elements p(a), a ^ L.DEFINITION. A representation p of a modular lattice I in a space V

over a field k of characteristic zero is called completely irreducible ifp(L) ^ P(Q, m), where Q is the field of rational numbers andm = (dim^. V) — 1.

We first construct the representations pi, i of the first kind in thespaces Vt th t 6 {!> • • •» r}> J = 1, 2, . . . Clearly, a representation p ofDr in a space K is completely determined by the subspaces

178 /. M. Gel'fandand V. A. Ponomarev

p(ej) (/ = 1» • • •» r) o f v- F o r brevity we set pt, i(ej) = Eu t s V<f | .Let / = 1. Then p*,i, £ 6 {1, . • ., r} , is the representation in the one-

dimensional space F = k' such that pM(e;-) = 0 if / =£ r and P M ( ^ ) = F.Now let / > 1. We denote by Wtl the linear vector space over k with

the basis {r}a}, where a = (/j, . . . , /,_!, f) ranges over the whole setAt(r, I) {a E At(r, I) «-> a = (z l5 . . . , il_l, t) and t is fixed).

We denote by Zt l the subspace of Wt l spanned by all possible vectors

Sa.h = 2 ' t\it ..i fc...i, Au where 1 < k < / - 1 and y ' is summation over

those a = (/j, . . . , z^, . . . , //_!, 0 in which iu . . . , zjt_1, ik + l , . . . , //_!are fixed. Next, we set Vtl - Wtl\Ztl. The images of the vectors r?ain thefactor space Vtl are denoted by £a . Thus, Vtl is the vector space over kspanned by the vectors £a for which

. . . i , .1 = 0 for every k.

By Ej t we denote the subspace of Vt t spanned by all vectors £a suchthat a = (il9i2, • • • > ^- i» 0 = (/, «2> • • • » *'z-i> 0 (where the index z't = /is fixed). We define a representation p*. i in F r z by setting pt,i(ej) = ^y,/.

Now we define the representations po,i of the second kind(/ = 1, 2, . . . ).

For / = 1, Po.z is the representation in the one-dimensional spaceV = k' such that po.ife) = 0 for all i = 1, . . . , r.

For / > 1 we define a set A0(r, I) in the following way:

Ao(r, I) = {a = (iit . . ., it-v 0) | iK 6 J = {1, . . ., r} , ix ^ i2,

Clearly, ^ 0 ( ^ 0 — (r» ^ ~ 0- The representation p0, t is constructed onA0(r, I) in the same way as pt,i is constructed on At{r, I). Namely, wedenote by Wo l the vector space over k with the basis {T]a}, wherea E A0(r, I). Further, let ZOj be the subspace of Wol spanned by all

possible vectors ga,h = S ^- • -'w • -Vi0- We denote by Vo t the factor space

WOJIZOJ and by £a the image of r?a under the canonical mapWOj -• F o z. The subspace of F o ; spanned by all vectors £a witha = 0i> 2 • • • > ?/-i 5 0) = 0"> ?25 • • • 5 //__i, 0) is denoted by Ej 0 . We definea representation pOt z in F o / by setting p0, /(e;-) = Ejt0.

REMARK. Our definition of the representations pt, i of Dr makes sensefor all r > 1. It is known [2] that the lattices Dl, D2, D3 are finite,and each of these lattices has only finitely many non-isomorphicindecomposable representations. It can be shown that the number of suchrepresentations is 2, 4, and 9, respectively, and that they coincide withrepresentations pt, i, I ^ r. It can be shown by direct computation thatfor r 6 {1, 2, 3} and / > r the space Vtl is equal to 0, hence, the corres-ponding representation pt, i = 0.


The second main result of this paper is the following theorem.THEOREM 1.1. (I) For every t £ {0, 1, . . ., r) and all r, I > 1 the

representation pt,i' Dr-*X{k, Vttl) is indecomposable.(II) / / the characteristic of k is 0, if r > 3, / > 1, and if

(f, r, Z)£{(1, 4, 2), (2, 4, 2), (3, 4, 2), (4, 4, 2)}, r/tert the representationPt, i' Dr-+%(k, Vt, i) is completely irreducible.

Other properties of the representations pt, i are described by thefollowing propositions.

PROPOSITION 1.1. Let pt,i be a representation of the first kind(t (i {1, . . ., r}) in the space Vt x. We denote by vt l the elementfl ht(l) ofB+(l).

(I) For every element x G 5 + = U B+(s) such that x D vt j we have

pt,i(x) = Vt, i. In particular, this equality holds for all x E B+(m), m < I.(II) For every element y £ B+ such that y C ht(l) we have pt, i(y) = 0.

In particular, this equality holds for all y E B+(n), n > /.(III) For every a = (il9 . . . , il_l, t) E At{r, I) we have pt, i{ea) = k(la),

where k(%a) is the one-dimensional subspace of Vt z spanned by the vector

PROPOSITION 1.2. Let po,i be a representation of the second kindin the space Vo> t. We denote by vQ, ^ the element C\ ht(l — \) of B+(l~ 1).

(I) For every element x E B* such that x D vd !_{ we havepOi j(x) =V0, i. This means that p0, i(x) = Vo, z for x E B+(m), m I.

We shall now briefly describe the representations p7, i associated withthe lower cubicles B~(l).

DEFINITION. Let p be a representation of a modular lattice L in aspace V over a field k. We denote by V* the space dual to V. A repres-entation p* in F* is called dual to p if p*(x) = (p^))^ for every x G L, where(p(x))^ is the subspace of functional in V* that vanish on p(x).

We set pjf i = (pt, i)*' Thus, pjt i is the representation in Vf t such thatP7. i(et) = (pt, liei))1- for all / = 1 , ' . . . , r.

We do not describe in detail the properties of the representations pjy z,because they are dual to those of Theorem 1.1 and Propositions 1.1 and1.2.

1.5. As we have already mentioned, the p/, / describe all the indecompos-able representations of the lattices D1, D2, D3. For the lattices Dr withr > 4 this is not the case. We describe below how to split off from anarbitrary representation p of Dr, r > 4, indecomposable representationsPt, i and p7? z. By v+

Q t = vQ j we denote the smallest element of the cubicleB+(l). It follows easily from the definition of B\l) that

rve,l = n hi(0> where ht(l) is defined by (1.3). It can be shown/ = l

180 /• M. Gel'fandand V. A. Ponomarev

(see §3) that v+dl D i y 2 D • • • D v*eJ D • - \ Dual to the element

v+e,i ^ Dr is ujp which is the largest element in B~{1). Thenr

*>/, i = S &F(0> where hJ(J) is polynomial dual to ht(l). The elementsujz also form a chain i> x C uj 2 C • • • C u/-/ C • • • .

Let p be any representation of Dr. We write p(i>e, z) = Ve,i a nd

p(vjti) = Fj, i. We define F^ ^ = O Fa+

; and Fj ^ = U Vjv

We shall prove (§6) that Vj „ C K5fOO. Thus, the subspaces Fj-, andV*, form a chain

(7 ,1

THEOREM 1.2. Le/ p be a representation of Dr, r > 4, m a 5/?ace F,a«c? /ef 0 C ^"^ C V*^C V, that is, all the terms of this chain aredistinct. Then p is decomposable into a direct sum p = p" 0 px 0 p+,where p~ = p |v- is the restriction of p to the subspace Vf^ and

(p" 0 P0 = P lvjf00-/ / the representations p~ and p+ are decomposable, then p+ ^ 0 pt, i

and p~ & 0 p7, h where t 6 {0, 1, . . ., r) (Z = 1, 2, . . .)• //a&o r/ie

representation p% decomposes into a direct sum Px = 0 T , 5 o/ indecom-X, 8

posable representations rx s, then among the rx s there are no representationsisomorphic to pt, i and pT, i-

We shall prove this theorem in §6. For the lattice Z)4 the representationsp^were studied in [5], where they were called regular. The indecomposableregular representations of D4 are completely classified. For a specificationof these indecomposable representations one needs not only discrete invariants(of the type of a dimension), but also continuous parameters (analogous tothe eigenvalues of a linear transformation).

Very little is known about the regular representations of the latticesDr (r > 5). It is clear (see [5], [7]) only that the classification problem(up to similarity) of an arbitrary set of linear transformationsAi, . . . , An (n > 2), Ai : F -> F, reduces to a special case of theclassification problem of indecomposable regular representations ofDr (r > 5).

§2. The category of representations

In this section, which is of an auxiliary role, we describe some elementaryproperties of the category of representations. Also, we introduce the notion ofan admissible subspace, and we formulate a simple criterion for decomposabilityof representations.

2.1. The category ^?(£, k). Let px and p2 be representations of a modularlattice L in finite-dimensional spaces Vx and F2 over one and the same field


k. A morphism u: pt -> p2 is a linear mapping u: Vx -> F2 such thatupxix) s P2W f°r every x E Z,, where up^x) is the image of the subspacePi(#)- When there is no ambiguity, we denote a morphism u: Pi->• P2 byu: pi-^p2-

We denote by Hom(p!, p2) the set of all morphism from pi to p2. Notethat Horn (p1? p2) is a vector space over k.

It is not hard to verify that we now have a category °rt(L, k) , that offinite-dimensional representations of L over k.

REMARK. Let u: p] ->• p2 be a morphism in 3?(L, A;) and w: Fj -• F2

the corresponding linear transformation. It is not true that the set of allsubspaces up^x), x£L, defines a representation of L in F2. If x, y £ L,then upifc + y) = w(pi(aO + Pi(*/)) = upi(x) + up^y). However, in general,

up^xy) = uipiixfaiy)) =^ ("Pi(«))("Pi(y))« W e c a n o n ly a s s e r t t h a t

( (For any two objects px and p2 in ^(L, /c) there is the direct sum

p! 0 p2. Namely, let pi> p2 6 ^(L, A:) be representations in spaces Fj andV2. We set K = F! © F2. For every x £ L we definepW = PiW © P2W ^ Fx 0 F2. It can be shown that this defines arepresentation p in Vx © F2.

It is not hard to check that fi{L, k) is an additive category.We now return to the category M{Dr

i k) of representations of a freemodular lattice Dr. For brevity, we denote ^(D11, k) by # .

It is easy to show fl is additive, and that every morphism u: px ->• p2

has a kernel and a cokernel,1) that is, the category >? is pre-Abelian. How-ever, it is not Abelian, because the canonical mapping2) Coim u -> Im uis not an isomorphism for an arbitrary morphism u .

EXAMPLE. Let 91 = #(D3, A:). We define representations px and p, inspaces Vx and F2 in the following way:

V1 ^ F2 ^ /J1,

Pi(«i) = Pi(«2> = V\> Pi ( 3) = 0, pofe) = p2(e2) = p2(e3) = F2.Next let u: Vx -> F2 be any isomorphism (in the category of linear spaces),and let u: px -> p2 be the morphism corresponding to the mapping u. It isnot hard to check that Ker u = 0 and Coker u = 0. Consequently, in thecanonical decomposition px -> Coim u -> Im iT -> p2 we have

^ We recall that a fern*/ of a morphism M: px -»- p2 is a subobject fi: p ' - • p2 of px such that forevery T of R there is an exact sequence of vector spaces0 -* Horn (r, p') -> Horn (T, p^-* Horn (r, p2). In other words, /T is a monomorphism such that ifu w = 0 with u )e Horn (r, p , ) , then there exists a morphism M)' e Horn (r,p') such that u; - jH w'.The cokernel of a morphism u: pl -* p2 is a factor object TT: p2 -»• p " such that for every object T of/? there is an exact sequence of vector spaces Horn (PX,T) «- Horn (p2,r)+- Horn (p", T) •- 0.

' We recall that Im u is the kernel of the cokernel of u, and Coim u the cokernel of the kernel of u.

182 I' M. Gel'fandand V. A. Ponomarev

Pi ^ Coim u and p2 ^ Im u. Therefore, the mapping Coim u -> Im u isnot an isomorphism.

2.2. Decomposable representations and admissible subspaces.DEFINITION. Let p be a representation of a modular lattice L in a linearspace V. A subspace U of Fis admissible relative to p if for any x, y E Lone of the following conditions is satisfied:

(I) U(p(x) + p(y)) = Up(x) + Up(y);

(II) U + p{x)p(y) = (U + p(x))(U + p(y)).

The equivalence of (I) and (II) follows from a more general statement.LEMMA 2.1. Let L be a modular lattice and let x, y, z E L. The sub-

lattice generated by x, yy z is distributive if one of the following conditionsis satisfied: 1) z(x + y) = zx + zy and 2) z + xy = (z + x) (z + y).

A proof of this assertion can be found, for example, in Birkhoff [2].Three elements x, y, z of a modular lattice L satisfying the conditions ofLemma 2.1 are called a distributive triple. It follows from Lemma 2.1 thatthe following equalities are equivalent:

z(x + y) = zx + zy, x(y + z) = xy + #z, #(z + z) = i/a: + z/z, z + ^ =(z + z)(z + y), x + yz = (x + y){x + z), y + xz = (y + .r)(z/ + z).

PROPOSITION 2.1. Le? p be a representation of a lattice L in a spaceV. Let U be a subspace of V and let U" - V/U the factor space. Let6 : V -> U" be the canonical mapping. Then the following conditions areequivalent:

1°. The subspace U is admissible relative to p.2°. The correspondence x -> Up(x) defines a representation in U.3°. The correspondence x H-> dp(x) defines a representation in U".PROOF. Let us show, say, that 2° follows from 1°. Let U be an admiss-

ible subspace. Then Up(z + y) = U(p(x) + p(y)) = Up(x) + Up(y). Moreover,Up(xy) = Up(x)p(y) = (Up(x))(Up(y)). Consequently, the rule x *-+ Up(x)defines a representation in U. This representation is called admissible; it isalso called the restriction of p to U and is denoted by pu or p 1 . Theproofs of the remaining parts of the proposition are elementary.

The representation in V/U, where U is an admissible subspace, is calledan admissible factor representation.

We say that a representation p 6 M{L, k) is decomposable if it is iso-n

morphic to a direct sum 0 p., n > 2, of representations pi =£0.i = l

PROPOSITION 2.2. A representation p 6 3^{L, k) in V is decomposableif and only if there exist non-zero subspaces Uu . . . , Un such thatV = Ux © • • * © Un, and if for every x E L

(2.1) P(*)=S Utp{x).


PROOF. The necessity of (2.1) is clear. To prove the sufficiency weshow that every subspace Uf is admissible, that isUj(p(x) + p(y)) = Ujp(x) + Uj9(y) for any x, y € L.

By (2.1) we have p(x) + p(y) = % Utp(x) + 2 Utp(y). Using Dedekind'saxiom, we find t = 1 l==1

Uj (p (x) + p (y)) = Uj(j] (Utp (x) + Utp (y))) =

= U, (Uj9 (X) + UiP (y) + S (Utp (x) + UiP (y))) =

= Ujp (X) + Ujp (y) + Uj S (Utp (x) + L p (y)).

Note that C/; ( 2 ^/P W + k7;P (y)) s ^ ; S Ut = 0. Consequently,

()This proves that every subspace U}- is admissible and means that the

correspondence x H-> Ujp(x) defines a subrepresentation p^' in Uj. It is

easy to check that P = ® p u s so that p is decomposable.

We now assume that V = Ux ® * * * © Un, and that each of the subspacesUt is admissible relative to p. The following example shows that we cannot,in general, assert that p is equal to the sum of its restrictions pUi.

EXAMPLE. Let D2 be the free modular lattice with two generators. (Notethat D2 consists of four elements el9 e2, exe2, ex + e2.) Let V denote the8-dimensional vector space over a field k with the basis £l5 . . . , £8. Wedefine a representation p of D2 in V in the following way. We set

pta) = kit + kib + k(i2 + y , p(e2) = ki3 + ki7 + HI, + y ,where k%t and k(^- + £z) are the one-dimensional subspaces spanned by the

4 8

vectors £,- and £• + ^, respectively. Let U1 = 2 &£; anc^ ^2 = S ^?f-i= l i=5

It is easy to check that each of the subspaces Ut is admissible relative to

p,but that p ^pUi + pU2-In conclusion of this section we state a simple criterion for the decom-

posability of a representation p of Ef.PROPOSITION 2.3. Let p be a representation of Drin a space V. Then

np is decomposable into a direct sum p = © pi of representations pi if and

i= lonly if there exist non-zero subspaces Ul9 . . . , Un with V = 0 Ut such

n i=l

that p(et) = ^ p(et)Ui for every t 6 {!> • • .» }» where the et are

generators of Dr.


§3. Construction and elementary properties of the cubicles^+(/) andi?~(/)

3.1. Definition of the cubicles B\l) and B~{1). The sublattices B\l) andB~(l) were defined in §1. To make the text independent, we repeat herethe definition and introduce some notation.

The sublattice of Dr generated by the elements ht (1) = 2 ei *s' bydefinition, the upper cubicle B+(\). Hi

We define the sublattices B*(l) for / > 2. First we construct importantpolynomials et ir Let / = {1, . . ., r}. We set A(r, 1) = /, and letA(r, I) be the subset of all sequences {iu . . . , /,), ix E /, withh ^ *2> h ^ h> • • > t n a t is, ( ^ Z) = {a = (it1 . . ., ii)\i£l andVXifc, =^^+1}.From an element a. = (il9 . . . , zz), / > 2, we construct a setF(a) C y4(r, / - 1) in the following way:

Note that from 0 = (fclf . . . , k^) e A(r, / - 1) it follows thatkx =£ k2, k2 ^ k3, . . . , kx_2 ^ kx_x. The polynomials ea = ei t ared e f i n e d b y i n d u c t i o n o n / :

if a = ( / j ) G ,4 ( r , 1) , t h e n ea = e(i)= eix ;

if a = ( / l 5 / 2 ) G y l ( r , 2 ) , t h e n ^ a = e{i = eti 2 *e;1 2 P£r(a)

if a = ( / j , . . . , it) G yl(r, / ) , then <?a = ^ . . . . = e, YJ e3.1 J P£r(a)

E X A M P L E . L e t a = ( i l 9 / 2 ) ; t h e n T ( a ) = {ft = {kx) \kl£{i1, j 2 } } , . t h a ti s ,

ea = eiii2=eu S «p = î1 S eJ-PET(a) ^ h ^

Now we define generators ht(l) (t = 1, . . . , r) of i?+(/). We setAt(r, I) = {a = (iu • • ., *M» 0 I a 6 (r» 0. ^ is fixed },

r, Z) i=^t

The sublattice of D1" generated by h^l), . . . , /zr(/) is denoted by B\l) andis called the /-th upper cubicle. The sublattice dual to B+(l) is denoted byB~(l) and is called the /-th lower cubicle.

3.2. A structural lemma and its consequences. In this section we provethat B+(l) and B~(l) are Boolean algebras.

LEMMA 3.1. Let L be an arbitrary modular lattice, and {ex,. . ., er) afinite set of elements of L. Then the sublattice B generated by the elements

hj = 2 *i (/ = 1» • • •» r) is a Boolean algebra.

PROOF. Let C be a non-empty subset of / = {1, . . ., r}. We claimthat the following identity holds in L:


(3.i) n to = 2*«k«+ 2 «*•

If C consists of a single element, C = {/}, then (3.1) takes the form

(3.2) hj-=ejhj+ 2/*«

By definition, 2 ek = hj. Thus, in the case C = {/} we must prove that

hj = ejhj + hj, which is obviously true.Suppose that (3.1) is proved for every subset C of m elements (m < r).

We show that then (3.1) holds for every subset C\ containing C and con-sisting of m + 1 elements. Suppose, for example, that Cx --= C U {s}, wheres £ C. Then

( S + 2 + )

It follows from s ^ C that 2 ^ ^ 2 ei ^ S eiî and 2 et ^ 2 *ft.<#=« t£C i£C /^s k£I-Ci

Consequently, by Dedekind's axiom,(3.4) n hi = 2 etht + 2 ^ + ( 2 et)e8 -

2 ek+eh=T eihi+ 2 eA.£ J C

We have denoted by B the sublattice of L generated by the elements

hj — 2 et- We claim that every element v £ B can be written in the form

2 *j/ij,j £ '

where 0 ^ a and a = I - a.Note that in the case u = hj we have proved in (3.2) that

hj = 2 *,- + £,-//>. Now let Ui and y2 be two elements of L such that

vq= 2 ei+ 2 ejhj (g=l ,2) .

It easily follows from the identity e(h + et - e( that

2= 2 î+ 2

Applying (3.1), we can write in the case a

186 /. M. Gel'fand and V. A. Ponomarev

i£a j£a' fea*

In accordance with this identity, in case aa =£ /,

Since a\ U a'2 = (ax O a2)', we have

It is not hard to verify that the formula

2= 2 ** + 2remains true when tfj = / or a2 = I.

It follows from the relations we have proved that every element v G B

can be written in the form u = 2 ef + 2 ejhj- We denote such an elementby va.

i£a jea'

We denote by % (I) the set of all subsets of / = {1, . . . , r}. It is wellknown that $B (I) is a Boolean algebra with 2r elements. We have provedthat va + vb = vaUb and vavb = vanb for any a, b£$9(I). This shows thatthe correspondence a -> va is a morphism of 98 (I) onto B.

It is not hard to prove from this that B is a Boolean algebra, and thenumber of elements of B is 2m , where m < r.

COROLLARY 3.1. Each sublattice B\l), B~{1) (I = 1, 2, . . . ) of Dr isa Boolean algebra.

COROLLARY 3.2. Every element v of B+(l) can be written in the form

where a is an arbitrary subset of I ( 0 ^ a ^ / ) . / / a =£ /,

va.i= n M Z ) .i

minimal element of B\l) is vQtl = S ^{1)^(1) = fl MO,^ iGJ" iGJ"

maximal element is vjtt = > e^Z).

We shall prove later that each sublattice B+(l) and B~(l) consists of 2r

elements, so that B+(l)^&(I) and B'{1) ^ &(I).Occasionally we denote an arbitrary element of B+(l) and B~(l) by v+(l)

and v~(l), respectively.3.3. Ordering of the sublattices B\l) and £"(/). PROPOSITION 3.1. Let

v\l) e B\l) and v\m) G B\m). If I < m, then v\l) D v\m). Similarly, ifI < m, then iT(/) C v~(m), where v~(i) G B~(i).

The proof of this proposition rests on two lemmas.LEMMA 3.2. Let a = (/,, . . . , /,) G A{r, I) for I > 2. We write


ir(ot) = ( * ! , . . . , / /_i) . Then the elements ea and en^ of Dr can be orderedas follows: eâ) D ea.

T h e p r o o f is b y i n d u c t i o n o n / . L e t 1 = 2, t h a t i s , a = (il9 i2). T h e n

via) = ii^ and ea = eili2 = e i l ^ e, and en(a) = et Clearly, ea C en,a).

Suppose that the lemma has been proved for every a E A{r, X) withX < /. We prove it for a = (il, . . . , /,). By definition,

where

Similarly,

where

Clearly, for any j3 = (fc1} . . . , ^/_i) ^ F(a) we can find an element

ff ^ r(7r(a)) such that 0' = TT(/3) = (ku . . . , kt__2\ By induction on such

0 and |3r = 7r(j3) we have e6 C e .flV Consequently S eP ^ S P'» hence

LEMMA 3.3. L^r htil - 1) G ,g+(/ - 1) /or / > 1 and t G /. Thenea Q ht(l - 1) for every a E A(r, I).

PROOF. By definition,

S, l - l )

We consider first the case when a = (i1, . . . , z ^ , /z) with / ^ j == Z1. Then7r(a) = (/l5 . . . , I ^ J ) E ^ ^ ^ r , / - 1), and so

*««x) s 2j ( r , l - l )

= ht (I — 1).

By the preceding lemma, ea C eff(a). Consequently ea C /zr(/ - 1).Now we consider the case a = {iu . . . , il_1, /z), where ii_x = t. By

definition, ea = e\x S eP> and A;/ - I^{JZ-I» i/} f ° r everyper(a)

0 = (&!, . . . , fy.j) E r ( a ) . In this case, / ^ j = t. Therefore kl_l ¥= t.Consequently, F(a) cz 2 ^;(r» J — 1)» ^nd so

188 I.M. Gel'fand and V. A. Ponomarev

A fortiori ea = eh 2 e$ ^ ht (I —1).P6r(a)

PROOF OF PROPOSITION 3.1. It was proved in Corollary 3.2 that themaximal element of B+(l) is vIti = 2 ^ ( 0 = S e<x., and the minimal

i£/ £A( I)element of B+(l - 1) is va , , = n /z,(/ - 1). It is clear from Lemma 3.3

that for any a G ,4(r, /),

Therefore,

t£I a£A(r, Z)

Now if u+(/ - 1) and u~(/ - 1) are arbitrary elements of B+(l - 1) andB\l), respectively, then v\l - 1) D vdJ_1 D vIfl 3 u+(/).

The corresponding statement for the cubicles B~(l) and B~(m) is obtainedby duality.

We denote by B+ the subset of Dr that is the union of theB\l), / = 1, 2, . . . . Similarly, B~ = U 5"(/).

COROLLARY 3.3. £+ a«c? 5" flre sublattices of Dr.3.4. Fundamental properties of cubicles. We have, thus, proved the

following theorem.THEOREM 3.1. (1) Every sublattice B+(l) and B~(l) (/ = 1, 2, . . . ) is a

Boolean algebra. Any element va t G B+(l) can be written in the followingform:

i£a

where a is an arbitrary subset of I = {1, . . ., r}, and a' - I - a. Ifa ¥* I, then va z = mCi h^l).

(2) Let v\l) E B\l) and v\m) G B\m). If I < m, then

(3.5) v+(l) ^ v+{m).

Similarly, if I < m, then

(3.6) v~(l) c= v-(m),where v~(j) G B~(J).

REMARK. Later we shall prove the stronger statement:

where v+(j) G JB+(/') and iT(/) G J?"(/).CONJECTURE. Let v+(l) G ,8+(/) fl«^ v~(m) G ^"(m

u"(m) C u+(/) /or a// / and m.3.5. In this subsection we introduce another definition of the elements


ea E Dr, which appears rather clumsy, but leads to shorter formulae thanbefore. First, however, we examine an example.

Let r = 5, and let D5 be generated by the five elements ex, . . . , es. Leta = (2, 1, 5), that is a E ,4(5, 3). Then it is not difficult to compute thatF(a) = {(3, 2), (3, 4), (4, 2), (4, 3), (5, 2), (5, 3), (5, 4)} hence

*a = e2> 1. 6 = e2 2 ^ = *« (e3, 2 + *3, 4 + e4, 2 + *4. 3 + *5. 2 + *5. 3 + *5. 4) •

If in this formula we substitute eitj = ef 2 ^ , then we obtain the finalt=hh i

formula for the polynomial ^2 1,5 • However, we do not write it down be-cause of its extreme length.

It can be shown that in Dr

«2.i.5 = ^2( 3 + e5 + eii2 + e4t3) == ^2^3 + 5 + ^4( 1 + e3 + ^5)+ ^4^1 + 2 + O)»

and also^2.1.5 = «a(^4 + e5 + e3>2 + e3i4) =

= e2(*4 + 5 + ez(ex + ei + e5) + ez(ex + e2 + e5))-

It is clear from this example that our method for writing the formulae

for the elements ea, a E ,4(5, 3), as ea = ^ l i 2 i3 = eii 2 P is n ° t ve rYeconomical. |36r(cc)

Let a = 0*!, . . . , /7) G ^4(r, /) with / > 3. We describe an inductivemethod of constructing polynomials e'a from an element aGA(r, I). For everya E A(r, 1) with / > 3 we construct an entire family of polynomials {e«}.We associate with « = ( / ! , . . . , /z), / > 3, a fixed sequence (fc1? . . . , fc/_2)of numbers A:z- G / such that A:x ^ k2, k2 =£ fc3, . . . , fc/_3 ^ kl_2, and*i?{ii» *2» ':3}» k2f{h^ *s. *4>» • • •» ^z-2?{*/-2. */-i» *'}• W e s e t

= {6 = (A:lf . . ., **_!, /0 Mx 6 / — {^

= (S = (*i, • • ., kt.2, / M ) I ^_x 6 / -

We denote by H the disjoint union of Hx, . . . , i/j_ j . We set

i 26£H

where 5 ranges over the whole of / / , that is, Hx, . . . , ^ _ j . This definitionis ambiguous. It depends on the choice of the numbers kl9 . . . , kt_2, and,of course, on the choice of e'b. For example, e\ t t , depends on the choiceof ku that is, there are r - 3 (or r - 2 if il= i3) polynomials e\ i t . Thepolynomial e\ t f- ^depends on the sequence {ku k2) and on the choice ofe'd, where 5 = (ku k2i t), and so on.


We propose the following conjecture.CONJECTURE. For every a E A(r, I) with I > 3, and every e'a

We are able to prove this only for / = 3. For / > 3 we can prove onlythe weaker assertion that, for every a and every e'a9 the elements ea ande'a are linearly equivalent.

§4. Representations of the first upper cubicle

By definition, the cubicle B+(J) is the sublattice of Dr generated by the

elements /zi(l), . . . , hr(\), where ht (1) = 2 *i- I n this section we prove that

all elements of B (1) are perfect.4.1. Atomic representations and their connection with representations of

B+(l). We define the most trivial among the indecomposable representationsof Dr - the atomic representations pj, i for / 6 {0, 1, . . ., r} .

a) The representation po,i in the one-dimensional space V = kl is de-fined by po.i(î) = 0 for all i 6 {1, . . ., r} . It follows that pOtl (x) = 0for every x E Dr.

b) The representation pt, i for t 6 (1, . . ., r} in V = kl is defined byPt.ife) = 0 for r = / and pM(e f) = F.

Presently we describe the connection between the atomic representationptrl and those of i?+(l). In §3 we provide that i?+(l) is a Boolean algebrawith the minimal element vdl = O ht(\). Now we prove that u 0 1 is perfect,

that is, the restriction of p to p(vQtl) is a direct summand of p. We denote

p(ê.i) bY ve,\-PROPOSITION 4.1. Let p be a representation of Dr in a space V over

k, and let vdl be the minimal element of B+(\). Then p decomposes intoa direct sum

P = ( © PJ\ i) © *e. i,iGIU{0}

where rQl is the restriction of p to the subspace F e > 1 = p(^ e , iX an<^

~pjt j w a multiple of the atomic representation pj, ly that is,

p}.i == P J . I © • • • © Pi.i, where m;- > 0.

PROOF. We indicate how to choose subspaces Uj such thatV = ( © Uj) © F 6 t l , where /° = / y {0} = {0, 1, . . ., r} and p

decomposes into a direct sum relative to these subspaces.We claim that the Uj can be chosen as subspaces satisfying the following

relations:


(4.1) Uoj]p (e,) = 0, Uo + S P (e,) = F,

and for any / ^ 0, / E / ,

(4.2) ff,p(fy) = 0, Uj + 9(ejhj) = p(e,),

where /^ = &y (1) = 2 ef.

Step 1. We recall that any element of B+(l) can be written in the form

(4.3) i>«.i = 2 2i

where a is a subset of / = {1, . . ., r} and a = / - a.We write Vatl = p(va,i) and claim that if in V subspaces

Uj, j 6 {0, 1, . . ., r}, are chosen to satisfy (4.1) and (4.2), then for anya CI

(4.4) Va.i^Ûj + Ve.i.

We prove (4.4) first for the case of one-element subsetsa = {*}, * 6 {1, • • ., r}, that is, that

(4.5) F m , i = ^, + F9(1.

Let p(ei) = Ei and p(hi) = Hi. It follows from (4.3) thatr

F{t)f i —Et + S ^ i^ i and Fe>1 ~ 2 î^*z- Then we find that

r

Ut + VQ,1=.Ut+ 2 îî = Ut + EtHt + S î^f-

By construction (see (4.2)), Ut + EtHt = Et, consequently,

This proves (4.5).In §3 we have proved that i?+(l) is a Boolean algebra, and that

vaub l = va l + vb l ôr a ny subsets a, b C I; in particular,

ô.i —S y{t>, !• ^ follows that every subspace p(vatl) = Va<1 can be repres-

ented as a sum Fa§1 = 2 {<}, i- Putting F{t}> i = C/* + VQ>1 in this

formula, we obtain

t£a

This proves (4.4).Step 2. We show that our chosen subspaces Uj are such that

F s F 9 f l 0 t/0 © C/i 0 . . . © f/r We write a, = {* + 1, . . .,


and W^ = Vahi = p(va). Note that (see (4.3))

Wt •-= yj EtHi + S £/. It follows easily from the relations EiHi C £,.

that the subspaces PF*, t £ {0, 1, . . ., r} form a chain

Wr c= PF,.., <= . . . c= TF2 cz c= JF0 c= F,

where PFr = 2 £,#, = F9(1, PF0 = j] Et = F7 i l .i= l i= l

We claim that the Uh j £ {0, 1, . . ., r}, subject to (4.1) and (4.2), areconnected with the Wj by the following equations:

(4.6) TF0 + f/0 = V, WQU0 = 0,

and for every t 6 {1, • • ., r)(4.7) ut + Wt = Wt-V

(4.8) f /^ i = 0.

Note that (4.6) is the same as (4.1), because Wo = V, Et = 2 p(*i)- Wei= l i= l

have proved earlier (see 4.4) that Va>1 = ^] Ut + F 6 t l for every a C /.

Consequently, PFt = Fa, t = S UJ + ê.i = S ^ + ^ . Now (4.7)*' i=t+i i=*+i

evidently follows from this equation. Note thatt r

W* = S EtHi + S ^ s ^/^« + S £/ = ^ ^ + Ht = //,. From this,i=l ;=*+l j^t

using (4.2) (t/r^r = 0), we obtain WtUt C HtUt = 0, that is, UtWt = 0.This proves (4.8).

It follows easily from (4.6)-(4.8) thatr r

V-=y] Ut + Wr = ^] Ut+ Ve 1 and that this sum is direct.

Step 3. We claim that for every i £ (1, . . ., r}(4.9) Et = S^C/y- + ^ F 0 l l .

To prove this we show first that EtUj = 0 for / == /, and - ^ = f/z-. For by

construction, f/0 21 î ^ ° a n d ^ 7 ^ = ^ S Ei = °» ^ + ^*#* = i-Consequently, for every / ^ 0 we have: 1) U0Ei = 0, 2) UJEi = 0 for/ =£ 0, / * i, and 3) ^ - ^ = £/,,

Thus, we can rewrite the right-hand side of (4.9) in the following form:

(4.10) jj EiUj + EtVe.ÊiUi + EiVe.Ût + EtVe.i.j-=0

Let us find EtV6 x. By definition, Fe>i = p(ê i) =P(h M = flîi w h e r e

il ii


Hj—^\Et. Hence Efl^^j - Et, and therefore

EiVQ,iÊiCn Hj) = f| EtHj = EtHt.

We have to show that Et = Ut + Vel Et = Ui + £,//,, which is true byconstruction (4.2). This proves (4.9).

Step 4. We combine the results of steps 2 and 3. We have proved that

V = ^] Ut + VQtl and that this sum is direct. Further, we have proved thatt=0

every subspace Et (Et = p{et)) is representable as a sum

2£j = j\EiUj-\- EiVQtl. Consequently, by Proposition 2.3, p splits into the

direct sum p = © py, x © Te,i, where pj9l=puJ and t 9 f l = pve.i.

Step 5. We claim that pjtl is a multiple of the atomic representationp;l t, that is, p/fl = p/,i © . . . © Pi.i, with rrij > 0.

First we study the representation pOil. It follows from Proposition 2.3that the subspace Uo is admissible, therefore, p0>i (et) = Uop(et) = U0Et.We have just proved in b) that U0Et = 0 for every /. Thus, the sub-representation po,i in Uo is such that po.ife) = 0 for every /. Ifdim Uo = m0 > 0, then po,i is different from zero, and, clearly, splits

into the direct sum po.i — ^ x © . . . © p ^ °f atomic representationsPo.i-

Similarly, for the pj,i with / =£ 0, we obtain

0 if i¥=hUj if i = /•

If dim Uj = mj > 0, then it is easy t o see tha t pj.i splits in to a direct sum

of a tomic representa t ions pj . i = P J . I © • • • © P ; M -

PROPOSITION 4.2. Let val be an arbitrary element of B*(\) correspond-ing to the subset a £ /. Let p be a representation of Dr in V such that thesubspace Va x = p(â.i) *5 different from zero and from V. Then p splitsinto a direct sum

P = ( © P / . i ) © T a , i ,

where a = I - a\ each of the representations p;,i is a multiple of an

atomic one: P;,i = P./,i © . . . © PJM with m;- > 0: Also, ral is therestriction of p to Val. The representation Tal is such that if it splits

into a direct sum of indecomposable representations xa>1 = 2 TJ, tnen


none of the atomic representations pitl with i £ a' [} {0} occurs in this

sum.

PROOF. In Proposition 4.1 we have shown how to choose subspacesUti t £ {0, 1, . . ., r}} such that p splits into a direct sumP = ( © P;\i) © Te.i relative to Uo, Uu . . . , Ur, and VB , . Herepjtl = p \v. is a multiple of the atomic representation pjtl, and

Te = p |Vfl ^ where F e > 1 = p(ye i l ) , is the image of the minimal elementveA G £ + ( l ) . The subspaces Uf are such that Val = 2 # ; + F 6 l l .Consequently, 3â

P=( © P;.i) © ( © PA.I) ©*e.i = ( © p l i l e v i ,;6a'U{0} fe£a iea'LKO}

where xa>1 = (© pfetl) © xe>1 is the restriction of p to Ffl x .

We now claim that rfl ! does not contain as direct summands the atomicrepresentations p ; f l , / 6 «' U {0}. We assume the contrary. For example,let xa > 1=pJ- ,1 © xit i, where / G a\ and let Ffl>1 = Wj + F j f l be the corres-ponding decomposition of Va x into a direct sum of subspaces W}- == 0 andV'a x ¥" Va j . Let us find out what the subspace Ta x (hj) is, where

hj = 2 et- ^y Proposition 2.3, Ffl j is admissible, therefore

*a.i(h3) = F a , l P ( ^ ) = p(i;a.i)p(^) = P(â.i^). We recall that vatl= (] ht

and, by assumption j £ a . Consequently, hjVa>1 = hj (] hi = f] ht = i;a t l .iga' igo'

Finally, rfl j (/z;) = Ffl x. On the other hand, we have assumed thatTa>1 s* p;>1 + x^ 4. Therefore, ia,i(hj) = pj.^hj) + T;, I(/I7). Since therepresentation Pj, x is atomic, we see that p;\ i(et) = 0 if / ¥= /. Therefore,pj.i(^j) = PJM S e* = S Pi.i(g«) = 0- Consequently,Ta.i(hj) = 0+ i'at i(hj) g 7^ j ^ Fa,x. So we have obtained a contradiction.

It can be proved similarly that ra l does not contain the atomic representa-tions po.i-

COROLLARY 4.1. Every element va r G £+(l) is perfect.We have already mentioned that every subspace Vatl = p(vail) is

admissible. But we can prove the more general assertion that every elementva x G B\\) is neutral1) in Dr.

COROLLARY 4.2. The Boolean algebra B+{\) consists of 2r elements.PROOF. We choose a representation p which is a direct sum

Pi,] © . . . © pr.i of the atomic representations f>f(1for t £ {1, . . ., r}.Then it is easy to see that the representation space V of p is r-dimensional:

r

V ~ k?. Here all the subspaces p{et) are one-dimensional, and 2 P(ei) — V-i l

> An element a of a modular lattice L is called neutral if a(x + y) = ax + ay for all x, y G L.


It is easy to see that the sublattice p(B+(l)) generated by the p(ef) is aBoolean algebra with 2r elements, that is, p(£+(l)) ^ # ( / ) , where &(I)is the Boolean algebra of subsets of / = {1, . . ., r}. Earlier we haveproved that B*(\) is a factor algebra of &(I). Now we conclude thatB+(i) s* # ( / ) .

In the following sections we show that every element in the cubicleB*(l) is perfect. However, to prove this we need more complicated techniques(the functors ®+ and ®~).

4.2. Representations of the first lower cubicle. The results of this sub-section are dual to those of 4.1. The first lower cubicle B~(l) is the sub-lattice of Dr generated by the r elements ht~ = n et {t ~ 1, . . . , r).

In addition, we write h$ = O et. Clearly, HQ C hj for every / =£ 0 and

ho = /2zrfy~ for all /, / £ /, / =£ /. Thus, /*o is the minimal element of B~(\).

Then any element ufl { G B~(l) can be written in the form

vZ, I = ho + S r»

where a C /. Since / o = ye i is the minimal element of B~(\), we see thatif a =£ 0,then

Let p7, i be dual to pfil.Namely, pr, i is the representation in V ^ kl

such that po, i(et) = V , pf, i(^) = V for all /, t ^ i, t ¥= 0 andPi", i(*«) = 0. The representation pr, i is called O~-atomic, and P M = pj", iis called O+-atomic.

The dual to Proposition 4.2 is the following assertion.PROPOSITION 4.3. Let p be a representation of Dr in V such that the

subspace Vz,i = p(â,i) is different from zero and V. Then p splits into adirect sum p = p' © xa>1, where â.i = p \v- . Here ra x can berepresented as a sum xa>1 = 0 pj u where'each pj i w a multiple

e U ( 0 }

o/ ^ e atomic representation pjtl. If p' 5p//Y5 /nto J direct sum, then theatomic representations pj, i , /6 « ô «or occwr m ^ w 5wm.

We have also proved that each element v~ x G 5~(1) is perfect.

§5. The functors O+and <D"

In this section we define functors O+and O~ from M{Dr, k) intoM{Dr, k)y and use them to study representations of the cubicles B*(l)y

I > 1. In essence, these functors were first defined in [5]. A modificationof them, under the name "Coxeter functors", was used effectively in [ 1 ] .They also play a decisive role in the proofs of the main results of thispaper. However, we still do not understand their connection with the latticeoperations sufficiently well.


5.1. Definition of the functors and their fundamental properties. Let pbe a representation of Dr in F. We define a space F1 and a representationp1 in F1 in the following way:

t" = {(l,, ...,Er)|6i6p(e«), S16« = 0>,

where e,- (/ = 1, . . . , r) are generators of Dr.

rIn other words, let R = 0 pfo) and let V: # -> F be the linear trans-

it 1 r

formation defined by V(£i, • • •> Ir) = S £i- Then F1 = Ker v. We set

Gl = {(Slt . . ., £*_!, 0, £,+1, . . ., lr) e /?}, that is, G\ = 0 p(ej). ThenThus, from the representation p £ J2 we have constructed another

representation p*£ M. It is not hard to check that the correspondencepH>-pl is functorial. We denote this functor by 3>+. r

A representation p"1 is constructed dually. We set Q = 0 (F/p(e,)). We

denote by P*: F-> V/p(et) the natural map, and by JU: F -> Q the mapdefined by JU£ = (0i ? , - - . , ]3r?). Further, we set F" 1 = Coker M = Q/Im M-We define a representation p"1 in F"1 in the following way. We writeQt - {(0, . . ., 0, pf | , 0, . . ., 0) | I e F}. Let (9: Q -• F"J= Coker /xbe the natural map. We set

def

where 6Qt is the image of the subspace Qt (6Q( C F"1).It is not hard to check that the correspondence p *-*> p~l is functorial.

We denote this functor by (p-(d>-: &-+<%) and we write p"1 = O"p.We describe first some simple properties of the functors O+ and O".PROPOSITION 5.1. Let p and p1 == O+p and p"1 = O"p be repres-

entations of the lattice Dr in the spaces V, F \ and F"1, respectively.

r

(I) / / p is such that 2 p(et) = ^. ^^«r

dimF1= ^] dimp(ei) — dim F.

(II) / / p is such that S.pfo) = F, then

dim p1 (et) = 2 P (e7") — ^ i m V.


(III) / / p is such that (1 p(et) = 0, then

r

dimF-1^: 2 dim 7/p (a,-) — dimF.

(IV) / / p is such that (] p(et) = 0, then

dim p~l(et) = dim F/p(ef).

PROPOSITION 5.2. Let p be a representation of Dr.

, vv/zm? P* w r/ze dwa/ to p

(II) / / p = 0 p^ then <D+(p) ^ 0 (cJ)+Pi) p (&)

i==l i ^ l 7=1

The proof of these propositions is not difficult and reduces to a directverification.

Next we prove a number of important properties of O + O".(I) There is a natural way of defining a monomorphism /: <[)~O+p—>- p

such that O~O+p is isomorphic to the subrepresentation p |ve { — therestriction of p to Fe, x = p(i;e, i)» where vd x is the minimal element of£+O).

(II) There is a natural way of defining an epimorphism p: p-Ô+ O~psuch that O+O"p is isomorphic to the factor representation p/x7i, whereTj i is the restriction of p to Vjti = p(vjt i), where v] x is the maximalelement of B'{\).

If p is indecomposable, then so are, as a rule, ®+(p) and <£>~(p) . Moreaccurately, we have the following assertions.

(III) If p is indecomposable and O+p =^0, then O+(p) is alsoindecomposable; in this case the monomorphism i: O-ct)+p->p definedabove is an isomorphism. If p is indecomposable and O+p = 0, then p isO+-atomic: p ^ p;\ i for some / 6 {0, 1, . . ., r}.

(IV) If p is indecomposable and O~p =7^0, then ®~(p) is alsoindecomposable, and the epimorphism p: p -> O+O~p is an isomorphism.If p is indecomposable and O"p = 0, then p is O'-atomic: p = pj, i forsome ; G (0, 1, . . ., r}.

We prove (I) and (III) below in the framework of the more generalProposition 5.4. (II) and (IV) are dual to (I) and (III).

5.2. The elementary maps q>j. We denote by cp the linear mapcpf. V1 ->• V from the representation space V1 of p1 = ®+p to therepresentation space V of p that is defined by the formula

(5-1) <Pi(£lf • • -, Si-i, In h+i ?r) = h.

We call these maps elementary. It follows at once from the definition ofp1 that for any i g {1, . . ., r)


(5.2) p*(ef) = Ker <pf,

(5.3) Im cp, = p (et) 2 P (et) = P (etht),

where ht = 2 **• Note that cpf does not define a morphism from p1 to p.

The cpi have another property, which could also serve as their definition.First some notation. We note by n the embedding fx: F lc—>/? = © pfe).

It follows from the definition V1 = Ker V that the following sequence of

vector spaces is exact: V+- R +- F1*— 0. We set

G,= {(0, •.. ., 0, g,,0, , . .,0) \tiep(et)}.Then R = S G, and this sum isi=i

direct. We denote by JI,- the projection Jt,: R -* R with kernel GJ = 2 ^

and image Gt.PROPOSITION 5.3. (I) q)i = VJiiM- «< (II) p1^*) == Ker cpf = Kerjt|(.i.

The proof is elementary.We now describe an important "construction." Later (§§6, 7) we shall

prove that this construction "builds up" from V\it = p1(va> t), va, t g B+(l),the subspace Va, Hi = p(va,i+1)1 va l+1^B+{l-\-i), where a is some subsetof / = {1, . . ., r).

CONSTRUCTION 5.1. Let p be a representation in V, T1 a representationof p1 = O+p, and U1 be the representation space of x1.

We set U = 2 îUx(U ^ V). Then a representation x in U is given by

i(et) - (p^1 .PROPOSITION 5.4. Let x1 Z?e a ^/>ecr summand of p1 = O+p. Then the

representation x defined from x1 Z?y r/ze construction 5.1 /20s ^ e followingproperties:

(I) x ;* O-x1;(II) x1 s O+x;(III) x w fl direct summand of p.Before proving Proposition 5.4, we illustrate it by one of its consequences.

r r

In §3 we have introduced the element ve<1 = f| ht — 2 ethu wherer i=i i=i

î == 2 ej- This ve.i is the minimal element of B+(\). We denote by %Q 1

the restriction of p t o F 9 l l = p(ê.i)-COROLLARY 5.1. xe, x = O-O+p.PROOF OF THE COROLLARY. We have proved in Proposition 4.1 that

x6(1 is a direct summand of p. Therefore,


Since hj = 2 et> w e s e e that eA- = e{ if / =£ /, and hence

In Proposition 5.4 we set /71 = V1. Then £/ = S <Piyl a n d t h e

i—1

representation x in ( / i s given by x(et) = cp^F1. By Proposition 5.4,x ^ O-p1 = ®~®+p.

We claim that T = T8 f l . It follows easily from the definition of theelementary map q>< that q^F1 = p^,-/^). Therefore,

Moreover, by definition xfo) = q^F1 = p(e^j). Thus T 8 ( 1 = T d)-O+p7

and the Corollary is proved.PROOF OF PROPOSITION 5.4. Suppose that p1 = O+p splits into a

direct sum p1 = x\ 0 Tg. We denote by U \ and C/J the representationspaces of xj and x\. Then V1 = U\ + t/J, (/} C/J = 0, and for every

rNext we set Rj = S jiî/J (/ = 1, 2). Note that Rj c /? = © p(^.).

i=i t=i

We list some properties of Rx and 7?2-a) Firstly, RXR2 = 0. WQ recall (see Proposition 5.3(11)) that

p1^.) = Ker jtjfx. Consequently, we can rewrite (5.4) as:

(5.4') Ker ji pi = U\ Ker jt£jx + U\ Ker jijî.

By construction, the subspaces U\ and L J do not intersect. Using this factand (5.4'), we deduce easily that for any i G / the subspaces nt\iU\ andTCtnUl also do not intersect:

(5.5) (nt\iUl)(niViU\) = 0.r

Note that ni\iUj C Im Jij = Gt. Also, R = 2 ^f > a n ^ this s u m is direct.

From these properties, the definition Rj = V ji/jxZ/J, a n ^ (5.5), it is nothard to show that R^2 = 0. M

b) Next we claim that R^V1) = iiUj (/ = 1, 2). Since /x: V1 -• /? isan embedding, we set /xF1 = F 1 and fiUj = Uj. By definition of

ifiCSi, • • • , £,-, • • • , ^ ) = (0» • • • > 0, f,-, 0, . . . , 0). Consequently,r

2 3xjJJ. f/} ^ U), in other words, £/y C jRy-. Using this relation, the equationi= l

V1 = U\ + U\, and applying Dedekind's axiom, we obtain


Since Ul2C R2, we have RXU\ C RlR2 = 0, hence RiU\ = 0. Consequently,

(5.6) R{Vl = U\.

The equation R2 V1 - U\ is proved similarly.r

c) We denote by x/ the representation in Uj = 2 9;^) (/ = 1> 2)i = l

by t;(^j) = cpff/J. We now claim that x;- ^ O"x}.r r r

We note that vRj = v ( 2 J I ^ C / } ) = 23 V^^f / J = 2 q>*C/} = £/,.i l i l i i

It follows from this equality that we can define the restriction ofy: R -+ V to Rj C R and Uf C F. We denote this restriction byVj: /?./ ->• f/j. Since V-ff; = Uj, it is an epimorphism. Also,Ker V; = Rj Ker V = RjV1. We have shown (see (5.6)) that RfV

l = U}.Consequently, Ker Vj = U). We denote by \ij the embedding Uj C _> /?;-.Clearly, ji; is the restriction of |i to L and Rj. Thus, the followingsequence of vector spaces is exact:

By definition, the representation T) in Uj is a direct summand of p1 hencei)(et) = f/Jp1^). According to Proposition 5.2, p 1 ^ ^ = Ker nt\i. Thereforex} (e() = Uj (Ker JI,JA). It clearly follows that ^ £ 7 } ^ C/J/xJfo). By

definition, Rj = ^nt\iUji and as we know nt\iUj C jtfi? = Gz-. Thus ther

sum T] JtjjLit/} is direct, hence i?7- ^ © C/j/xJ-î).i i = l

We denote by (jifjji)y: £/y - /?;- the restriction of nt\i to £/;? and Rj. By

definition of nti V ^ = i# Therefore 2 /M- = ( S nt) M- = H- ^n particular,T i i

it follows that UJ = 2 (jtj^)y.So we have shown "that (I) Rj i S « i ^ } = © U)/^(ei); (II) ^ = 2

i i l i l(Ill) Uj = VjRj ^ ityKer Vj = Rj/Imiij = Coker \ij\ (IV) the representation

def

ij in f/y is such that T/(ef) = cp C/} = SJnÛ) = vCrt^t/}) = Vji^tiiU}).

By comparing these properties with the definition of ®~, it is not difficultto convince ourselves that T,- <I)~x}. This proves (I) of Proposition 5.4.

(II) We outline first a proof of the isomorphism 1} = O+x;-. It is easyto verify that VJ: (n>i\iU))-+ Uj is a monomorphism, consequently,

r 1 / , d c f v i

nt\iU) ^ Vtn^c/j) == x;(e,). Therefore, Rj — ZJÎV'UJ ^ ® x;(ef). Weomit the rest of the proof.

(III) We claim that each ty (/ = 1, 2) is a direct summand of p. To prove

' e . ithis, we show that xx 0 x2 = xe>1, where x 9 i l = p | v ^ and


VBtl = p(vQtl). Note that in the proof of Corollary 5.1 we have used onlyProposition 5.4, (I). Thus, we may now assume that Corollary 5.1 has beenproved (that is, t h a t T 9 i l ^ O~O+p).

By assumption, V1 = U\ + U\, and this sum is direct. In Corollary 5.1r r

we have proved that 53 Vivl = Ve. j . Therefore, Ve. i = 53 Vi(u\ + UD =•

53 ^We claim that Ul and U2 do not intersect. In (I) we have shown thatUj = vi?; . Consequently, Ux + U2 = V(#i + R2)- There we have alsoshown that Rx and R2 do not intersect, and that

Ker V = V1 = U\ + U\ = VlRt + F1i?2.Thus, we can write :Ker V = Ri Ker V + R2 Ker V- From this and from RXR2 = 0, it followseasily that (Vi*i)(Vi*2) = 0, that is, Ux U2 = 0.

Thus, VQl s Ux 0 £/2- Consequently, Te > 1 = p | VQ i splits into thedirect sum iQ}1 = Tj + x2, where Xj = <l>"xj. Now T6[ is a direct summandof p: p =p( l ) © xe >i .Thus, p = p(l) 0 xx 0 x2, and Proposition 5.4 isproved.

COROLLARY 5.2. Let p1 = O+p Z?e decomposable:n

pi = 0 TJ? w >> 2, x} ^ 0. 77ze« p splits into the direct sum

p = p(i) 0 (® x^), where T;- W constructed from x/ fl5 m 5.1. Moreover,

a) T, ^ O - T } ;

b) 0 T; ^ O-O+p ^ xe, i, where x 9 l = p | v •

c) O+p(l) = 0 and p(l) = 0 ^ 0 ^ , w/z^re p<" 1 w fl« atomic<o

representation.PROOF. It follows from the proof of Proposition 5.4 that

p = p(l) 0 ( 0 TJ), where x, s <l>"x} and 0 x;- = xe , i ^ O-(D+p. Byj=i J i=i

Proposition 4 .1 , p(l) is the direct sum of atomic representationsr

P?, i» * € {0, 1, . . ., r} , that is, p = 0 ktpt i- It is not hard to check

that <&+pi, i = 0 for every atomic representation p^ lB Therefore, O+p(l) = 0.PROPOSITION 5.5 (I) O " O + p ^ p < = > /or 0// / ; J p(e.) = v, where

F w ^ e representation space of p.

(II) (D+O-p ^ p <=> fl pfo) = 0 /or a// /.

PROOF. (I). By Corollary 5.1, O"d)+p ^ p is equivalent t o i e , i = p.By definition, x e t l = p I v0t ^ where Fe>1 = p(^'e.i)

Thus, O-(D+p ^ p <=> F = fl p(^)- Clearly, V = f] p(hj) <=>v = p(hj) = p ( V. et).

202 /. M. GeVfand and V. A. Ponomarev

for all /.(II) This assertion is dual to (I).PROPOSITION 5.6. Let p be an indecomposable representation.(I) / / O+p = =0 then p1 = ®+p is also indecomposable, and

p s* O-p1 = ®~(D+p;(II) / / O+p = 0, then p is ®+-atomic: p p*, i, •* £ {0, 1, . . . , r};(III) / / ®~p = =0, then p"1 = ®~p is also indecomposable, and

p ^ O+O~p;(IV) / / <D-p = 0, then P ss p7, i, t 6 {0, 1, . . ., r}.PROOF. (I) and (II) clearly follow from Corollary 5.1; (III) and (IV) are

dual to (I) and (II).5.3. The maps <p?. From a given representation p we can define not only

the representations p"1 and p1, but also an infinite series of representations. . .p-', p-l+1, . . ., p - \ p \ . . ., p U l , p', • • ., where pz O V " 1 a n d

p-i ^ 0~p-'+1.To avoid loss of generality, we occasionally denote p by P°,and the representation space of p° by V°. Thus, if / > 1, then

and for any / E /,

We define the elementary map <p|: Ff-> Vl~\ I > 1, by settingtpKSi"1, • • .» l\'\ • • ., Sr"1) = Hi"1. Sometimes we denote (pi by<p,y It follows from this definition that

(5.7) Ker <p{ = p'(^),

(5.8) Im cp} = 9l-KeA) s p ' " ^ ) -

Next, we set (p(ife, . . ., it) = (PôCPift+1°- • • ° i^ 1 & ^ Z. Thus,

cp(4, . . ., it): Vl-+ Vh~K It follows from (5.7) and (5.8) that if

if = ij+l, then <p(ift, . . ., ih ij+l1 . . ., ij) = 0.We now describe a generalization of the Construction 5.1.CONSTRUCTION 5.2. Let pl = (®+)zp°» and let V° and Fz be the

representation spaces of p° and p*. Let T' be a subrepresentation of p' ,and U1 the representation space of %l{Ul C Fz). We set

U° = . *- y(h, • • ., ii)Ul, where the summation is over all sequences?i» • • • » i[

iu . . . , ij with ij£I = {1, . . ., r} . We define a representation T° in

U° by setting T°(e;-) = cp} 2 cp(/2, . . ., iz)(7'. Thus, x° is a subrepres-

entation of p°.PROPOSITION 5.7. Suppose that T1 is a direct summand of

Free modular lattices and their representations 20 3

pl = ((fr+y p°. Then the subrepresentation x° ^ p°, which we constructfrom T1 by the method described above, has the following properties:

(I) T° 5* (O-)V;(II) x< ^ ((D+)'T°;

(III) T ° is a direct summand of the representation p°.REMARK. We shall prove later (in §6) that the Construction 5.2 has

another property which is, perhaps, for us the most essential. Let

T' = p' |V a where Vatl = p(uOtl) and va x G B\\). Then x°, which isconstructed from xz, is such that T° = p° \Va /+ j , where Va, i+i = p{va, l+1)and va / + 1 £ #*(/ + 1). Here va 1 and ufl / + 1 are the elements of 2?+(l) andB*(l + 1) corresponding to the sanie subset a C I.

PROOF. We introduce the following notation:

In the space Ux C F x we define a representation x by settinge,) = F/l

We claim that for all X with 0 < X < / - 1

(5.9) x gg

and that x^ is a direct summand of p* = (O+)^p°.For X = / - 1, this assertion follows from Proposition 5.4, if we replace

p by p u i and use the fact that pl = O + p U l .Suppose now that (5.7) has been proved for X = k + 1, . . . , / - 1. We

prove it for X = k. By definition ph+l = ® + p \ and we take it as provedthat rh+1 is a direct summand of p f t+ l. We denote by %h the representation

in £/ft = S yhi + lUh+l for which xft(ef) = $+1Uh+1. Then according to

Proposition 5.4 xft is a direct summand of ph and %h = ®~x*+l. We claimthat xft = x \ Indeed, we take it as proved that

Uh+1= 2 <P(*fe+2> • • •» i / ) ^ . Therefore,

cpf+1C/'*+1 = <pift+l 2 <P(*fc+2» • • •» ^ ^ = ^ i - Consequently,

^ + 2 * - • • • U

r rUfa = y* (pj 6 / = = ^ j CPJ Z J P V ft+2? • • • » ll) U ==r

_

204 I- M- Gel'fand and V. A. Ponomarev

Thus, xft ^ ®~Tft+1, and fh is a direct summand of p*. Also, byProposition 5.4, xfe+1 ^ <&+vh. By induction we conclude that theseassertions are true for any k > 0. Consequently,x° ^ O-x1 ^ O - ( O - T 2 ) ^ . . . & (d>-)lxl and, similarly, x1 ^ (<D+)'x°.

We write VQ, * = 2 <P(*i» • • •» i j ) ^ By Te, z w e denote the_ 2, xj =^0, ê« p° is also decomposable:i l

np° =L p(/) 0 ( 0 x;.), w/zer^ x;- w constructed from xj &y ^ e Construction

3=1

5.2. ^ / - e Ty ^ (<D-)'TJ, fl«^ (O+)fp(0 = 0.The proof of this Proposition is, essentially, a combination of the argu-

ments in Proposition 5.6 and Corollaries 5.2 and 5.3.

§6. Proof of the theorem on perfect elements

6.1. In this section we prove the main theorem: that all elements ofB+(l) and B~(l) are perfect. We assume that the following assertion hasbeen proved: let p be an arbitrary representation ofD r , r > 4 , pl = (®+yp, and let V and V1 be the representation spaces ofp and p* , respectively. Then

(6.1) P(*,. . .V) = <P(*i, . . . , i i )p ' (e«) ,

(6 .2 ) P ( / i i . . . i l o ) = <P(*i, . . . . ii)Vl.

The proofs of these assertions, which are in §7, are the central andmost complicated part of this paper.

THE REPRESENTATIONS pt+, / AND p7, i-We define representations pl i andP/" i (t 6 {0, 1, . . ., r} , Z = 1, 2, . . .) by means of the atomic represent-ations Pi, i and pr, i that were constructed in §4. We set, by definition,

Pt+,i = PM, tii = (Q-)l-ltftu PF.i = (<J>+)l-1pF.i.

We shall prove later (§8) that pt,i&-Pt,i, where Pt.i are the represent-ations defined in § 1. The functorial definition we have given just now ismore convenient in those cases when we need not go deeply into the"inner structure" of Pt, i-

Let p be a representation of Dr in V. We writedim p = (TI; TO1, . . ., TO1"), where n = dim V and TO* = dim p(et). For


example, the atomic representations pj", i and pF, 1 havedim pj t j = (1; 0, . . ., 0) and, for t =£ 0,dim pi i = (1; 0, . . ., 0, 1 , 0 , . . ., 0), wherem* = i ; dim po, i = (1; 1, . . ., 1) and, fort =^0 dim pf, i = (1; 1, . . ., 1, 0, 1, . . ., 1), where nt = 0.

PROPOSITION 6.1. (I) pjtl ^ (p?, i)*, where * denotes the conjugaterepresentation.

(II) The representations pt,i and p7, i of Dr, r > 4, are indecomposablefor all * 6 {0, 1, . . ., r} am* / = 1, 2, . . . .

PROPOSITION 6.2. We ser dim pjTj = (n,, ,; mj>z, . . ., mj t,).(I) The sequence nt t satisfies the recurrence relations

(6.3) nt, i = (r — 2)nt, z_x — nt. i_2, Z > 3,

e initial conditions nt x = 1, nt 2 = r ~ 2 for t ^ 0 and

no l = I « o 2 = r - 1. Hence it is clear, in particular, that, for t =£ 0,nt t

does not depend on t.(II) TTzere exzs^ intergers mx and m0 t such that

!

i \ ^ if i

mt>l~\ mi + ( — l)Ui if ;=^*numbers satisfy the relations

!

mi ~ mo, i = 0,

rrrn = nt + n M + (-1)',rm0, i = n0, t + n0, z_x.

«/ stands for nt l with t =£ 0.We prove first Propositions 6.1 (II) and 6.2 by induction on /.Let t =£ 0, so that pf, z is a representation of the first kind. For / = 1

we have already proved that dim pit\ = (U 0, . . ., 0, 1, 0, . . ., 0) andthat pt, I is indecomposable.

By definition p£ 2 = O~p£i. It is not hard to verify thatdim pi 2 = (r — 2; 1, . . ., 1, 0, 1, . . ., 1), where mj 2 = 0. It was shownin Proposition 5.6 that if p is indecomposable and O~p =^=0, then O~pis also indecomposable. Consequently, pt, 2 is indecomposable. We have shownthat nt 1 = \ and nt 2 = r - 2 for t ¥= 0. It is also easy to check that(6.4) and (6.5) hold for the numbers m\ x and m\ 2-

Suppose that Propositions 6.1 (II) and'6.2 have been proved for allk < /, so that the pt+,k, t ^0 are indecomposable, and that the numbersnt k ~ nk (* ^ 0) satisfy the relations

* = (r - 2)nfe.x - nh_2 (k = 3, . . ., / ) .


We now prove these assertions for k = / + 1, where / > 2.a) We show first that pt+,z+i, I > 2, is indecomposable. Note that from

(6.6) for r > 4 it follows that

(6.7) nk > 1 for k = 2, 3, . . . , /

We claim that ptj+i = $>~pt,i (t =£0) is different from zero. Assume thecontrary, that Q>~pt, i = 0. Since pt, i is indecomposable, pt", i is atomic(see Proposition 5.6): p £ ^ p ~ i . By definition, p7, i is a representation ina one-dimensional space, dim V] x = 1. Since dim F / , = nh we havereached a contradiction to the fact that nx > 1 for I > 2 (see (6.7)).

Thus, pt^ + i = ^~Pt,i = =0. Hence, by Proposition 5.6, pt,z + i and pt,iare indecomposable, and pt, i ^ O+pt

+^ + 1, that is, pj",; O+d)-p^.b) We now prove (6.3)-(6.5) in the case / + 1. In Proposition 5.5 we

have proved that if ®+®-p ^ p, then p(f] ej) = 0 for every /. Hence it

follows elementarily (see Proposition 5.1) thatr

O~p(et) = dim V — dim p(et) and d i m F " 1 = 2 d i m ®~P(ei) — d i m ^-Apply-

ing this to p j ^ and pt,i+i — ®~Pt,ii we obtain

(6.8) mj, z + i = TZf. / — ATZJ, z,

(6.9) »*.z+i = 2 wit, i+i — ^ . z -

Since by the inductive hypothesis ntl = nt and mlt t = ml if i ^ t and

raj , = / ? ! / + ( - 1 ) / + 1 , we see that

. r nt — mi if i ^ ^ ,(6.10) m*.'+* = \ n z _ T O z + ( _ i ) ^ if i = «.

Thus, we have proved (6.4) in the case / + 1.Substituting (6.10) in (6.9), we find nt l+l = (r - \)nt - rml + (-1)7.

By the inductive hypothesis we have rml = «/ + «/_i + (~l)1- Consequently,

nt i+i = ( r ~ 2)wj - «/_!• This proves (6.3) in the case / + 1.We set, by definition, ml+1 = nx - mx. Then from (6.9), (6.10), and

(6.3) just proved for / + 1 it follows that rmi+i = «/ + «/-i + (~1)/+1.Thus, we have proved (6.5) in the case / + 1.

Combining the results of a) and b), we see that we have proved Propositions6.1 (II) and (6.2) for / =£ 0. The proofs for t = 0 are similar.

Now we prove that ptti and p^ t are conjugate. Now p^ i and pjt {

are conjugate by definition. Suppose that (pr, i) = (pt, /)* • Then(91 i+i)* = (^+pr, /)* = (by Proposition 5.2) ^ O-(pr, z)* ^ ^"pt4; i =pi, i+i-

6.2. The lattices D19 D2, D3 are known to be finite. Therefore, each of

these lattices has only finitely many non-isomorphic indecomposable repres-entations, namely 2, 4 and 9, respectively. Each of these representations


can be written either in the form pt, u t 6 {0, 1, « < ,, r} , / £ {1, , . ., r},or in the form pr, *.

For example, the 9 different indecomposable representations of D3 arethe following: pt, i, pf

+, 2» where 2 6 {0, 1, 2, 3} and pj, 3. Each of theserepresentations can also be written in the form pr, n* Namely, if / =£ 0, then9t, 1 ^ pr, 2 and pt, 2 = PF, 1 and pj, { ^ po, 3, Po, 2 = Po, 2, po, 3 = po, \ • The"dimensions" dim p , 1 of these representations are given in the followingtable:

(1; 0, 0, 0), (1; 1, 0, 0), (1; 0, 1, 0), (1; 0, 0, 1),(2; 1, 1, 1), (1; 0, 1, 1), (1; 1, 0, 1), (1; 1, 1, 0).(i; 1, 1, l) .

Thus, among the indecomposable representations of D3, four are®+-atomic, and four are CD "-atomic, so that the spaces of these eightrepresentations are one-dimensional. Only the representation pj, 2 = Po, 2is such that dim pjf 2 = (2; 1, 1, 1).

6 . 3 . With each representation p of D in F there is associated in a natural way an oriented graphT:

V• • \

where Ei = p (^ ) . The diagram of this graph with unoriented edges is called a Dynkin diagram [3] :

Then dim p is an integer-valued function on the set r 0 of vertices of the graph (for Dr, the set r 0

consists of r + 1 points). Following the methods of [1] , it can be shown that the numbers dim p^ t

and dim p ~ l correspond to positive roots of the Dynkin diagram.

6.4. Theorem on the perfectness of the cubicles. As we know alreadyfrom §3, the /-th upper cubicle B+(l) is generated by the r elements

ht(l) = J) eAl), where ej(l) = 2J e* = S e , - . , - * . We have also

proved there that B+(l) is a Boolean algebra. Every element v+(l) E B+(l)can be written in the form:

2 M0 M0 = M0 (n M0),

where a is a subset of / = {1, . . ., r} and a - I - a, and where

M0 = 2 MO- (Note that M 0 D fy(0, / ^ 0.)The minimal e lement of B+(l), which we deno te by vdl, corresponds to

208 I M. Gel'fand and V. A. Ponomarev

the empty subset a — 0 . Thus,

We now prove a theorem that generalizes Proposition 4.1 to an arbitrary/.

THEOREM 6.1. Let p be a representation of Dr in V, and let vel bethe minimal element of B+(l). Then p splits into a direct sum

where T6, t is the restriction of p to p(vQt j) = VB, i< and each pj, k is amultiple of the indecomposable representation p~jik, that is,

;\ ft

Pi. ft = Pi, k © • • • © pj, ft» rnit fc > 0.The proof is by induction on /c. For A: = 1, the proof was given in

Proposition 4.1. We assume that the theorem has been proved fork = I - \ and prove it for k - I.

Proposition 4.1 applies to the representation p*-1 = (O+)Ulp. This meansthat we can choose subspaces iflr* C Vl~l with the following properties:

1) V1-1 = 2 tfjrl + Fe"}, where VeTi1 = P^Mê.i), and this sum is direct;i=o ' r

2) p u i splits into a direct sum: pl~l = © p],"} © TJ;} , wherej=o

p j i = P 1 ' 1 I i-i> Te~i = P1""1 I i-u 3) Pi~i is a mult iple of the a tomicuj, i Fe, i

representation p£i. The subspaces (7/711 chosen in this way are such thatany subspace p ' " 1 ^ , 2) s F1"1, corresponding to yfl j E ^+(1), can beexpressed as a sum

With the help of the construction in 5.2 we build up subspaces Ut / andVel such that £ / , . , = 2 (p(ilf . . ., i ^ ) ^ 7 } and

*i *l-i

ê.z= S <P(*i, • . ., ii-i)V\r\. ^ §5 (Corollary 5.4) we have proved** h-\

that p then splits into the direct sum

(6.12) P « P ( J - l ) 0 ( 0 P , M ) 0 T e i i .i=o

where pJt t = p | ^ ^ and x0,, = p |^ . Also py, z s ((l)~)Ulpj,~iI.

a) We claim that p;-, z is a multiple of the indecomposable representationpj" /. By construction, pj"}^p/ t 1 0 . . . 0 pjf 1. Consequently,


b) Now we prove that for any a C / and va l E i?+(/)

(6.13) 9(vati) = Ûul + VBtl.

We denote the subset /— {/} by a.-. Then va.ti = &;-(l) = 2 et. Hence,

in accordance with (6.11), we can write

PI"1(^(1))= S U\;}+VlfX

We also write Xj = ^ cp (il9 . . ., it_^) p U l (hj (1)). Let us determine Xj. On

the other hand,

But

x,= 2

We assume it as proved that cp(t1, . . ., iiî)pl~1(et) = p(î...il_1t) if

t ¥= //_!, and that q>(il9 . . ., j ; _ i ) p u i ( ^ t) = 0. Consequently,

^ = 2 2 p ( s . . . i M t ) = p ( S 2 . ^ . . . < M t )

So we have proved that

(6.14) P(M*))= 22It is known (see §3) that va t = n hAl) for every a =£ /. Therefore,

p(l;at ,) = p ( f| ^-(0) = (1 p(hj(l))- Hence, using (6.14), we find that

P(Va,i)= fl ( 2 ^,^-f-Pe.z) .jca' i£l-{j)

It follows from (6.12) that the sum 2 UÎ+VQ, t is direct. Consequently,i

the sublattice of X (V) generated by U}- / and VQ / is distributive. Usingthese facts, we can show easily that


fl ( S ff«.i + Ve.i) = 2ff«.i + F9.j.

Thus, if a ¥= /, we have proved that

(6-13') 9(va.i)-J]U,.l + Ve.l.

It remains to prove (6.13) when a = I. To do this, we have to use thefact that for any /, s G / with / #= 5 we have u7 , = hj(l) + /zs(/). From this

and (6.14) it follows easily that p(vI>?)=. 2 Uit j + F e , z. This proves

(6.13). In particular, for a = 0 , we obtain p(ve. i) = ê . i« fiy definition,

S 9 ( 1 , ' M ) P ( a . i )

Thus, we can rewrite (6.13) in the following way:

(6.15) 9(va.i)= S l = p(VcLi j) is such that ra, t = p |ya> z is a direct summand of p. Forit follows easily from (6.12) and (6.13)'that

P = P(Z — 1 ) 0 ( © P.M) © t a , j .i£a'U(0}

c) We claim that

(6.16) T O i M = ( e P ; , Z ) © T 6 > Z ,

where xe, k = p | v , F e . ft = p(ê. &)• BY definition, 0 p;. , 0 T6> J is thee, k j = 0

r

restriction of p to 2 t/^ z + F e , z. Clearly,

S c/;. J+F 9 . «= S cp (i,, . . . , i,.,) (y, u];! + vi:l) =

= S q>(£i ^-i) V - 1 = S q> (it, . . -, i « ) S <PI, .v'-*.r

In §5 we have proved that 2 < P i y l = = Z: p(eM = p(ê i)- Applying

this result to p U a and p U l O+p1-2, we find that 2 Ûiyl"1 = pz~2(ê,i)-

Thus,r

S tfj. z + Ve. i = 2 P (*i, • • •, iz-2) P'"2 (ve. i).i=o *i *i-a

By induction, we assume it as proved that2 <P(*i> • • •» ii-2)pU 2(ê,i)= p(ye. i-i). W e w r i t e Te. z-i == P lv

*i h-2 Q>1~1


7 e t_% = p(ve> î). Now (6.16) is proved. By induction, we assume it asproved that p = 0 p;- k © xe, i-t. Comparing this with (6.16), we obtain

;, ftP = 0 pj,h © x e ^ - 1

j , h

MAIN THEOREM I. In the lattice Dr, r > 4, every elementv E B+ = U £+(/) oru G 5 " = U £"(/) w perfect. This means that for any

representation p in any space V over a field k, the subspace p{v) is suchthat the restriction T = p |P(U) is a direct summand of p.

PROOF. In §3 we have proved that B+ = U B\l) is a lattice. In the

course of proving Theorem 6.1 we have shown that every elementv~ i E B~(l) is perfect. The corresponding result for elements v~j E«5~(/) followsfrom the duality principle.

The following proposition refines the main theorem.PROPOSITION 6.3. Let val E B\l) C Dr', r > 4, and suppose that

Va, i = p(â. z) w different from zero and from V. Then p splits intoa direct sum _

P = ( © P ; . A ) © ( © P ; , z ) © x a > z ,

where xa, i = p | va z» « f ^ c/z representation pj.u and (py, i) w fl multipleof the indecomposable representation p£ ^ <3«<i (p/, 0, respectively.

It is elementary to deduce from our results the following Propositions6.4 and 6.5.

PROPOSITION 6.4. Let pfti be an indecomposable representation ofthe first kind (t £ (1, . . ., r}) in Vt v We set vt , = D ht(l) (vt t eB+(l)).

i*t(I) pt,i(x) = Vt,i for every element x G B+ = U B+(s) such that

x D vt i. In particular, this identity holds for every x E B+(m), m < I.(II) pj", i(y) = 0 for every element y E B+ such that y C ht(l). In

particular, this identity holds for every y E B+(n), n > I.PROPOSITION 6.5. Let pjt i be an indecomposable representation of

the second kind in F o / . Let fe/_i = 0 ni(l — I) be the minimal element

of B\l - 1) and vZt, = S et(l) the maximal element of B\l).(I) Po. ifr) = Vo, i for£Ievery element x E B+ such that x 2 vdl_l. This

means that pjt t(x) = VQ, z for every x E B+(m), m < / - 1.(II) po, z(y) = 0 for every element y E B+ such that y C u7 z. r/z/5

mâ«5 //wfr f>o, i(y) = 0 /or ev^rj/ y E i?+(rc), ^ > /.PROPOSITION 6.6. The element fo(l)= S A i o is linearly

equivalent to the minimal vdl_l E ^+(/ - 1), that is, p(/0(0) = p(ê. z-i)/or every representation p.

PROOF. In the course of proving Theorem 6.1 we have shown that

S <P(*i» • • •» ^-i)^""1 = p(ê. z-i)- We assume it as proved (see§7) that

^ M. Gel'fandand V. A. Ponomarev

ii-xW1'1 = p( / i l . . . i z_ 1o) . Consequent ly ,

1-1) = H P( / i i . . . i , to) = P( 2 /*,. . .*, 4o) - p(/o(0).V - - . * i - i *i 2i-i

PROPOSITION 6.7. £+(/) W £"(/) are Boolean algebras with 2r elements.PROOF. We consider the representation p, the direct sum 0 p£ l

t=i '

of all indecomposable representations pit, i with one and the same /. Itfollows easily from Proposition 6.3 that all the subspaces p(va, t) fora C / are distinct. Thus, p(B+(l)) as %\I). It clearly follows that£+(Z) ^ #(7).

6.5. Connection between the upper and lower cubicles in Dr, r > 4.We have proved in §3 that the lattices B+(l) can be ordered in the follow-ing way: u* z D v+

bj+l for every / and any a, b C I. In particular, theminimal elements v+

d z E 2?+(/) form a descending chainU0 l ^ U0,2 ^ y0,3 ^ • • • • It follows from Proposition 6.7 that all theelements of this chain are distinct.

Dual to the v^ j are the elements vjt GB~(l). They form (with respect to/) an ascending chain vj x C vj 2

c vj 3 C . . . . Let p be an arbitraryrepresentation of Dr. As usual, we write Fg, i =p(ê,z) and 77, z = p(^7, /)•In addition, we set F >0O = O F ^ and Vf^ = U F/ / .

PROPOSITION 6.8. VJ^ C F ^ /or every representation p of Dr, r > 4.PROOF. The representation space F of p is finite-dimensional. Therefore,

there are integers / and m such that Fg, «> = p(ê, *) = Fg, i and" 7,00 = p(vjt m) = F j > m . It follows from the results of this section that psplits into a direct sum p = p(l) 0 xe, z, where T8 I t = p |y+ i and(O+)'p(Z) = 0. We denote the representation space of p(Z) by U. Since pis decomposable with respect to U and VQJ = Vj^, every x G Dr satisfiesp{x) = Up(x) + Fe, oopW. In particular, for the element x = vj m E B~(m),we can write Ff, TO - p(yjf m) = f/p(yj, m) + Fg(0Op(i;7, m ) .

We claim that (Jp(vjt m) = 0. Hence it follows at once thatp(i>ifm)= ^7,m = FJ,ooFj,m, that is, F ^ = F^m C F0

+ >oo. Suppose the

contrary: that f/p(^7, m) =7 =0. It follows from the properties of a directsum that p(l)(x) = Up(x) for the restriction pl^ = p(Z) . Thus,p(Q(*>j,m)= ^p(^7,7w)- If p(0(y7, m) is non-zero, then, according to theproposition dual to Theorem 6.1, p(l) splits into a direct sum:

p(Z) = T(Z) 0 ( 0 pj k), where ( 0 pj h) is the restriction of p(Z) to

p(0(?;7,m)- Here "each pj, & is a multiple o fp j^ .By assumption, (O+)zp(Z)=0.On the other hand, (O+)lp(l) = (O+)rx(Z) 0 ( 0 (O+J'pj ft), and we assume

U h

that the sum 0 pj ft is non-zero. This means that there are / and k such

that pjk = pjh 0 . . . 0 pjki where m^k > 0. Then


(®+)lPJ,h => @rnj.h(®+)lPlk. By definition (<S>+)lpJtk ^ Plh+i- Since pj, fe

is a representation of Dr, r > 4, as we have shown at the beginning ofthis section, p j 5 ^ 0 for all s > 1. Consequently, (®+)*pj?ft =^0, andhence (O+)lp(Z) = 0. But this contradicts the fact that (O+)fp(Z) =£0.

PROPOSITION 6.9. Let p be a representation of Dr, r > 4, />z F, andfef 0 C F/>oo C F0

+)OO C F 2?<? a chain of distinct subspaces. Then p splits

into a direct sum p = p~ © px © p+, where p~ = p \Vj ^ and p" © px"=P ly- •Note that for p+ and p~ there are integers / and m such that

(0)+)lp+ = 0 and (O-)mp- = 0. As regards the px, it is not difficult toshow that (O-)1 (O+)[ p x ^ p* and (O+)'(O-)<px ^ px for every / > 0. ForZ)4, the representations of type px were studied in [5 ] , where they werecalled regular. Very little is known about the regular representations ofthe lattices Dr, r > 5. It is clear (see [5 ] , [7]) that the classificationof regular indecomposable representations contains as a special case the problemof classifying up to similarity an arbitrary set of linear transformationsA l 9 . . . , A n , n > 2,At: V -> F .

§7. The subspaces p (ea) and the maps cpi

As usual, let p and pl = (®+)lp be representations of Dr, r > 4, in Fand V1, respectively, and let (p(il9 . . ., ii): V1 ->• V be the linear map(pijo(pi2 o. . . o <pfj, where cp1fc: Vk -^ V*1'1 are elementary maps (see §5).The basic aim of this section is to prove the formula

(7.1) p(î,...i,() = 9(h» • • «i if) pl (et)i

where e i i t E Dr is the lattice polynomial defined in §1 .

7.1. A lattice in the space J B = © p(e<)* We repeat the construction ofthe representation p1 = O+p from a representation p in V. Let

rR = 0 p (a), and let V: R ->• V be the linear map defined by the

i=^l r

formula V(5i, • • ., lr) = S i/i where lt £ p(et). We denote by Gi the

subspace of R consisting of the vectors (0, . . . , 0,£,, 0, . . . , 0) with

It 6 P (ei)> a n d 1^ GJ ~ 2 £/• Then p1 = O+p is defined as the representation

in V1 = Ker V for which p 1 ^ ) = V 1 ^ .Let X (R) be the lattice of all linear subspaces of R. We denote by M

the sublattice of X (R) generated by the subspaces G1? . . . , Gr and F 1 .It is easy to see that these subspaces satisfy the following relations:

1 ° ^ C 7?

2°. Gi^Gt— 0 for every /.

3°. G{Vl = 0 for every i.

214 I-M. Gel'fand and V. A. Ponomarev

From 1° and 2° it is obvious that the sublattice of M generated byGu . . . , Gr is a Boolean algebra, which we denote by $. Clearly, themaximal (unit) element of $ is R.

We define two representations v° and v1 of Dr in R by the formulae

(7.2) ^(efi^Vi + Gu vi(ei) = ViG'i = V*yiGt.

Thus, v°(Dr) E M and v1 (Dr) G M. It turns out that v° and v1 are almostthe same as p and p1. More precisely, the following is true.

PROPOSITION 7.1. (I) The map V: R-+.V defines a morphism ofrepresentations v : v° -> p. Moreover, p(x) — Vv°(x) for every x E Dr.

(II) v° splits into the direct sum v° = vj + v°, vv/zere

vj ^ Im V = p/,i PJ.I W J/ze restriction of p ro Fj , i= p ( 2 î)^ i= l

^fl/ of v° ro F1. êre vj ^ Ker V ^«^ vi is a multiple of the atomicrepresentation p^,.

PROOF. We prove first that v° splits into the direct sumv° = vj + vj. Let U be any subspace complementary to V1 = Ker V (thatis, UVl = 0 and U + F1 = tf). By definition, v°(^) = Gt + V1 D V1.Consequently, v°(et) = vê^R = v°(e (C7 + y 1 )^ v°(e-)f/ + vê^^.This means(see Proposition 2.3) that v° is decomposable and that V1 is admissiblerelative to v° (that is, Vîx) + v°(z/)) = Vlv*{x) + F^v^y) for anyx, y E Dr).

Since V1 = Ker V is admissible, by Proposition 2.2 the correspondencex>-+ Vv°(.r), x 6 Dr, defines a representation in the space ImV = R/V1.Note that Vv°(ef) = V(^f + F1) = p(ef). Consequently, Vv°(x) = p(x) forevery x E Z)r. Thus V defines a morphism of representations: V: v0 -> p.Obviously, Im V = P/,i, where pj.i is the restriction of p to

* 1 i l ^

Let us verify that Im V ^ vj, where vj is the restriction of v° to U.First of all we note that the linear map \/v: U -> Im V is an isomorphism.( Vc; denotes the restriction of V to U.) Also, we proved above that

(7.3) v° (a) = G« + F1 = v° (Ci) U + v° (e«) F1 - vJJ (e,) + F1.

As we have already mentioned (see 3°),GtVl = 0; moreover, it follows

from UV1 = 0 that v J ^ F 1 = 0. Consequently,

(7.4) dim p(ef) = dim Gt = dim vjjfo).

Thus, Im F = vj.To finish the proof, we note that v}(ef) = F1 for any i 6 {1, . . ., r}.

Consequently, vj is a multiple of the atomic representation po, i (We recallthat po, I is the representation in the one-dimensional space W = kl forwhich po, i(et) = W for all /).


We denote by \i the embedding \x: V1 CL> R.PROPOSITION 7.2. (I) The embedding \i: V1 -> R defines a monomor-

phism p.: p1 -> v1 of representations. Also, v1(x) = jup1^) for any x E Dr.(II) v1 splits into the direct sum v1 = v\ + vj, where v\ is the restriction

v1 | y l vx ^ p1, a«d vj /s 0 multiple of the atomic representation po, i-The proof of this is similar to that of the preceding proposition.7.2. The connection between the maps (p* and the lattice operations.

In this subsection we use as definition of the maps cp* the formula(?i = Vî^t, where nt is the projection onto R with kernel Gj and imageGt. This formula was proved in Proposition 5.2.

It is useful for us to have a lattice definition of a projection.LEMMA 7.1. Let R be a finite-dimensional vector space, X a subspace

of R, and let n: R -> R the projection with kernel G' and range of valuesG. Then nX = G(G' + X), where nX is the image of X.

COROLLARY 7.2. For every subspace Y1 C V1 and every i £ (1, . . ., r}

n^Y1 = GAGi + [lY1).

The basic results of this section are formulated below in Theorems 7.1,7.2, and 7.3.

THEOREM 7.1. Let p be a representation of Dr, r > 4, and p1 = O+p.Then

1°. (PiiP'te,...^) = o.

2 ° . <Pjp1(ei1...tz) = p(*ji1 . . . t J) if j =£ rt.

Here, e^ t E Dr are the lattice polynomials defined previously.We shall show that the proof of 2° follows from the next Proposition

7.3. Its proof is very laborious, and is given in §7.3, 7.4.PROPOSITION 7.3. Let p1 = O+p, and let v°, v1 be the representations

of Dr, r> 4, in R = 0pfe) defined by v°(et) = Gt + F1, vx(^) = G\V\i

Then for every ea = e ^ t and every j == ix,

(7.5) V*l+ Gj (G; + vl(ea)) = v°(eJa).PROOF OF THEOREM 7.1. 1°. By definition,

Furthermore, Ker (pt = p1(ei) (see §5.2). Consequently, cp^p1 (e^.,.^) = 0.2°. Suppose that (7.5) has been proved. For any Y C V\ according to

Corollary 7.1, we have cp F = Vnj\iY = V{Gj(G'j + \xY)). By definition,V1 = Ker V- Consequently,

(7.6) <p,Y = V(T71 + G}(G- + ^y)).Now let Y = p 1 ^ ) . In Proposition 7.2 we have proved that

^P'fox) = vHfa). Therefore, ^ p 1 ^ ) = V(FX + Gj(G- + v ^ J ) ) . F r o m (7-5)and Proposition 7.1 (Vv°(x) = p(x)) it then follows that

216 I.M. Gel'fand and V. A. Ponomarev

cpipHO = Vv°(e7-a) = p(eJa).

THEOREM 7.2. Let pl = (O+)'p &e fl representation of Dr. Then forevery sequence il9 . . . , it, where ij G {1, • • •» r}» fy ^ *;+ir 0 « ^ everyt ^ it, we have

P R O O F . By definit ion, pk+1 = O + p \ Therefore , applying Theorem 7.1

to p* and p f t+1 , we find tha t for any a = ( / l 5 . . . , im ) and any / == /

p* (^ . . . . i m ) = q>?+ 1P*+ 1K.. . im) ,

where q)j + 1 : F f t + 1 - > F/l>. By definit ion, q>(/3, . . ., / /) = cp i jo. . , o < p i r

Therefore,

cp(i1, . . ., ^) pl (et) = cpi, o . . . o cpiz pf fe) = cp{l o . . . o cpi^p^"1 (eiz<)

= (Pii° ••• ° ^î-aP1"2 ^z-iM*) = • • • = P K . . J , t ) -

7.3. Lattice lemmas. Here we prove three lemmas, which will be used in

the proof of Proposi t ion 7 .3 .

LEMMA 7.2. Let aua2,b, c be four elements of a modular lattice L suchthat ai C a2. Then

(I) fll (b + a2c) - fll (a2b + c);

(II) «2 + &(fli + c) = a2 + (fll + fe)c.

PROOF. (I) Since ax C #2, we have a! = «1<22 and hence,fli (^ + aic) = d\î{b + <22

C)- It follows from Dedekind's axiom thata2(b + a2c)â2b + a2

c==fl2(«2^+c)- Therefore, ax(6 + a2c) = axa2(a2b + c) ==aâjb + c).

(II) Since 0j C a2, we have «2 = a\ + a2- Therefore

«2 + h{flx + c) = a2 + fl2 + 6(a! + c) = a2 + (a± + ^(aj + c) == a2 + «! + (ax + fe)c = a2 + (ax + 6)c.

LEMMA 7.3. Let a, a , xu . . . , xn be elements of a modular lattice Lsuch that aar = 0 and a + a' = 1, where 0 and 1 are the minimal andmaximal elements of L, respectively. Then

PROOF. Let x be an arbitrary element of L. Then

a ' + x = \{a + a-) == (a' + a) (a' +x) = a' + a(a' + *).

Using this equation, we can write

Since a 2 S a(a' + ^ ) , applying Dedekind's axiom we find


a (a' + S xt) = a {a' + I a (a' + xt)) = aar + â (a' + xt) =% i i

= O + â(a' + Xi)--=â (a' + **).

LEMMA 7.4. Lef 38 be a Boolean algebra with generators gl9 . . . , gr

r

such that 2 gi = 1 and ftSft = O /or ev^r^ /. We sef g* = S ^»i = l <=j«=i <^=i

Then ^

7.4. Proof of Proposition 7.3 ( F 1 + ^ ( ^ + v1(6a))=v°(c / a))- Tobegin with we prove a lattice proposition equivalent to the equation

l • • -i) = 0.PROPOSITION 7.4. Let p1 = O+o a«(i /e/1 v1 Z?e the representation of

Dr in R - 0 p(^) e/e/in^ by vl{et) =

F1 + Gi4 (G-j + v1fel...iz)) = F1 /or every a = (iu . . . , /z) G ^(r, /).PROOF. By definition, v1(ei....i.) = v1(e.) S v1(^) = F1G'il 2 v\e

1 Per(a) per(a)Consequently, G^ + v1 (g^ •) = G\^ and so

V1 + ^.(Gi, + v 1 ^ . . . ^ ) = V1 + G i ^ - F1 + 0 = F1.

We prove Proposition 7.3 by induction on /, therefore, it will be moreconvenient for us to prove the following stronger Proposition 7.5, whosefirst part is equivalent to Proposition 7.3.

PROPOSITION 7.5. (I) Let p be an arbitrary representation ofDr, r > 4, let p1 = ®+p, and let v°, v1 be the representations of Dr in

r\R = 0 p(et) defined by the equations vo(et) = Gt + V\ v1^) = GJF1.

i l

/ o r every (/1} . . . , / / ) G ^4(r, /) and every j ^ ilt we have

(7.7) F 1 + G ; ( ^ + v ^ . . . ! , ) ) = vo(ejiv..h).

(II) Fo r every a = (z1? . . . , / / + 1 ) , / > 1, and/ every ^ $ {i1? ?'2} ^ ^

elements eii 2 eP ^"^ ii ta + S ^P) a r ^ linearly equivalent, that is,Perxcc) 3er(a>

/or every representation i

(7.8) x( e i l S eP) = T ( e i l ( e , + SPer(a) P£r<

w/zere

r(a) = r(i1T . . . , tl+1) == {p = fe . . ., A:,) £ A(r, I) \ kx $ {il9 i2}, k2 $

fl«^ gfl 2 ep = ea.pena)

REMARK. Apparently, in D '

2^g I. M. GeVfand and V. A. Ponomarev

(7.9) eu S e^ = eil(et+ 2 *()psr(a) per(a)

for every t ^{i^ i2). However, we can prove this only for 1 = 2 and/ = 3. A proof for / = 2 is given below. Note that (7.9) is a special caseof the more general conjecture stated in §3.5 (that ea = e'^y

Our proof of Proposition 7.5 is by induction on /.Step 1. We prove (7.7) for a = (i^), that is, we show that if/ =£ iu

then V1 + Gj{Gj + v 1 ^) ) = v 0 ^ ) .By definition, v 1 ^ ) = F 1 ^ = V1 2 Gt. It follows from / # ix that

G'^ D Gj. Hence, applying Lemma 7.2, we can write

Gj(G'j + VlG'{i) = GjiG'fi'i, + V1). Also by Lemma 7.4, GjG^ = S G«.

Using these equations, we find

t=j, ii

Step 2. We prove (7.9) for / = 1, that is, we show that if t ± iu i2,then

By definition T(ilt i2) = {k\k${i1, i2}}. Thus,

ûfe+ 2 ^p)=îi(^+ 2 efe). Since t ¥= /1? z2, we have

e< + 2 *h = 2 ' eh hence, ^^(^ + 2 e§) = eti{ 2 h) = en, iv

as required.Step 3. Now we suppose that Proposition 7.5 (I) is proved for all

a - (*!, . . . , i\), X < / - 1, and Proposition 7.5 (II) for alla = (il9 . . . , / x + 1 ) , X < / — 1. We show that then Proposition 7.5 (I) isalso true for any a = (iu . . . , it).

Step 3a. We perform some manipulations with the subspaceV1 + Gf(Gj + v H O ) - By definition,

(7.10) v i ^ ^ v * ^ 2 ^)=vi(eil) 2 v ^ e ^ ^ F ^ 2 v*Psr() perc) per()

We set(7.11) X = 2

pr(By definition v 1 ^ ) = VlG\ C F1 for every /. Consequently, vx(x) C F1

for every element x E Dr. In particular, 2 vl(ep) ^ ^ - Thus, we canPerc)


write vHâ) = V^G'^ 2 ^M = G'ux- F o r brevity, we also setperx)

(7.12) F1 + G,(Gi + v 1^)) = F(ea).Thus, F(ea) = F l + G;(GJ + Gyr).

def

By assumption, / =£ iu hence G^ = 2J Gt ^GJ. Using Lemma 7.2 as

we did in step 1, we obtain(7.13) F(ea)

Step 3b. We carry out the proof for r ~> 4 and, by assumption, / =£ il.Then the subset / — {/, ix) = {1, . . ., r} — {/, ij} is not empty, and wecan Write / — {/, ix} == {s3, . . ., sr}. We show that

(7.14) F(ea)=Vi + Gj(r^GSk(G'Sk + X) + G8r + X).

In (7.13), replacing X by the equal subspace X + G's X and applyingLemma 7.2, we find that

, (X + G^ {GSi + J 4 G,k + X)).r

Obviously, GL ZD 2 Gsft. Consequently, by Dedekind's axiom,

3 (^3 + S ^ + I ) = i l Gsk + G;3(GS3 + X). Thus,fe=4 fe=4

(«a) = V1 + Gj(X + G;3(GS3 + X) + S 6 5fe). Note that

+ G'S3(GS3 +X) = (X + G'S3)(GS3 + X) = X + GS3(G'S3 + X). Therefore,

(7.15) F(ea) ( ^

Applying transformations similar to those we have used to get from(7.13) to (7.15), we obtain

(7.14') F(ea)=Vi + GJ(^G.h(Gh

Step 3c. We show that

(7.16) F(ea) = 7i + G3 ( S GSk (Gs'fe + X) + GSr r

We assume that Proposition 7.5 (II) has been proved for all X < / - 1,that is, that for every representation x and everya = (i19 . . ., ix+1), X < I — 1, and every t ^ il9 i2, we have

220 I- M. GeVfand and V. A. Ponomarev

*(eii 2 ef) = x(eil(et+ S *(0-PSr(a) tJgr(a)

Replacing T by v1 in this equation, we see that for alla= (,*!, . . . , ij) EA(r, I)

v1 (e«) = vi (eh) S ^ v* (ep) = v* (^) (vi (et) + %

We have written 2 v l W = X. Now we setper()

Then we can write v1(ea) = v1(e\^)X = vl(e'u)X. Sincev1(ei ) = VlGf

t, X C V1 and X C V\ the preceding equation can berewritten in the following form: vl (ea) = Gi X = Gt X. From this itfollows easily that F(ea)

d= V1 + Gj(G] + G'UX) - F1 + Gj(G- + G^X). In

the derivation of (7.14), the only property of X we have used is thatX C V1. Consequently, we may replace X by X and as a result we obtain

(7.17) Vi + G, ( 2 3 GSfe (G;fc + X) + G^ + X) =

= V* + Gy ( S 3 GSfe (G',k + X) + GSr + X).

By definition, X C X. Consequently, we also have

F (ea) = F + G, ( S G (G; + X) + G + X) <=

<=Vi + G} ( j j GSft (C;fc + X) + GSr + X) = T" + G; ( j ( G*h (GSk + X) + G.r + X).

In (7.17) above we have shown that the extreme terms of this inequalityare equal. Hence so are the first two terms, that is,

F (ea) = V* + G, ( £ G$h {G'Sh + X) + GSr + X).ft=3

By definition, X - v 1 ^ ) + X = VlG't + X, where / =£ / t , /2. We didnot impose any restrictions on sr except that sr # /, ix. Now we requirethat sr ¥= /2• Then we can choose X = V1G'S + X, that is, if sr ^ {/, /1? i2},we obtain

^ + X + V*G;r).(7.16')

Step 3d.

(7.18)

F(ea) = l

We claim

F(e,

ri + Gj

that

r-1

k=3 h

+ GAV

(Czs -J- A ) -\- Ors

r

+ 2J Gsk (G$k

k=3 "• k


Since sr =£ /, we have G'Sr D G;, and so, applying Lemma 7.2 (I) to theright-hand side of (7.16'), we find that

F (ea) = 7* + Gj (Vi + G'Sr (GSr + J ? G&k (G'Sk

By definition, all the indices s3, . . . , sr are distinct, therefore,r - l r - l

GI 5 2 Gs 3 2 £* (Gi + x)- U s i nS t h i s relation and Dedekind'sr k=3 h fe=3 h h

axiom, we obtainF(ea) = F» + G, (7» + ' S GSft (G;k + X) f G; (G,r + X)).

h—3

Since V1 D X, in accordance with Lemma 7.2 (II) we haveV1 + G'Sr(GSr + X) = F1 + GSr(G;r + X). Thus,

(7.18') ^ M = j

Step 3e. Now we prove that F(ea) = v°(^ ;a). By definition, Gr and G'tdetermine a partition of the identity in M, that is, GtG't = 0, G* + G\ = /Therefore, applying Lemma 7.3, we find

2 vi(ep))= 2 ^ ( G H - V 1 ^ ) ) .p e r )

Using this equation and also the fact that / — {/, h} = {s3, . . ., sr},we can transform (7.18) to the following form:

(7.19) F (ea) = 71 + G, (F« + S S G (G; + v1

ft3 p e r ( ) ft A

2r

By definition, F(a) = {P = (A:x, . . ., ki^) | ^ ^{ii, J 2 }, . • •}• Conse-quently, two cases for the pair (tfi) - (t,kl9 . . . ,f/_i) are possible:1) t = kXi or 2) t =£ kx. Earlier, in Proposition 7.4, we have proved thatVl + Gki (G'ki +v1(eki,.mkl_i)) = V1. I f t ± kl9 t h e n b y t h e i n d u c t i v ehypothesis V1 + Gt(Gi + vêp)) ="vofap)f where fl3 = (^ fc1? . . . , A:^^.

Therefore, we can rewrite (7.19) in the form

F(ea)=V^ + GJ(Gi + ^(ea))^(y^+GJ)( 2 2 ^ M =

= vo( ) 2*¥=i, ' i , hi

Per(a)Let ja = (j, i1} . . . , i z ) . T h e n

r(/a) = {7 = (A:o, /c1? . . ., &M) | A:o ^ {/, ^J , ^ ^ {ilf j 2 } , . . .}

It is not hard to check that

(7.7') ( 2 )

222 £ M. Gel'fand and V. A. PonOmarev

So we have proved that if (7.7') F1 + Gj(Gj + vl(ea)) = v°foa) holds forall a = (zl5 . . . , i\), X < / - 1, and (7.8)

T (ei, 2 *P) = T (îi (** + S *p)) for all a = (z'i, . . . , i\), X < /, thenperca) pgr(a)

(7.7') is true for any a = (il9 . . . , /,).Step 4. Now we show that if (7.7') holds for any

a = (i"i, . . . , i\), X <_/, and (7.8) for any a = (iu . . . , i\), X < /, then(7.8) holds for every a = (/, iu . . . , it) G A(r, I + 1); that is, we provethat for every t ¥= /, /x

(7.8') p(e-)=p(ej S «T) = p(e;)(^+ S ey).V 6 r ( ) Y 6 F ( )

To do this we show that

(7.20) Vi + Gi(G'i + vL(ea))=V>(eJ(et+ S eY)).ero')

From this (7.8) follows almost immediately.Just as we have proved (7.14'), so we obtain

F (ea) = F* + G, ( j ] G.fc (G;fe + X) + GSr_t

where X = S v1^).

We can number the subset / — {/, ix} = {s3, . . ., 5r_1? sr} so that5^.! ^ i2, that is, 5r_x ^ {/, i l t i2}. Then, by arguments similar to those instep 3c, we obtain

(7.21) F (ea) = V + G,- ( j f GSft ( ^ + X) + 6 ^ + GSf + X + F ' C j .

Using the same techniques as in step 3, we see that (7.21) can be trans-formed into:

(7.22) F{ea) = Vi+GJ(Gi+v1(ea)) = 'tf>{ej(ehi+ 2 *,)),vero'a)

where jot. = (/', h, • • • , //)•By construction, 5r can be chosen subject only to the condition

/» ii}. Therefore, comparing (7.7') and (7.22), we obtain

(7.23)

where t fc {/, ^J .By Proposition 7.1, for every x in ZX we have Vv°(o:) = p(x). Applying

the map V: R ->• F to both sides of (7.23), we obtain

P (ej 2 «v) - P (^ (^ + S ^v)) , ^ { / , «i}-veroa) ver(ja)By construction, p was an arbitrary representation in V. We have nowproved (7.8) and with it Proposition 7.5.

7.5. Proof of the formula <p(ii, . . . , it) Vl = p (/*,...ifo). The element


fix ...i/O °fDr i s constructed from a sequence a = (ix, . . . , il9 0) in thefollowing way:

Pe(wherer(a) = (P = (*lf . . ., kt) £ 4(r, 0 | *x £ {»lf *2}, k2

Thus, the polynomial of the second kind / ^ / / 0 is defined in terms of thepolynomial of the first kind e^, where ]3 E {ku . . . , kj) E ,4(>, /). Note thatby the definition of A(r, /), all kt E / = {l, . . ., r). For example,

etio = eit S îi2o = îi 21 0*1*1. where /cj ^ i l s /2 and A:2 ^ /2.

THEOREM 7.3. Let V1 be the representation space of pl = (®+)lp-

The proof of this theorem easily reduces to that of the followingformulae:

(I) cpJF1 = p(/i0),(II) q>}(p1(/ii...«Mo)) - P (/iii...iMo), ^ > 2.

By definition, / ;0 = ej 2 ^- Consequently, (I) can be rewritten in

the following form:

The truth of this equation follows easily from the definition of the ele-mentary map cp).

Now (II) is obtained from the following assertion, which is analogous toProposition 7.3.

PROPOSITION 7.6. Let p1 = O+p, and let v°, v1 be the representationsof Dr, r > 4, in R = 0 p(e ) defined by the formulae

= Gf + F1, vx(^) = GIV1. Then for every fa = f^ //0, l>\,and every

F1 + Gj(G- + v1^)) = v°(fja),

where ja = (/, ^'i, • • • , //, 0).The proof of this proposition is, essentially, a repetition of that of

Proposition 7.5, with // changed to 0 in a = (il9 . . . , it). A differenceoccurs only for / = 1, when we have to prove thatV1 + G,(Gj + F1) - v°(/j0). Here is a proof:

V* + Gj (G] + F1) = (F1 + Gj) ( 2 (V' + Gt)) =

224 ^ M. Gel'fand and V. A. Ponomarev

All further steps in the proof of Proposition 7.5 remain valid whenix = 0, that is, when e^ , M / / i s changed t o / ^ . . . ^ o -

§8. Complete irreducibility of the representations p*, i

8.1. Equivalence of the representations pt, i and pt,i> ^n §6 we have givena functorial definition of the representations pt, i, namely,Ptti = ((I)~)z"1p?, i for / ^ 2. In §1 we have given a constructive definitionof the representations pt, i for / > 2, which we repeat here.

By Wtfi (t 6 {0, 1, . . ., r} , I = 2, 3, . . .) we denote the vector spaceover k with the basis r)a, a €z At(r, I) where

At(r, I) = {a = fe, . . ., i M , 0 Mfe e / = {1, • • .. r}

and ix =£i2i i2 ^ i3 , . . ., iz_2 =£ii-x, ii-

By Zr / we denote the subspace of H r z spanned by all vectorsr

ga;k= S T l i , . . . i f t . . . i z_1 i , w h e r e a = ( i i , . . . , / A , . . .Aû t),

and the summation is over all a' = (il9 . . . , ik, . . . il_l, r) in which all iswith s =A /: are fixed.

We set Vt l = Wt x\Zt h and denote by 0 the canonical mapft: H r / -» Fr ; ; we denote the vectors ftr]a by Jo . Thus, Vt z is the spacespanned by the vectors £o , o: G ^4r(r, /), for which S iii...ife...iz_1t = 0.

Now let / be a fixed index / E / . By F;- r t we denote the subspace ofWt j with the basis {t]a}, where a has the form a = (/', *2> • • • » ' /-l» ^*We define a representation pf,; in F r / by setting pt,i(ej) = ft^j, t,/-

To establish the isomorphism pt.i ^ p?, z, J > 2 , it is convenient tointroduce auxiliary representations vt x in Wt x. We setvt i(ej) = Fj t [ + Zt j . We now list some of the simplest properties of thevt t. Obviously, the map ft: Wt x -> Fr / defines a morphism ofrepresentations ft: vt) t -+• pt, i- We introduce the trivial representation Tin Zr z by setting %(et) = Ztl for all / G /. It is easy to see that T is amultiple of the atomic representation po, i. It turns out that

(8.1) vttls* p u ® T.

To establish this isomorphism, it is sufficient t o choose in Wt x any sub-space U such tha t UZtj = 0 and U + Zt x = Wt i- It can be shownelementari ly tha t vtti = vt, i \v + vttt |Zf z and that vt,i \v ^ pt,i andVf. | Z t i ^ T. Thus , v/, z differs from pt,i by the trivial representa t ion T.

It follows from (8.1) that <3)-vt>l ^ ^ "p f . z © <D"T, and it is clear fromthe definit ion of T that O ~ T == 0. Consequent ly ,


(8.2) Q'Vt.i £< ®~Pt,i-

I n a d d i t i o n , w e s e t , b y d e f i n i t i o n , vtti = p t f i = pt, i -P R O P O S I T I O N 8 . 1 . For every I > 1

PROOF. We proceed first with an auxiliary construction.a) We denote by Yj the subspace of Wt l spanned by the vectors

gjfok = 2 I}™.. -\"-h i«» w n e r e 2 < k tjZt /. We setr

y = S y ; . Clearly, F C Z f / . W e denote the factor space Wt tY by G and

the canonical map PVr z -> Q by 5. It follows from the relation F C Zr z

that i>: ^ j -> Ff / splits into the compositum of the epimorphisms

where 0 is the epimorphism with kernel Ker 6 = dZt /# From the proof,r

which we give repeat below, we can deduce that Q ^ © p*. i{et).

b) We write <2;- = 5F;- r /# By definition, PFt, t = 2 ^ i , *, i a n d t n i s s u m

is direct. Clearly, Q = 6 ^ , z = 6 S ^ i , *. z = S ^ i , *, z = S <?;• W e h a v e

i ' 3 ' 3=1

also defined Ker 6 = Y = S ^;» where y;- C FJ>tl. Hence it is easy to

deduce that the sum (? = 2 (?./ i s direct.

c) Let / be a fixed index. We define maps fij>: Wt / - A -> Wr z by meansof the system of equations

f 0 if y = i2,(8'3) ^ - ^ ^ i ^ . . , ^ , if ,*=«, .

r

We set jut = 2 Pi- Thus,3=1

r

f A T l i 2 . . . i z _ 1 t = ^ 2 T l s i 2 . . . i M t = ^ a ; l ,

where a = (/l5 /2, . . . , //_!, 0- Next we set y = 8JJL and 7;- = 5JU;-. Thus,the diagram

commutes.


LEMMA 8.1. The maps y and 7;- defined above have the followingproperties'.

(I) Im 7 = 5 Z r / ;(II) imyj^Qj;'(III) Ker y, = vtt ,-ifo) = Fjt tt ,_i + Zt, n .A proof of this lemma is given later; first we complete the proof of

Proposition 8.1. From (II) and (III) in Lemma 8.1 it follows thatQj g* Wt, z-i/Ker yy = Wt, i-i/v,. i-i(ej). Thus, Qe*®Wtt ^ / v , , ^(ej). It is

5also clear that the map y: Wt t_x -» Q is such that

y = SJA == 6 2 MJ = S ^^J == S 7J- Consequently, O-v<} z - 1 is a representa-3 i j

tion in V = Coker y = Q/lm y. From (I) in Lemma 8.1 we obtainV = Q/lm y = QISZti t = (?/Ker 6 = Vt% t. Therefore^"v*. z_i(^j) = e(?j = Q(6Fjt t, i) = QFJt t, i = pt, zfe). We have now provedthat ®-v t | Z_i s* p/, .

PROOF OF LEMMA 8.1. From the definition of the maps \XJ and \i itcan be verified immediately that

f a) Im HJ = ^-, <t j , b) Ker ^ = Fjt tt z_ l f

( 8 > 4 ) I c) Z,,i = y 1 ) ^ ^

(I) From (8.4) c) we find dZttl = b(Y += 6(Ker 6 + \iWt, i-i) = 6\iWtt z-i = yWt, z-i = Im y.

(II) From (8.4) a) we find Im 7; = yjWt, 1-1 = $\XjWt, i-\ == SFJt tt t = Qj.

(III) Obviously,x G Ker yj = Ker 6fx;- <=> ji (a;) 6 Ker 6 <=> jut o:) 6 (Ker 6)(Im \i3). Using

r

the equations Im /i/ = ^} f /> Ker 6 = 2 ^ j a n^ the relation Y- C F;-1 /5

it is easy to show that (Ker 5)(Im ju;-) = Yj. Thus, we can write

(8.5) z g K e r Y , - ^ ^(x) 6 Fj .

As we have mentioned, Ker /x;- = F;-1 t_x and y;- = \XjZt l_l. Thus,

From this and (8.5) it is easy to see that Ker yj = Fjtl_l +Z r / _ 1 .Th i sproves the Lemma.

PROPOSITION 8.2. pt,i^pt,i.The proof is by induction on /. By definition, pt, 1 = 9t, i« Suppose that we

have proved that pt.k = Pt, k for all k < / - 1. Then we show thatp*f 1 ^ pt,i- ^y (8.2), Q>~pt,i-i = O~v^.M, and by Lemma 8.1,


0>~vt, i-i = Pt, i> Consequently, ®~p*. ;_t ^ p/ , ; . By induction, we assumethat p,, w ^ p£ z_j. Hence pi, t ^ O~pf, M ^ ®-p£ IM = pt

+t z.

8.2. Complete irreducibility of the representations ptt {. In this subsectionwe explain the basic steps in the proof of the theorem on completeirreducibility. A full proof of the theorem will be published separately.

THEOREM 8.1. Let pt, i and (pT,i) be indecomposable representationsof Dr, r > 4, in spaces Vtl over a field k of characteristic 0, withdim Vtl > 3. Then pt, i and (pt,i) are completely irreducible, that isp±z(Z)r) ^ P(Q, m), where P(Q, m) is the lattice of linear submanifolds ofthe projective space of dimension m = dim Vt l - 1 over the field Q ofrational numbers.

REMARK. The restriction dim Vt l > 3 occurs because the followingindecomposable representations are not completely irreducible: 1) all theatomic representations p?", i and pt, I for t £ {0, 1, . . ., r} for any latticeDr', and 2) the representations p£ 2 and (pjt 2), t £ {1, . . ., r}, for D4. Forthe latter representations, dim Fr 2 = 2.

We describe the basic steps in the proof of Theorem 8.1. We denote byL the sublattice of X (Vt, 1) generated by the one-dimensional subspaceskla, a G At(r, 1). Since kla £pj"t / (D

r), it can be proved elementarily thatL^p i , z ( £ r ) .

The proof of the isomorphism L = P(Q, m) when dim Vt t > 3 is basedon the following assertions.

Let {£i}i£A be a set of non-zero vectors in V. We call the set {h}i£Aindecomposable if for every subset B of A (B =7 0 , B ~^ A) the intersection

of the subspaces VB—-^2}k\i and V-B = 2 &!./ is non-empty:

FB F5- ^ 0 and VB + F^ = V.

A set {IJigA in a finite-dimensional space F over a field /c ofcharacteristic 0 is called rational if we can choose a subset B C A oflinearly independent vectors {lt}i£B such that for any / € A

lj = 2 ailt > where a*£Q.i6B

PROPOSITION 8.3. Le/1 pt, 1 be an indecomposable representation ofDr (r > 4) in a space Vt t over a field k of characteristic 0. Then for/ > 1, the set of vectors £a, a E A(r, /), is indecomposable and rational.

The proof of the indecomposability of the set {la}a£At(r,i) follows easilyfrom the indecomposability of pf,j. The proof of rationality is alsoelementary.

In establishing the isomorphism L = P(Q, m), the central fact is thefollowing theorem, which is of independent interest.

THEOREM 8.2. Let {lt}i£A be an indecomposable and rational set

228 ^ M. Gel'fand and V. A. Ponomarev

of vectors in a space V over a field k of characteristic 0. / / dim V > 3,the lattice L generated by the one-dimensional subspaces k%t is completelyirreducible, that is,

L ss P(Q, m), where m = dim Vt x - 1.

A proof of this theorem will be published separately.

References

[1] I. N. Bernstein, I. M. Gel'fand and V. A. Ponomarev, Coxeter functors and Gabriel'stheorem, Uspekhi Mat. Nauk 28:2 (1973), 19-33.= Russian Math. Surveys 28:2 (1973), 17-32.

[2] G. Birkhoff, Lattice Theory, Amer. Math. Soc, New York 1948.Translation: Teoriya struktur, Izdat. Mir, Moscow 1952.

[3] N. Bourbaki, Elements de mathematique, XXVI,Groupes et algebres de Lie, Hermannand Co, Paris 1960. MR 24 # A2641.Translation: Gruppy ialgebry Li, Izdat. Mir., Moscow 1972.

[4] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103.[5] I. M. Gel'fand and V. A. Ponomarev, Problems of linear algebra and classification of

quadruples of subspaces in a finite-dimensional vector space, Coll. Math. Soc. IanosBolyai 5, Hilbert space operators, Tihany (Hungary) 1970, 163-237 (in English).(For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765.= Soviet Math. Doklady 12 (1971), 535-539.)

[6] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in thecollection "Investigations in the theory of representations", Izdat. Nauka,Leningrad 1972,5-31.

[7] L. A. Nazarova, Representations of quivers of infinite type, Izv. Akad. Nauk SSSRSer. Mat. 37 (1973), 752-791.= Math. USSR - Izv. 7 (1973), 749-792.

Received by the Editors, 10 June 1974

Translated by M. B. Nathanson

Dedicated to P. S. Aleksandrov,who has done so much for the development

of general ideas in mathematics

LATTICES, REPRESENTATIONS, ANDALGEBRAS CONNECTED WITH THEM I1


In this article the authors have attempted to follow the style which one of them learned fromP. S. Aleksandrov in other problems (the descriptive theory of functions and topology).

Let £ be a modular lattice. By a representation of L in an .4-module ./I/, where A is a ring, we mean amorphism from L into the lattice X (A, M) of submodules of M. In this article we study representationsof finitely generated free modular lattices Dr. We are principally interested in representations in thelattice £(K, V) of linear subspaces of a space V over a field K (V= Kn).

An element a in a modular lattice L is called perfect if a is sent either to O or to V under anyindecomposable representation p : L -*• X(K, V). The basic method of studying the lattice Dr

is to construct in it two sublattices B+ and B~, each of which consists of perfect elements.Certain indecomposable representations p+

t / (respectively, p~tj) are connected with the sublattices B+

(respectively, B~). Almost all these representations (except finitely many of small dimension) possess theimportant property of complete irreducibility. A representationp:L -*- £(K, V) is called completelyirreducible if the lattice p(L) is isomorphic to the lattice of linear subspaces of a projective space over thefield Q of rational numbers of dimension n - 1, where n = dim^ V. In this paper we construct a certainspecial K-algebrdAr and study the representations p ^ : Dr -> XR (A r) ofDr into the lattice of right idealsof Ar. We conjecture that the lattice of right homogeneous ideals of the Q-algebra/4r describes (up tothe relation of linear equivalence) the essential part of Dr.

Contents

§ 1. Basic definitions and statements of results 229§2. The category ek{L, K) 235§3. Perfect elements. Elementary properties of the lattices B* and B~ . . 237§4. Proof that B\\) and B~(\) are perfect. Atomic representations . . . 240§5. The functors 3>+ and <£>" 243References 246

§ 1. Basic definitions and statement of results

This article is a further development of the authors' paper [7 ] , but canbe read independently.

1 The second part of this article will be published in these Uspekhi 32:1 (1977), 85-106.

229


1.1. Lattices. A lattice I is a set with two operations: intersection andsum. If a, b E L, then we denote their intersection by ab and their sum1

by a + b. Both these operations are commutative and associative, and,moreover, satisfy the axioms of absorption: a(a + b) - a and a + ab = a.

An order relation is defined naturally in a lattice L:aCb*=*a + b= b.It is easy to deduce that aa - a and a + a = a for every a E L.

A lattice L is called distributive if for any a, b, c G L(1.1) a(b + c) = ab + ac,(1.2) a + be = (a + b) (a + c).It can be shown that a lattice is distributive if it satisfies at least one of

the equations (1.1) or (1.2).A lattice L is called modular if for all a, b, c £ L

a C b =• b(a + c) = a + be

EXAMPLE 1. Let A be a ring and let M be a left (right) A -module.Then the set of all submodules of M is a lattice with respect to the operationsof intersection (O) and sum (+). We denote this lattice by X(A, M).

EXAMPLE 2. In this article we shall most often consider the latticeX(K, V) of linear subspaces of a finite-dimensional vector space V over afield K. When dim V = n (that is, V = Kn), we also denote this lattice byX(V) or X(Kn). If Ux and U2 are subspaces of F, then we denote theirsum ana intersection by U1 + U2 and Ux U2.

EXAMPLE 3. Let P(F) = Yn(K) be the projective space corresponding toV = Kn+l. Then X{V) is well known to be isomorphic to the lattice oflinear submanifolds of the projective space P(F). We call this lattice aprojective geometry (PG(F)). Thus, if V = K3, then the elements of thecorresponding geometry are the points and lines of the projective planeV2{K). (If a and b are points in Y2(K), then a + b is the line passingthrough a and b. If A and B are lines, then AB is their point of intersection.)

The basic objects of our investigation are the free modular lattices Dr

with a finite number of generators (el9 . . . , er). The lattices D1 and D2

are obviously finite. It is not difficult to show that the lattice D3 is alsofinite (see Birkhoff [3]). The lattices Dr, r > 4, have a very complicatedstructure. We are by now only close to an understanding of the structureof Z>4 (the factor lattice of Z)4 with respect to linear equivalence, whichwe define in 1.2).

1.2. Representations. Let I be a modular lattice and X (A, M) thelattice of submodules of a left A -module M. A representation p of L in Mis a morphism2 p: L -+ X(A, M). Here, for any x, y G L we havep(x + y) = p(x) + p(y) and p(x>>) = p(x) n p(y), where p(x) and p(y) aresubmodules of M.

The intersection of the elements a and b is often denoted by a n 6 or a A &> and the sum by a u ft ora\J b.

A morphism p:Lx -*L2 of lattices L x and Z,2 is a mapping such that p(xy) = p(*)p(y) andP(* +y) = p(x) + pO) for all x, y e L,.

Lattices, representations, and algebras connected with them 231

Throughout this article we are concerned with only two types ofrepresentation.

1) A representation of a lattice L in a finite-dimensional vector space Vover a field K (p: L -» X{K, V)). Such a representation associates withelements x, y S L subspaces p(x), piy) C F.

2) A representation of L in the lattice X(A) of left ideals of a ring ,4.We introduce in L an equivalence relation by setting x ~ y if

p(jc) = p(y) for every representation p: Z, -+£(K, V) in any space F overany field K. It can be shown by examples (even in D4) that there existlinearly equivalent, but unequal elements.

We consider the set whose elements are the classes of linearly equivalentelements o f / / . T h e operations (•) and (+) carry over naturally to this set.We denote by Dr the lattice obtained in this way. The aim of this paper isthe study of the lattices Dr.

An important technique for the study of Dr is the construction of thesublattice B of perfect elements, to whose definition we now turn.

1.3. Perfect elements in Dr'. An element i; in a modular lattice L is calledperfect if for every indecomposable1 representation p: L -+%{K, V) eitherp(v) = V or p(v) = 0.

In the free modular lattice Dr with generators e1? . . . , er we constructtwo sublattices B+ and B~, whose elements, as we shall prove later, are allperfect.

For every integer / > 1 we construct a sublattice B+(l) consisting of 2r

elements. We shall call this sublattice B+(l) the /-th upper cubicle.It is quite simple to construct B+(\). Namely, we set ht(\) - 2 e,-.

i*t

Then the upper cubicle B+(l) is the sublattice of Dr generated by theelements hx{\), h2{\),..., hr(\). It is not difficult to prove (see §3) thatB+(\) is a Boolean algebra with 2r elements. Thus, B+(\) is isomorphic tothe lattice of vertices of an r-dimensional cube with the natural ordering.

We proceed to the definition of the cubicles B+(l). The elements ofB+(l) are constructed with the help of polynomials e^ iy which are ofindependent interest. We denote by Air, I) the set whose elements are thesequences a = (ilf i2, . . . , //) of integers 1 < ij < r such that ij ¥= iJ+l

for all 1 < / < / - 1. We set, by definition, Air, 1) = / = {!, • • ., r). Theelements ea = e^ t are defined by induction on / as follows. If / = 1 anda = (z'i), then ej = et ; if / > 1 and a = (il9 . . . , /z), thenea = et ( 2 eB), where F(o:) C A(r, / - I ) consists of the sequences

1 / * e r ( r. . . , A:/_1) constructed from a fixed a in the following way:

A representation p in a space V is called decomposable if there exist non-zero subspaces Ult U2 in Vthat are complementary to each other (Ul U2 = 0, Ux ~v U2 = V) and such thatp(x) - p(x) Ux + p(x) U2 for every x e L.


T (a) = {p == (kt, . . . , ki-i) I kt ^ ii9 i2; k2 =£ i2, ia, . . . , k^ *£ U_u it}.

For example, if a = (ilt /2)> then T(u) = {?> = (ki)\ki=£iu i2}, and soîii2 = eh( 2 ej)'

Now we define the elements ht(l). We set

a£At(r, I)

where At{r, I) is the subset of A(r, I) consisting of all sequencesa = (/l5 . . . , tz_l5 0 in which the last index is fixed and equal to t.Further, we set

ht{l) = %ej(l).tet

Then we define B+(l) to be the lattice generated by the elementsMO, -.., W

It is not difficult to prove (see §3) that B+(l) is a Boolean algebra andthat elements from different cubicles B+(l) and B+(m) can be ordered in thefollowing way: for every vt E B+(l) and vm £ B+(m) it follows fromI < m that Vi D vm. Thus, the set

B+ = u B+ (I)

is itself a lattice.oo

A second set B~ = U B~(l) consists of the elements dual to those of/ = i

B+. (We say that a lattice polynomial g(el9 . . . , er) is dual to a latticepolynomial /(e1? . . . , er) if it is obtained from / by interchanging theoperations (+) and (H). Thus, for example, ei(e2 + e3 + e4) is dual toex + (e2eze4).

One of the main theorems of this article is the following.

THEOREM 1. The elements of the lattices B+ = U B+(l) and

B~ = U 5~(/) are perfect.i=i

1.4. Characteristic functions of an indecomposable representation. LetL be an arbitrary modular lattice. Then the set B of perfect elements inL is a sublattice of L (see §3). If p is an arbitrary indecomposablerepresentation of I (p: L -+%(K, V)), then every perfect element has,by definition, the following property: either p(v) = 0 or p(v) - V(V is the representation space of p). Thus, to every indecomposablerepresentation p there corresponds a function xp

o n the set B ofperfect elements, which is defined as follows:

0 if p(i>) = 0,

1 if p{v) = V.


We call xp the characteristic function of p.In the lattice Dr we have defined two sublattices of perfect elements B+

and B~. We claim that p(iT) C p(u+) for any representationp{p: Dr -> X (K, V)) (r > 4) and any iT G 5 " and u+ G £ \ We denote byB the sublattice of perfect elements in Dr generated by B+ and B~. FromP(V~) C p(i/) it follows that every characteristic function xp defined on Bbelongs to one of the following three types: 1) Xp(u

+) = 0 for somev+ G B\ hence xp(*O = ° for all v~ G B'\ 2) XpOO = 1 for somev~ G B~, hence, xp(V) = 1 for all v+ G B+; 3) x p 0 O = 0 for all v' G 5"and Xp(^

+) = 1 for all v+ G £+. We denote the last function by x i ; In §7we prove the following theorem.

THEOREM 2. Let p be an indecomposable representation of the latticeDr (p: Dr -> «£(#, V)) (r > 4). / / the characteristic function \p of p isof the first or second type, then p is defined by its characteristic functionuniquely up to isomorphism.

We shall find all indecomposable representations corresponding to thevarious characteristic functions of the first or second type. In the followingsubsection these representations will be constructed explicitly.

As for the indecomposable representations p whose characteristic functionsare of type 3 (xp = Xo)> w e know at the moment only that there areinfinitely many of them. In the case of D* the classification of all suchrepresentations is known [6]. For the lattices Dr (r > 5), the classificationof the indecomposable representations with xp

= Xo contains as specialcases such problems as the determination of a canonical form for severallinear operators Al9 . . . , An (At: V ->- V).

1.5. The algebra Ar and the representation pA . Let K be any field. Wedefine the i^-algebra Ar as the associative i^-algebra with unit element egenerated by £0> £i> • • • > £/- with the relations

(1) E ? = 0 (i = l f . . . , r ) ,

(2) Io2=&),

(3) Ui=h (* = 1, . . . , r ) ,

The standard monomials in Ar are the products ^ . . . ^ %t such that1 < ij < r, ij * i /+1 for all 1 < / < / - 1, it_x ¥= t, and 0 < t < r. Thus,in a standard monomial £0

c a n occur only in the last place. It is easy tosee that any non-zero monomial can be brought to standard form. Thestandard monomial ^ . . . £/, , £r is also denoted by £a = £Zi A t, wherea =(*! , . . .Ji_xt).

The degree of the monomial £a is the number d(i~a) defined in thefollowing way: d(e) = d(%0) = 0, d(£t) = 1 for everyi ¥= 0, d(Za$p) = d(ia) + d%) if ? a ^ ^ 0. The degree of the element 0is left undefined.


We denote by V{ (Vl C Ar) the space of homogeneous polynomials ofdegree /. It is not difficult to show that this introduces a grading in Ar:Ar = Vo © Vx © . . . © Vx © . . . (ViVj C Vi+j), where1 Vo = KE © £ £ 0 .

In §8 we shall show that the algebras Ar (r > 4) are infinite-dimensional,and that dim Vl > 0 for all / > 0. The algebras A\ A2, A3 are finite-dimensional, and their dimensions over K are 3, 5, and 11, respectively.

We denote by XR{Ar) the lattice of right ideals of Ar (with respect tothe operations of intersection O and sum +).

We define the representation pA\ Dr -+ XR(Ar) by setting pA(et) = %tAr,

where £tAr is the right ideal generated by £z-, and the et(i = 1, . . . , r) are

the generators of Dr.In §8 we shall prove the following interesting theorem, which establishes

a connection between the lattice polynomials ea (which were defined in 1.3)and the monomials £a in Ar.

THEOREM 3. For every a = (zl5 . . . , il_l, t) E At(r, I) (I > 1) we havepA(ea) = %aA

r', where %aAr is the right ideal generated by the monomial

£a = £/, • • • £/z_ x Sf -This result is due to Gel'fand, Lidskii, and Ponomarev.1.6. The representations ptl. Let A%t be the left ideal generated by the

element £ r We introduce the following notation:

(* = 0, 1, . . . , r ; Z = 1,2, . . . ) .

According to this definition, Vi 0 = 0 if / # 0 and Fo 0 = ^? 0 - F°r

/ = 1, F,- j = K%t if / ¥= 0 and K01 has the dimension r- 1 and is thesum of the one-dimensional subspaces Kô. For I > 2, every subspaceVtl is generated by one-dimensional subspaces K%a%t, where £a£r is amonomial of degree /.

We define a representation pt l of Z)r in Vt i as follows. We set

where ^^4 is the right ideal generated by £,- (1 < / < r).We define a representation p e of Z)r in i e (where 8 is the unit element

of Ar) by setting p z{ei) = 0 for every et G Dr.It is elementary to prove (§8) that pA is isomorphic to the direct sum

p A ^ p e 0 p o , o © ( 0 ( 0 p#. i))-

It turns out that the representations ptj so constructed possess thefollowing remarkable properties.

THEOREM 4. (i) The representation p 0 0 and the representationspt j(t = 0, 1, . . . , r\ I = 1, 2, . . . ) of Dr(r > 4) are indecomposable. Anytwo representations ptl and pt>j> such that (t, 1) =£ (t\ I') are not isomorphic.

(ii) p 0 0 and pt j(t = 0, 1, . . . , r; / = 1, 2, . . . ) #re the only representa-tions whose characteristic functions are of the first type (that is,

By Ke (respectively K£o) we denote the subspace generated by the element e (respectively, £0).


Xp(V) = 0 for some v+ G B+).Let V be a linear space over a field K of characteristic 0. A representation

p of a modular lattice L in V is called completely irreducible ifp(L) C #(F, K) is isomorphic to the lattice ^(Qn) wheren = dim^ V (n > 3) (that is, p(L) is the projective geometryP ^ - i ( Q ) in > 3) over Q.

The following result holds.THEOREM 5. All representations ptl\ Dr ^X(VtiI; K) (r > 4) over a

field K of characteristic 0, except finitely many, are completely irreducible.The only representations that are not completely irreducible are the

following:a) p0 0 and pt x (i =£ 0) for any r > 4;b)p/ ;; (iÔ)forr = 4.1.8. The lattice F*. We introduce an equivalence relation R in Dr by

setting x = y (mod R) if ptj(x) = ptj(y) for any representationptl\ Dr -> X{VttU K). We denote the factor lattice Dr/R by F*.

We now state an important conjecture about the structure of thelattices F*(r > 5). Let Ar

Q be the Q-algebra A\ where Q is the field ofrational numbers. A right ideal in ^4Q is called homogeneous if it is equalto a finite sum of ideals ftjA

rQ, ftl G Vtl. The lattice of right homogeneousideals of AQ is denoted by o//f {Ar

Q).CONJECTURE. The factor-lattice F* of Dr (r > 5) is isomorphic to the

lattice QM(AQ) of right homogeneous ideals of the Q-algebra AQ.

We shall soon publish some results obtained jointly with B. V. Lidskii,which bring us close to a proof of this conjecture.

§2. The category M (X, K)

2.1. The category ^ (X, K). Let px and p2 be two representations of amodular lattice L in spaces Vx and V2, respectively. By a morphism^ : Pi ""* P2 w e mean a linear transformation w: Vx -+ V2 such thatuPi(x) 5 P2(-x:) f° r aH x £ L, where wpi(x) is the image of the subspacePJ(X) under the transformation u.

We often denote a morphism w simply by u (the corresponding lineartransformation).

We denote by Horn (pl5 p2) the set of all morphisms from px to p2.It is not difficult to verify that this determines a category %(L, K) thatof finite-dimensional representations over K. In ffl(L, K) the direct sumPi e P2 of any pair of objects px and p2 is defined in the natural way.

The category M(Dr, K) is the object of our study. It is easy to showthat M is additive but not Abelian.

Let p E M(L, K) be a representation of a lattice L in a space K. Wedenote by K* the space dual to V. We define the representationp* G m(L, K) in F* by p*(jc) = (p(x))1 for all x G L (where


(p(jc))1 C F* is the subspace of functionals that vanish on p(x)). We call p* therepresentation dual to p.

2.2. Decomposable representations and admissible subspaces. Let p be arepresentation of a modular lattice L in a linear space V. A subspace U of V iscalled admissible with respect to a representation p if for all x, y €i L

(I) tf(p(s) + p(z/)) = Up(x) + ffp(y).

It is not difficult to show that (I) is satisfied if and only if(10 U + p(x)p(y) = (U + p(s))(ff + p(y)).

PROPOSITION 2.1. Z,e p be a representation of a lattice L in a space V.Let U C V be a subspace of V, U" = V/U be the quotient space, and6: V -> U" be the canonical map. Then the following conditions areequivalent:

1. U is admissible with respect to p.2. The correspondence x •-> Up(x) defines a representation in U.3. The correspondence x *+ 6p(x) defines a representation in U".The proof is elementary (see [3] ).•The representation in an admissible subspace U defined by the corre-

spondence x *-» Up(x) is called the restriction of p to U and is denoted byPit/-

PROPOSITION 2.2. A representation p G M(L, K) in a space V isdecomposable if and only if there exist non-zero subspaces Uli . . . , Un

such that V = Ux © . . . © Un and for every x G L

REMARK l . I f a representation p G M(L, K) is decomposable, withn

p = © Pj and if Ut the subspace corresponding to p,-, then the Ut are

admissible. The converse however, is false, that is, if V = © Ut and if eachi= l

Ut is admissible, then it does not follow that p splits into the direct sum oftheir restrictions.

QUESTION. Is it true that if U is an admissible subspace, then there isa subspace U' complementary to U (that is, UU' = 0 and U + U' = V)such that £/ and Uf define a splitting of p into a direct sum?

PROPOSITION 2.3. 4 representation p G # (Z)r, ^) ^pto into a directn

sum p = ® Pi of representations pt if and only if there are non-zero sub-i=1 n n

spaces Ux, . . . , Un such that V = © Uj and p{et) = 2 p(et)Uj for everyj= l /= l

/ G {1, . . . , r) , where the ef are the generators of Dr.*


§3. Perfect elements. Elementary properties of the latticesB+ and i T

3.1. In this section we prove that the set B of perfect elements in amodular lattice I is a sublattice.

PROPOSITION 3.1. Let a and b be perfect elements of a lattice L. Thenso are a + b and ab.

PROOF. If an element a is perfect, then there exists a subspace Vcomplementary to p(a) (that is, p(a)V = 0 and p(a) + V' = V) such that psplits into the direct sum of representations p = pq + p', wherePa = P\p(a) anc* P' = P\v- In particular, for x = b

(3.1) p(b) = 9(b)p(a) + p(b) V.

Note that p(b)p(a) = p{ba) and, by Proposition 2.1, p(b)V = p\b).Thus, we can rewrite (3.1) in the following form:

(3.2) p(b) = p(ab) + p'(6).

Now Z? is perfect, consequently, in V there is a subspace V" such thatF ' = p'(b) + K" and p = p'b + p" , where p'b = p'|p'(Z>> and p" = p'\v».Thus, p = pq ® p' = pa® (p'b © p") = (pfl e p^) e p".

We claim that pa © pj, is the restriction of p to the subspace p(a + &).Indeed, by definition, pa ® p'b - Pf\P(a)+p\by Moreover, using the fact thatp(ab) C p{a) and p(Z?) = p(ab) + p'(^)3 we can write

P (a) + 9' (b) = (p (a) + p (ab)) + p7 (&) = p (a) + (p (a&) + p' (6)) -

Thus, p = pa+b © p".In other words, a + b is perfect. The fact that ab is perfect can be proved

similarly."COROLLARY 3.1. The set S of perfect elements of a modular lattice L

is a sublattice of L. •PROPOSITION 3.2. Let L be a modular lattice, and p G M(L, K) an

arbitrary representation. Then the image p(a) of a perfect element is aneutral element of the lattice p(L) C X{V, K),that is, for any xt, x2 £ L

PROOF. The element a is perfect, hence there exists a subspace Vsuch that F ^ p(#) © F ' and p = pa ® p' = P\p(a) ® P\v'- Consequently, for anyxt G I we can write p(x() = p{xt)p{a) + pix^V.

Using this identity, we obtain p(a)(p(xl) +p(x2)):= p(a)(p(xx)p(a) +

+ p(xt) V + p(x2)p(a) + p(x2) V) = p{xx)p{a) + p(x2)p(a) + piaXpix,) V ++ P(^2)F'). By construction p(#)F' = 0, and a fortiorip(a)(p(xl)V

f + p{x2) V) = 0. Thus, p(tf)(p(*i) + p (^) ) = p ( ^ i ) + P(a)p(x2).mCOROLLARY 3.2. Let S be the sublattice of perfect elements in a lattice

L. Then p(S) for any representation p E fi (L, K) is a distributive sublattice ofneutral elements of p(L)M

CONJECTURE 3.1. Let L be an arbitrary modular lattice. Then an element


a is perfect if and only if it is neutral.3.2. Elementary properties of the sublattices B+(l). Now we study some

properties of the sublattice B+ in Dr, which, as we shall prove later,consists of perfect elements. The lattice B+ is the union of the sublattices£+(l), B\2), . . . , B\l), . . ..which are called cubicles. The cubicle B\l) isconstructed with the help of special elements ea in the following way. Weset

M0 = 2 ea, MO = 2 MO,<x£At(r, I) ij=t

where a. = (fl9 . . . , it_lt t) E At(r, I) is a sequence such that/;., t E {1, . . ., 7*}, /;. =£ z/+1, /,_! T£ r (the definition of ea is on page 69).

Now B+(l) is, by definition, the sublattice generated by the elements

MO, .. . , ^(0.THEOREM 3.1. B+(l) is a Boolean algebra.This theorem is made more precise in the following proposition.PROPOSITION 3.3. (I) Every element vaJ of B+(l) can be written in the

following form:

(i) vatl= 2 M 0 + 2 MO MO,

where a is an arbitrary subset of I ={1, • . ., r).IfaÎ, this can also be written

(ii) vCtt= fl h}(l).

(II) Let $(I) be the Boolean algebra of all subsets of I. Then the corres-pondence a <->• va i defines a morphism vf .^(/)-> B+(l) (that is,

%ub),l = va,l + vb,l u %nb),l = va,l n vb,l for any a, b C I).REMARK. In §7 (Corollary 7.2) we shall prove that the mapping

vf. J?(/)-> B+(l) is an isomorphism.The proof of Proposition 3.3 is based on the following lattice-theoretical

lemma.LEMMA 3.1. Let L be an arbitrary modular lattice, and {e^ . . ., er)

a finite set of elements of L. Then the sublattice B generated by theelements hj = 2 ef (j' = 1, . . . , r) is a Boolean algebra.

i*iThe proof of this lemma reduces to a proof of the formula

2 ei + 2 ejhj == fl hj,

where <£> C b C /. This formula is easily proved (see [7]) by induction onthe number of elements in b. •

3.3. Structure of the lattice B+. We denote by B+ the lattice generated bythe sublattices B\l) (I = 1, 2, . . . ).

THEOREM 3.2. (I) B+ is the union of the sets B\l).


(II) The sublattices B+(l) in B+ can be ordered in the following way: letv+(l) and v+(m) be any elements of B+(l) and B+(m). If I < m, thenv+(l) 2 v\m).

REMARK. We shall prove in §7 that, in fact, for I < m strict inequalityv\l) D v\m) holds.

The proof of this Theorem is based on two lemmas.We recall that

A(r, I) = {a = (il7 . . ., i.) \ ij 6 / = {1, • • ., r},

where ij =fc //+1 for all / < /.LEMMA 3.2. Let p = (i!, . . . , /,) e A(r, I). We set pf = (/j, . . . , i/f /)

for j =£ it. Then ep D epf.LEMMA 3.3. Let ht(l) be a generator of B+(l). Then ea C ht{l) for all

OL e Air, I + 1).The proof of these Lemmas is elementary (see [7]).Now we prove Theorem 3.2. It follows from Proposition 3.3 that the

r

minimal element in B+(l) is vd l - n fy(/), and the maximal element in

B\l + 1) is U/f+i = S eAl + 1) = S ea. It follows from Lemma 3.3i=i af=A(r,l+l)

that O ht(l) D ea for every a G A(r, / + 1). Therefore,r y~s

ve l = ^ ^/(0 — ^ ea = u/ /+1 • Now if ufl / and uft / + 1 are arbitrary/=1 a(=A(r,l+l)

elements of B+(l) and B+(l + 1), then ^ / 2 ^ / 5 ^ / / + i 5 ^ /+1 J and thetheorem is proved.

3.4. The lattice ^~. By definition, the elements of the cubicles B~(l) aredual to those of the cubicles B+(l). For example e^ t = ei% 2 e^ hence,by definition, we set ej .• = e,- + ( n e7). Similarly,

w n r wr, Z) ;><

Each cubicle B~(l) is a Boolean algebra, and the elements v~(l) and v~(m)of distinct cubicles .&"(/) and B~(m) (/ < m) are connected by the relationv~(l) C u"(m). Thus, the lattice B~ generated by the B'(l) is the union0 £-(/) of the sets B~(l).

In particular, just as the maximal and minimal elements vfl and vQl ofthe cubicles B+(l) form a chain

r

so the maximal elements Vj j = 2 /zz- (/) and minimal elements

240 I- M. Gel'fand and V. A. Ponomarev

rve l = n ei(O °f t n e cubicles B (/) are ordered dually:

r _ _ _ _ _D ti = ve, i s i>i, i s i;e, 2 ^ z>i, 2 £ . . . S i>e, z s Vj S . • •

Note that the element v^j is dual to i>7/ and u£j is dual to vel.In §7 we shall prove the following proposition.PROPOSITION 3.4. Let v+ and v~ be arbitrary elements of B* and B~

Then p(u~) C p(u+) for every representation p: Dr -> X(K, V)We also believe that the following is true.CONJECTURE. For every v+ G B+ and v~ G B~

§4. Proof thati?+(l) and "(1) are perfect. Atomic representations

By definition, i?+(l) is the sublattice of Dr generated by the elements^ i ( l ) , h2(l), . . . , hr(\), where hAl) = 2 et. The maximal element in the

rcubicle ^+(1) is vf j = 2 ^-. We note that u7 j is the maximal element in

/= 1

the entire lattice Dr. The cubicle B~{\) consists of the elements dualto the elements of B+(l). (It is generated by the elementshj{\) = O et. In this section we prove that every element of i?+(l) and

B~{\) is perfect.4.1. Atomic representations and the perfectness of i?+(l). We define

representations p] t for t G {0, 1, . . . , r}, which we call (+) atomic. Bydefinition, p\ x is the representation in the one-dimensional space V*tl = Kfor which

1) if t = 0, then p+Oil(e{) = 0 for all 1 = 1, . . . , r;

2) if t ¥* 0, then p ^ ( ^ ) = 0 for i =£ t and p]yl{et) = V*tl.Note that the atomic representations are none other than the

representations p 0 0 and pix defined in §1 on page 72. Namely,

Po5i - Po,o a n d Pu - Pu i f * * °-THEOREM 4.1. Each element va x G J5+(l) 15 perfect.The proof of this Theorem is based on the following lemma.LEMMA 4.1. Let p be any representation of Dr in a space V. Then p is

isomorphic to the direct sum p = p 0 1 4- py- x + Tj, where Tj = P\P(h.(i)y

where p ^ and p~x are multiples of the atomic representations p+Ol and

p^j (that is, pQ1 = p+01 + . . . + p*Ofl, where m0l > 0, and, similarly,

PROOF OF LEMMA 4.1. We set p(ez) = Et and p(hj(l)) = Hj. Thus, thesub spaces Et and H* are such that Hj = 2 Et. We also set


r

Ho = X Ef. Clearly, Ho has the following property: for every / =£ 0

(4.1) EJ + HJ^HO^HJ^H,.

We claim that the element ht = £ e{ of Dr is perfect. We choose sub-7 /*/

spaces C/o and Uj in F to satisfy the relations(4.2) U0H0 = 0, Uo + Ho = F,(4.3) ffjfT, = 0, Uj + £ ,# , = Ej.

We illustrate the subspaces in the following figure:

We shall show that Uf + Hj = Ho. Clearly, EJHJ + Hj = /7;. Using this and(4.1), (4.3), we see that Uj+Hf = Uf +EfHf +Hj = Ej +#y =H0. Thus,Uj + i/y = /f0, and this sum is direct. It easily follows from this equationand the definition of Uo that V = UQ e Uj e Z ..

We now show that the representation splits into the direct sumP = Po1 e P/1 e Tj, where pr1 = p | ^ and r; = p^ . .

For this it suffices to show that for every subspace Et = p{et) (i = 1,. . . , r)

(4.4) p (et) = p («,) f/0 + P (e«) + p (et) Hj.r

Let us prove this. By construction, U0H0 = Uo( 2Ef) = 0. Consequently,/= l

EjU0 = 0 for every i. By construction, Ej = Uj + EJHJ, henceEj = Ej(Uj + JS1^) = EjUj + yZTy. This proves (4.4) when i = /.

If i ¥= /, it is clear that Et C X Ek = H-, that is, £,#,• = £,. Next, it

follows from (Uo + t/y)/^- = 0 that (Uo + f/,)^ = 0, and a fortioriUQE; = 6^,. = 0. Thus, Et = Eô + EtUf + Eflj. This proves (4.4). Itmeans (see Proposition 2.3) that p = p ^ © p|^. © p|^.. Consequently,hj(\) is perfect. ° y J

Thus, the generators /z^l) , . . . , hr{\) of J5+(1) are perfect. By Proposition3.1, this implies that all elements of B+(l) are perfect, and Theorem 4.1 isproved.

To complete the proof of the lemma, it remains for us to establish thatp\v and p$j. are multiples of the atomic representations p*0l and pjx.

r

By construction of Uo, U0H0 = Uo 2 Et - 0, and so £/0£z- = 0 for every

/. Thus, p\v (et) = p(ej)U0 = EtU0 = 0 for every /. It follows thatPo,i = P\u *s isomorphic to the direct sum of the atomic representations

242 /• M. Gel'fand and V. A. Ponomarev

Po,i> t h a t is> Po,i - Po,i © . . . © Po,i> where m0 = dim Uo.

The sub space Uj has the following proper t ies :

a) UfEf = Uf, and b ) UfHf = Uf 2 Et = 0, tha t is, UjEi = 0 for every

i * j . Thus ,

h if i-h

Consequently, the representation p ;-1 in £/7- is isomorphic to the directsum of the atomic representations pjl. Lemma 4.1 and Theorem 4.1 arenow proved.

COROLLARY 4.1. The morphism of Boolean algebras vx\ 38(1)-* B+($defined by the formula a -* va l (where a C / = {1, . . . , r} andva j = 2 et + 2 ejhj(l) G ^+(1)) w flw isomorphism.

i j lPROOF. We construct a representation p in F = Kr as follows. Let

£i , . . . , £r be a basis for F. We set p(et) = K%t. It is easy to see that p

is isomorphic to the direct sum © p] x of atomic representations. It is also/= l

easy to check that p(hA\)) = X K%f, that is, dim p(hA\)) = r~ 1. Hence it/*/

follows easily that p(va x) = X K^. Consequently, for any two distinct

subsets a, b E / the corresponding subspaces p(va j ) and p(vb t) are distinct.This means that p(£+( l)) = 38 (I) and so B\\) 9* 38 (I).u

As we know, any element va 2 G ^"""(l) can be written in the followingform: va t = 2 et + 2 ^ ( 1 ) , or, if a ¥= I, va 1 = n /z( l ) . It follows

/so /G/-fl ' /e/-fl

from Theorem 4.1 that every element va j is perfect. So we come to aproposition that refines Theorem 4.1.

PROPOSITION 4.1. Let p e<%(Dr, K) be any representation. Thenp = pa(\) e TQV where ral = P|p(Ufl 1}- Here

(i) p_(l) = 0 p; t , and each p,-, is a multiple of the atomici6(Ia)(J{0} Jf J'

representation p;- j = pj x e . . . e pf t(mj > 0);

(ii) if Ta j ,yp/z75 z>zto <2 direct sum TQIX— © Ty o / indecomposablerepresentations r;-, r/ze/t «o«e o / /ze r;- «re isomorphic to any of therepresentations p]x for i G (/ - a) U {0}. •

We shall use Proposi t ion 4.1 most often when val = ve>1 is the

minimal e lement of B+(\). The space p(vei) is, as it were, the sum

of all subrepresentat ions tha t are no t (+)-atomic. Namely, p = ( l )

Lattices, representations, and algebras connected with them. 243

where T61 = p | p ( u )5 and if r 0 1 = © ry-, where the r7- are indecomposable,

then none of the r;- are isomorphic to any of the pj x for / E {0, 1, . . ., r}.4.2. (—)-atomic representations and the cubicle i?"(1). We define representa-

tions p~t x as follows: pj t = (p+t j )* . We call them (—)-atomic. It follows

from the definition that each p't x is a representation in the one-dimensionalspace V~x s £ , and Po5lO/) = 0 f o r anY *; Pj,\êd = *Xi i f / ^ ' a n d

The lower cubicle B (1) is defined to be the sublattice generated by ther _

elements h~AX) - n ef. Here, v* , = O e.- is the minimal element in B (1) .1 i*j i=\

If any element v~ x G ^"(1) with a ^ 0 (6), can be written in the formv~ x = 2 hj(\). Arguments dual to those used in the preceding subsection

ye<z

show that each element v~ x G B~(\) is perfect. Also, the representation

Pa I ~ P|p(u" ) is a direct summand of p, and p~ j = © mtP~t,\' w n e r e

0.

5. The functors 4>+ and

In this section we define functors <l>+ and 3>~: ^ - > ^ , where^ = ^ (D r , X) is the category of representations of D r in finite-dimensionallinear spaces over a field K. These functors play an essential role in provingthe theorem that B+ and B~ are perfect.

Analogues to the functors <£+ and 3>~ appeared first in the author's paper[6] . Then a modified form of them, called Coxeter functors, was usedeffectively to study representations of graphs [2 ] . A generalization of theCoxeter functors was constructed by Dlab and Ringel [8 ] . Recent work ofAuslander [ 1 ] clarified their connection with the classical functors Ext andTor.

5.1. Definition of the functors + and 4>~. Let p be a representation ofjy in a space V. We define a space V1 and representations p 1 in V1 asfollows:

yMfo, ....Wi&epfo), 2E* = O}, P1 (*,)={&, . . . . u o . U - y e n

where the et are generators of Dr'. In other words: we denote byr

V: © p(et) -* K the linear map defined by the formula

V ( £ l 9 . . . , £ , . ) = 2 £,-. We set Ker V = K1. Then the following sequencei= 1

of vector spaces is exact:"K r V

0 -^ V1 -> 0 p(e^) —> F, where X: K1 -> © p(e.-) is an embedding.i i

244 /• M. Gel"f and and V. A. Ponomarev

rWe denote by 717 the projection onto the space R = © p(et) (IT: R -+ R)

1=1

with kernel © p(et) and range p(e ;). We set <y = V 7r7-X. Then1 * 7

pV/) = Ker^y. Thus, from a representation p E 92 = 92{Dr, K) we haveconstructed another representation p' G M. It is easy to check that thiscorrespondence is functorial. We denote by <£+ the functor p -+ p1.

A representation p"1 is constructed from p in a dual manner. We setr

Q - e (Vlp(et)). We denote by JJL the linear map \x\ V -+ Q defined by1=1

M£ = (|3i5, • . • , j3rg), where fy: 7 -• V/p(ef) is the canonical map. We setV~l - Coker JJL = Q/Im /x. Thus, the following sequence is exact:

V ^ © ( F / p ^ ^ F - ' - ^ O .1 = 1

We set i//y = 07Ty/x, where 7r;-: Q -> 2 is the projection into the space

Q = e (F/p(e,)) with kernel © (V/p(et)) and range V/p(eA. Theni=l i>/

p~1(eJ) = Im i//;-. It is not difficult to see that the correspondencep ^ p~l is functorial. We denote this functor by <f>~.

5.2. Basic properties of the functors 4>+ and 4>~.PROPOSITION 5.1. Let p G ^(2)r, #) . Then the following assertions

are equivalent: (i) <£+p = 0. (ii) The subspaces p(et) are linearly independentf _ _ _ _

in V, that is, p(e,)( 2 p(et)) = 0 /or every /. (iii) p = © p] {, where each1 * 7 r = e '

pj t w a multiple of the atomic representation p+t l (that is,

P+t,x = P+t,

The following proposition describes the dual properties of <J>".PROPOSITION 5.2. Let p G ^(Dr, K). Then the following assertions

are equivalent: (i) 3>~p = 0. (ii) p(e}) + ( n p(ez)) = F /or ever^ 7

(where V is the representation space of p). (iii) p = © p?~i where eachr=o '

p^j w 6f multiple of the atomic representation p ^ .The proofs of these assertions follow immediately from the definitions."

PROPOSITION 5.3. (i) If p ^ © pt, then $+p = © $+pz- and

$ p ^ © <|> (p{). (ii) 77zere exw^ « natural monomorphism i: & <b+ p -> p./= 1

(iii) There exists a natural epimorphism p: p -* <l)+<l)~p.(iv) <l>+(p*) = (<l>~p)*, where p* /s ^ e representation dual to p.

PROOF. Properties (i) and (iv) can be checked directly from the definitions.


Let us prove (ii). The map <£y: V1 -• F (where K1 is the representationspace of p1) is such that ty($lf . . . , fy, . . . , £r) = £;, where £; G p(ey).

r

Here the condition (£l5 . . . , %r) G V1 is equivalent to 2 £,- = 0, and so1=1

£• G p(e.) ( 2 p(ez)). Thus, Im <A- C p(e) ( 2 p(e,)) = p(eyfy), where

If Si e P(Cjhj) = pO;0 ( 2 p ^ ) ) , this means that £y- = 2 £,., where

5, G p(^.). Then « ! , . . . , S/-!, "£/, £/+1, • . . , 5r) e K1, hence,S;- G Im^y. Thus, the map <p;-: K1 -> K has the following properties:

Ker *pj =L pi(ei)i Im^- = p(ejhj). Consequently,

e (FVp1 (et)) = © p(e,-/i/)- This implies that the following diagram is

commutative:

II T t*v ' 0-^y1 ^ $ p (ef/^) ^ Coker pi' -> 0

Now it is not difficult to check that we can construct a linear map i:Coker JU1 -> V such that the right square of the diagram is commutative.This map / is nothing but the natural isomorphism

r

Coker /z1 = 2 p(^//i/). We set $} - dTrjX', where nj is the projection inr© piejhj) onto the ;-th component. It is easy to see that

i = i

Im #} = p(ejhj). We define a representation p in Coker /i1 by settingp(^) = Inupj. It follows from the definition of 3>~ that p = ^"(p1),where p1 = ^+(p), is a representation in F 1 . Consequently, the embedding/: Coker yi -+ V defines a morphism of representations /: >~ >+p -* p. Theproof of (iii) is dual to the one just presented.

rWe n o t e t h a t 2 eihi is t h e m i n i m a l e l e m e n t vex o f t h e cub ic le B + ( \ ) .

/= l

In §4 we have proved that this element is perfect. Thus, p splits into thedirect sum p = p(l) © r 0 1 , where r 0 1 = p|p(u )? and p(l) is the direct

rsum p(l) = e mtp] x (mt > 0) of the atomic representations p\ x. We

t=ostate as a separate corollary the properties of 3>""<f>+p, together with thedual assertions for 3>+<l>~p.


COROLLARY 5.1. (i) There exists a natural isomorphism$+<|>~p 9* p | p ( u . ), where vdl is the minimal element of B+(\). Here p is

risomorphic to the direct sum p = 4>"4>+p © p(l), where p(l) = © mtp\.

r=ois a direct sum of atomic representations p] x.

(ii) <!>+<£>~p is isomorphic to the factor representation p/p"(l), wherep"(l) = P|p(u- ), is the restriction of p to p(vj' x), the image of the maximal ele-

r

ment ofB~(\).Here p s p-(\)®($*$~p) and p"(l)= ®mtpjx (where pjt ist=i '

the atomic representation).*We state another proposition in a form convenient for a later application,

which is a simple combination of the properties of <J>+ and <£".CONSTRUCTION. Let p be a representation in F, let r1 be a sub-

representation in p1 = <J>+p> and let U1 be the representation space of r1.

We set U = 2 ^ t / 1 (1/ C F), where spt: V1 -> F is the standard map

^/(£i> • • • > 5/» • • • » 5r) = 5i- We define a representation r in C/ by setting

r(ez) = tPjU1.PROPOSITION 5.4. Suppose that p1 = +p w decomposable:

n np1 = © TJ-. Then p is also decomposable: p = p(l) © ( © ry), where Tj is

obtained from rj by the construction described above. Heren

' (mt > 0)p

of the atomic representations p] x, and 4>+pO) = 0."

References

[1] M. Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974),177-268; 269-310. MR 50 # 2240. - and I. Reiten, III, Comm. Algebra 3 (1975),239-294. MR 52 #504.

[2] I. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev, Coxeter functors and Gabriel'stheorem, Uspekhi Mat. Nauk 28:2 (1973), 19-33.= Russian Math. Surveys 28:2 (1973), 17-32.

[3] G. Birkhoff, Lattice Theory, Amer. Math. Soc, New York 1948.Translation: Teoriya struktur, Izdat. Inost. Lit., Moscow 1952.

[4] N. Bourbaki, Elements de mathematique, XXVI, Groupes et algebres de lie, Hermannet Cie., Paris 1960. MR 24 # A2641.Translation: Gruppy ialgebry Li. Izdat. Mir, Moscow 1972.

[5] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103.


[6] I. M. Gel'fand and V. A. Ponomarev, Problems of linear algebra and classification ofquadruples of subspaces in a finite-dimensional vector space, Coll. Math. Soc. IanosBolyai 5, Hilbert space operators, Tihany (Hungary) 1970,163—237 (in English).(For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765.= Soviet Math. Dokl. 12 (1971), 535-539.)

[7] I. M. Gel'fand and V. A. Ponomarev, Free modular lattices and their representations,Uspekhi Mat. Nauk 29:6 (1974), 3-58.= Russian Math. Surveys 29:6 (1974), 1-56.

[8] V. Dlab and C. M. Ringel, Representations of graphs and algebras, Carleton Math. Lect.Notes No. 8 (1974).

[9] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in the coll."Investigations in the theory of representations", Izdat. Nauka, Leningrad 1972,5-31 .

Received by the Editors, 9 April 1976


LATTICES, REPRESENTATIONS, ANDALGEBRAS CONNECTED WITH THEM II1


Contents

§6. The representations p] t and p ^ 249§7. The perfectness of the lattices i?+ andi?~. Characteristic functions . . 252§8. The algebras Ar and the representations pr / 258§9. Complete irreducibility of the representations p+

tl and p~tl 264References . ' . . . ' 271

§6. The representations p\A and pf~,

We define the representations pj"/ (/ = 1, 2, . . . ) as follows: p*tl is theatomic representation (see §4), and for / > 1 we set inductively

The representations p ^ are, by definition, dual to the p]/5 that is,

Thus, the p ^ are the (-)-atomic representations, and it follows fromproperties of the conjugation functor (see Proposition 5.3 (iv)) that

In §8 we shall show that the p*tl are essentially the same as the psl,whose definition in terms of the algebra A was given in § 1 . Moreaccurately, we shall show that pQtl_x = p©,/ and pjt = p j z if/ =£ 0.

The functorial definition of p+tj is more convenient when we are

interested in such categorical properties as decomposability and when thereis no need to investigate the intrinsic structure of the representation.

DEFINITION 6.1. The dimension of a representation p G M(Dr, K) in the1 The first part of this article was published in these Uspekhi 31:5 (1976), 71-88 = Russian Math.Surveys 31:5 (1976), 67-85.

249


space V is the sequence of integers: dim p = {n\ m l 5 . . . , m r), wheren = dim V and mt = dim p(ef).

For example, the atomic representations p+0 l and p* ^ rô n a v e

dim p+0 ! = ( l ; 0 , . . . , 0) and dim p*>1 = (1 ; 0, . . . , 0,' 1, 0, . . . 0), where

the 1 stands in the (t + l)st place.PROPOSITION 6.1. Let p G J2(Dr, K) be an indecomposable representation.(I) Then there are the following possibilities for <£+p:

a) 3>+p = 0 <*» p s p* z /or some f G { 0, 1, . . . r} ,b) $+p =£ 0 «-* (3>~Vp s p);

/zere, f/ze representation <l>+p = p1 w a/ô indecomposable, and its dimensiondim p 1 = (ft1 ; m}, . . . , m)) can be computed from dim p = {n\mx, . . . , mr)by the formula:

r

n{=z^]mi — n, m)= ^mi — n.

(II) There are the following possibilities for <J>~p:a) ($-/> = 0) <-=> (p s p - j /or some f G {0, 1, . . . , r}),b) (4>~p = 0) «==> (4>+<l>~p = p); here, the representation <£~p zs also inde-

composable, and its dimension dim <£~p = («"; m\, . . . , m~) ca« Z?e computedfrom dim p = (n; m1? . . . , m r) Z? ^ e formula

PROOF. (I) a) and b), except for the assertions about the dimensions,clearly follow from Proposition 5.4. From the same proposition we findthat 3>~<l>+p = p if and only if p(u0 1) = V. By definition

vdl = n h^l). Hence, p(u5 x) = p(n/zz.(l)) = K implies that p ^ d ) ) = F/= l ' /

for all /, and so p{eihi{\)) - p(e^)V = p(et).Therefore, we can rewrite the diagram (5.1) in the following way:

2

0p (eiht) > Coker p,' > 0.i

From this we find that dim V1 = nl = 1 dimp(^)-dim V= fi i= 1

The formula m) = 2 mt - n is also easily proved. Part (II) of Proposition6.1 follows by duality.

THEOREM 6.1. Let p be an indecomposable representation of the latticeDr', where the number r of generators of Dr is at least 4. Then there are the

Lattices, representations, and algebras connected with them. II 251

following three mutually incompatible possibilities:(I) ( ( $ + ) M p =£ 0 and (<$>y p = 0) *=> (there is a t G {0, 1, . . , r } i p S p ^ ) ,

(Ill) V /fw ((<*>+)'p * 0 <mrf (4>T P # 0).PROOF. Let p be an arbitrary indecomposable representation. There are

the following possibilities:1) there is an / > 0 such that (&)l~l p ¥=0, (4>+)zp = 0;2) there is an m > 0 such that ( r f ^ p ^ O , (3>~)mp = 0;3) (40 'p ^ 0 and (SO'p =£ 0 for every / > 0.We consider these cases separately.1) (&)lp =£ 0, (<&+)7P = 0. We write (&)*~lp = pl~l. It follows from

Proposition 6.1 that p7"1, as well as p, is indecomposable, and thatp s ($ - ) ' - ip ' - i =(4>-)/-1(^>+)/~1p. Since ^ p 7 " 1 = 0, it follows fromProposition 6.1 that pl~l = p*tl. Consequently, p = ($~)l~lp*tl

= p]j-2)(^')m~1p^0, (<J>~)mp ='0. Arguments similar to those used in 1)

show that ($>-)m-lp^p-s>l and p *± (&)m ~l p~s x = p " m .For the proof of the theorem it remains to show that if p E ^ (//, K)

and r > 4, then 1) and 2) are mutually exclusive.Let p = p+

tl. We claim that ($~)m p]t =£ 0 for every m. By definition,(<|)")mp*/ = p*/+m.Thus, we must prove that p\k =£0 for every k > 0. ByProposition 6.1, dim p\k ~ (jitk^m\k^ • • • ' mrJ t ) c a n ê computedrecursively from the formulae

rnt, h — ( r " ~ l ) ^ , f e - i — Z J ^ J , fe-Ti ^ ? , k = nt, fc-i — ^ J , / i - i -

i l

It is not difficult to deduce from them that for r > 4 the terms of thesequence {ntl} (/ = 1, 2, . . . ) can be found from the recurrence relation

nt, i = (r — 2)nt> i-i — ntt z_2, Z > 3 ,

and the initial conditions nt x = 1, nt 2 = r —2 for t =£ 0, andwo, i = !» ^0,2 = r" !•

For r > 4 the terms of {ftf>/} increase monotonically with /, and so allthe p+

tJ are different from zero. Thus, if ($ +) / - 1p ^ 0 and (4>+)zp = 0, then(4>-)mp^0 for every m, that is, 1) and 2) are mutually exclusive.

REMARK. The lattices D1, D2, Z)3 are finite and each has only finitelymany indecomposable representations (up to isomorphism). The numbersof these representations of/) 1 , D2, and D3 are 2, 4, and 9, respectively.

If r > 3 and p is an indecomposable representation of Dr, then there arepositive / and m such that (^>+)/p = 0 and (<£~)mp = 0. Therefore, eachindecomposable representation of Dr, r < 3, can be described both in the.form p+

tl and pf~m . For example, in D3 the following isomorphisms hold:Po,i — Po,3> Po,2 ~Po,2»Po,3 — Po,i- ^ n e dimensions of these representationsare, respectively', (1; 0,0, 0),'(2; 1, l', 1), and (1; 1, 1, 1).


§7. The perfectness of the latticesB+ and B~.Characteristic functions

7.1. Proof of the theorem on the perfectness of the sublattices B+ andB~. This proof is based on the following proposition. (As usual, we writep 1 = $>+p> where V1 and V are the representation spaces of p 1 and p ;pj\ V1 -* V is the standard map pj(^ , . . . , £ / , . . . , £r) = £7-; and et t arethe lattice polynomials in Dr(r > 4) (see § 1 , p. 70).)

PROPOSITION 7.1. For every p G M(Dr, K), r > 4,

O,p(eHv.Ai), when jîim

We omit the proof of this proposition. It is the central and most compli-cated part of [7].

We recall that the generators ht(l) of the sublattice B+(l) are defined inthe following way:

et(l)= 2 e{ x ht(l) = ^1ei(l),

where et(l) is the sum of all possible et f t in which the last index is t.

COROLLARY 7.1. For every p G @(Dr, K), r > 4,

p(MZ+l))= S W'ihtV)).j=i

The proof obviously follows from Proposition 7.1 and the definition ofht{l)M

FUNDAMENTAL THEOREM 7.1. All the elements of the lattices B+

and B~ are perfect.Before proving this theorem, we prove the following proposition.PROPOSITION 7.2. (I) . Every element hj(l)eB+(l) (j = 1, . . . , r) is

perfect, that is, for every representation p G £R{Dr, K) in V

where Tjj = p |p (/z.(/)) u pu = p \Ut and U is a space complementary top(hf(l))in V.

(II). The representation pu satisfies the relation

Pu = ( © Pt.h) © P o , Z © P ; , I,0<k<l

where each representation pt k and ps i is a multiple of the indecomposablerepresentation p] k and p+

s lt that is, pt k = p] k ® . . . © p] k> where mt k > 0.mt,k

(III). IfTj i splits into a direct sum Tj t = © Tf of indecomposable representa-i= 1


tions rt, then none of the Ti are isomorphic to any of the representations

The proof is by induction on /. For / = 1 the corresponding assertionwas proved earlier in Lemma 4.1. Now we assume that the proposition hasbeen proved for the elements hj(l - 1). Then the representation p1 = 4>+pcan be written as

where p* = p1 iî, U1 is a complement to pl(hj(l- 1)) in V1, and

We apply the construction of §5 to p 1 . By Proposition 5.4, p is isomorphic

to a direct sum p = p (1) © pu © r7- /, where p (1) = © mt p] x (mt > 0) andf = 0

the representations pu and r;- z are constructed from p* and rjA_x as in §5. Thenr

Tj i is the restriction of p to the subspace 2 ^ (p 1 (fy(/ — 1))). From Corollary 7.1/= r

it is clear that this subspace is nothing but p(/z;(/)). Thus, r;-1 is a directsummand of p. This proves part (I) of Proposition 7.2 (that is, we have shownthat the element hjj is perfect).

Before proving parts (II) and (III) of Proposition 7.2, we now proveTheorem 7.1.

PROOF OF THEOREM 7.1. Since the elementshx(l), . . . ,hr(l) are generatorsof the lattice B+(l), and since each of them is perfect, the entire sublattice B*(l)is perfect (that is, consists of perfect elements). We have proved in §3 that B*is the union of the setsi?+(/). Therefore, B+ is also perfect. The perfectness ofB~ follows by duality.

COMPLETION OF THE PROOF OF PROPOSITION 7.2 (II). We may assumeby induction that we have proved that

pi = © P*. * © Po, i-i © Pi, i-i.

where each of the ptk and ps l_l is a multiple of the indecomposablerepresentation p+

t k and p+sj_\. By Proposition 5.4, pu = ~ p^ , where pu is the

representation constructed from plu. Consequently,

0<fe<Z - 1

B y a s s u m p t i o n , p t k = p +t k © . . . © p \ k, m r k > 0 . M o r e o v e r ,

mt,k

^~P+tk = P+t)k+\ by definition. Consequently,


Pf,h w h e r e

tt,k

Kk<lpt k = $~pt k_l, and each such representat ion is a mult iple of the indecom-

posable representat ion p] k . Finally, bearing in mind tha t p ( l ) = ©f = 0

we can write

p p ( ) © ( e p u ) e p o , i e p M © ^ i © p*.* epo . iep^!<*<* 0<ft<l

This proves part (II) of Proposition 7.2. Part (III) is proved similarly (herewe have to use the isomorphisms r;-1 = ^~rjl_l 2LndTJl_l = $+ r /).

As we know, any element of B+(l) can be written in the formva / = 2 et(l) + 2 e^DhM) or, if a =t I, va t = n /2-(/), where a is a

zGfl /e/-a ' j&l-a

subset of / = { 1, . . . , r}. The following proposition makes more precisethe properties of a perfect element.

PROPOSITION 7.3. Ler p G M {D\ K) by an arbitrary representation.Then

p £* ( 0 p,, ft) © ( 0 P., i) © Ta, Z,

where ra x - p | p ( u ^ ^/7G? eac/z pt k and ps t is a multiple of the indecomposable

representation p+tk and p] h Here, ifral splits as Tal - © rt into a direct sum

i= 1of indecomposable representations Tt, then none of the Ti are isomorphic toany of the representations p] k (t = 0, 1, . . . , r; k = 1, . . . , / - 1) orp+

sl(se ( / - a ) U {0}).We omit the simple proof of this proposition."COROLLARY 7.2. (i). The correspondence a\-*-\)ai defines an isomorphism

of the Boolean algebras vt: % (I) ->• B+(l).(ii) Let va i and vb m be arbitrary elements of B+(l) and B+(m). If

I < m, then'val D vbm.P R O O F . We consider the representat ion p = p*t © p\ , © . . . © p+

rj. By

Proposi t ion 7 .3 , if a and b C / and a =£ b, then the subspaces p(va z) and

p(vb z) are distinct. Hence, if a ¥= b, then val # vbl.r

Let ve i_i = n hj(l) be the minimal element in B+(l~ 1), andi - 1

rVj t = 2 et(l) the maximal element in B+(l). It is not difficult to check that the

/= l

subspaces p(vd j_i) and p(U//) can be described as follows:


p (vQ. M ) = © ^ + , I H P (VI, 0 = 0 Vt, uf=O

where V]j is the representation space of p+tl. Thus, p(vel_l) 3 p(u//), and

so y0/_i is strictly larger than VJJ (vdl_l D VJJ). Now ifva j _ 1 GB+(l~ 1) and vb / EB+(l) are arbitrary thenua,/_i 3 u0,/-i ^ y/,/ 3 vbj. Consequently, vaj_l D vb / .•

7.2. The connection between the representations of B+ and B~. We haveproved that the elements ve j (where vQ l is the minimal element of B+{1))form a decreasing chain: vdl D vd2 D . . . D u0 z D . . .. Dual to the elementsvd i of Dr are the maximal elements vfj of the lower cubicles B~(l)

r

(v7,= 2 /*."(/), where/*."(/) = O ^7~(/)). These elements also form a chain:' — 1 ' - A '

y/",i C u 7 , 2 C - . c v f j C . . .Let p E ^ (Z)r, Z). We write Vel = p(ve j) and Vfj = p(vjj), and we set

V* = H F. , and FJ = n F/ , ./ = i ' / = i '

PROPOSITION 7.4. F ; C Vl for every representation pe M{Dr, K),r>4.PROPOSITION 7.5. Every representation p G & (Z7, K),r> 4Js isomorphic

to a direct sum p = p~®p@p+, where p~' = P\y^> ( P ~ 0 P X ) = P | F + - Here(^>+)/p+ = 0 for some I > 0, (4>")/p" = 0 for some / > 0, and for every l> 0

The proof of these propositions follows easily from Theorem 6.1 andProposition 7.3.• _

Note that in Proposition 7.5 p+ = 0 pj" /, where each p\ x is a multiple of the

indecomposable representation p+t j = (&mt l p]lt mt / ^ 0, similarly,

p" s 0 p ~/5 where each pf; is a multiple of the indecomposable representationt, i

p^j. A classification of the px is known at present only for the lattice D4 [6].The classification of the representations px ofDr, r> 5, seems at present ahopeless problem.

7.3. Indecomposable representations and characteristic functions. Let Bbe a sublattice of perfect elements in a modular lattice L. With each in-decomposable representation p: L -> ^ ( Z , F) we associate a function xp

on the set B in the following way:

0 if p(v) = 0,1 jf p (y) — y,

where v G B and F is the representation space of p. This xp is called the


characteristic function. The set with two elements {0, 1} can be regardedas a lattice (a Boolean algebra), which we denote by 2. The characteristicfunction xp then becomes a lattice morphism xp

: B ->• 2. It is clear fromthe definition of the characteristic function that if two elements vx and v2

of B are linearly equivalent, then Xp(î) = Xp(v2). Thus, xp is completelydetermined once its corresponding map xp

: B -+ 2 is specified.We do not know the entire lattice of perfect elements in Dr (r > 4).

We know only its two sublattices B+ and B~ C Dr. We denote by Bthe lattice generated by B+ and B~, and by B the factor lattice of Bby the relation of linear equivalence. We know that p(iT) C p(v+) for anyrepresentation p & 91 {Dr, K) (r> 4) and any v+ E 5 + and v~ GB~. Consequently,5 = £+U £~ (that is, if uGJ?, then either^ G B+ or v G 5"), and ifu+ G B+ and iT G 5", then v~ C D+ in i?.

We shall study the characteristic functions xp- B ^ 2 (B C Dr). Itfollows from the ordering of B that there are three kinds of morphismsX: B -* 2: Either (1) there is an element v+ G i?+ such that x(u+) ~ 0- ThenXOO = 0 for every v~ G 5"; or (2) there is a u ' G F such that x(V~) = 1-Then x(y+) = 1 f° r every v+ G B+] or (3) x(v+) ~ 1 f° r every v+ G 5+ andX(t>~) = 0 for every iT G B~. We denote the last function by Xo- Character-istic functions of the first kind are denoted by x+> and those of the secondkind denoted by x~-

THEOREM 7.2. Let p be an indecomposable representation of Dr, andsuppose that its characteristic function xp is of the first or second kind.Then p is determined by its characteristic function uniquely up to iso-morphism. Moreover, if xp is a function of the first kind, thenp ss p+j for some t G {0, 1, . . . , r } and I G { 1, 2, . . . } ; // xp is af u n c t i o n of t h e s e c o n d k i n d , t h e n p = p ~ m f o r s o m e s £ { 0 , 1 , . . . , r }and m G { l , 2, . . . } .

Before proving this theorem, we define morphisms xj" / : B ~* 2 andXr~/: B -> 2. For every / = 1, 2, . . . we define

1 if m < Z ,

0 if

where v*m is an arbitrary element of B*(m).

Z, or if m=i and yj, = fes(m), where

where hs(m) (5 = 0, 1, . . . r) are generators of the cubicle B*(m).The characteristic functions xf";: ^~* 2 are defined similarly, namely,


0, I (Vm) = j1 if

%b,l(Vm)=\ Q i f

1 if I < m or if m = Z and /*7 (w) s z; ,

0 if m < / or if m — l and v~n^hr&{m), where s^t.

LEMMA 7.1. Let x: B -> 2 be an arbitrary morphism. Then there are thefollowing three possibilities:

1) x is a morphism of the first kind { that is, 3v+ G B+\x{v+) = 0). ThenX = Xt,i for some t G {0, 1, . . . , r) and I = {l, 2, . . . } ;

2) x is a morphism of the second kind {that is, 3v~ G B~\ x(v~) = 0- ThenX = Xs,m f°r some s G { 0 , 1, . . . , m} and I = { 1, 2, . . . } ;

3) X = Xo> that is, x(v~) = 0 for every v~ G B~ and x(v+) = 1 f°r everyv+ G B\

The proof follows easily from the description of the lattices B+ and B~.*PROOF OF THEOREM 7.2. We claim that each morphism xtj- B -• 2 has

one and only one indecomposable representation p such that xp = Xt,i anc^

p 9* p+t j . We first consider the case t =£ 0. The equality Xp

= Xr / indicatesthat p(/zf(/)) = 0.

It follows from this and from Proposition 7.4 that

where each ps k and ps t is a multiple of the indecomposable representationp+

sk and p*j. But p is indecomposable, by hypothesis, and so p = p+sk,

where for the pair (s, k) either k < I and s is arbitrary, or (5, k) = (0, /), or(s, k) = (t, I).

The equality xp = x]ti also indicates that p(ht{l)) = V for any i ¥= t. It

follows from p{ht{l)) = V, from Proposition 7.2, and from the indecompos-ability of p that p cannot be isomorphic to any p] k with k < / or0, fc) = (0, /) or 0, fc) = (z, /). This reasoning applies to all / (z ^ 0-

Therefore, the only possibility is that p = p+t t.

This proves that (xp = xj,/) =* (p = Pr+,/)-

Together with Lemma 7.1 this means that (xp = Xr/)<===>> (P — P?,/)- The

proof for the case f = 0 proceeds similarly. The proof for characteristicfunctions of the second kind (xp

= Xr"/) proceeds dually."Little can be said about indecomposable representations p for which

Xp = Xo (that is, xp (v+) = 1 and x p 0 O = 0 for all v+ G B+ and v~ G 5").

They are precisely the representations that are not annihilated by any ofthe functions (^)+)/ or (^~)z. Their classification is known only for thelattice Z)4 [6]. There are infinitely many such indecomposable representationsp G J?(D4, K) and each of them has not only integer invariants, but also acontinuous invariant X G K (similar to an eigenvalue of a linear transformation).


One might suppose that the equality xp = Xp'

= Xo> where p and p' arenon-isomorphic indecomposable representations, is a consequence of thefact that we do not know the entire lattice B of perfect elements in D4 .

However, we believe that this is not the case. We offer the followingconjecture.

CONJECTURE. The lattice B of perfect elements in Dr 0 > 4) is theunion of B+ and B~.

§8. The algebras Ar and the representations pt j

8.1. The algebras Ar. Let K be a field. We define the ^-algebra Ar as theassociative ^-algebra with unit e and with the generators £0, £ l5 . . . , £r

satisfying the following relations:

r)f

We write 4 instead of Ar when this cannot lead to misunderstandings.By a standard monomial in Ar we mean a product £,- . . . £• J r, where

1 Z — 1

for every 1 < / < / - 1

+i and

We denote this standard monomial by £z- , . . £z- t = Sa,ex = (/x, . . . , / ^ j , 0 is a sequence of indices. Clearly, ev

whereevery non-zero

monomial in Ar can be put in standard form.The degree of £a is the integer d(£a) defined as follows:

dtto) = d(e) = 0; d&t) = 1 for every i ^ 0; and d(SJfi) = d(ga) + tf(^) if£a£/3 ^ 0- ^ n e degree of zero is undefined.

We denote by Vl (Vt CAr) the space of homogeneous polynomials of degree/. For example, V1 is the space that contains the monomials£i> • • • > £r> £i£o> • • • > Mo of degree 1, and dim Vx = 2r - 1.

It is easy to see that in this way ^ becomes a graded algebra:Ar = Vo © V1 © . . . © F/© . . .; KjFy C F / + / , where Fo =Ke ®K%0.

Direct calculation shows that the algebras^1, A2, and A3 are finite-dimensional over K of dimensions 3, 5, and 11, respectively.

Let us show, for example, that A2 = Vo © Vx, and that dim^ A2 = 5.

Note that ^ ^ = 5 ? + * i * 2 = ( 2 5 / ) t 2 = ( 2 ^ ) ( ^ 0 ^ ) = ( 2 ^ o ) f e = 0 .i= l /= l /= l

Similarly, £2£i = 05 and so V2 = 0. Thus, Vx = 0 for every I > 2. In every


algebra Ar we have dim Vo - 2 and dim Vx = 2r- 1. Therefore,dim A2 = dim Vo + dim Vx = 5.

In the same way it can be shown that A3 = Vo © Vx © V2, and thatdim Vo = 2, dim Vx = 5, dim V3 = 4.

As regards the algebras Ar, r > 4, they are all infinite-dimensional, as weshall show later, and the numbers dl = dim Vt form a sequence strictlyincreasing with /.

8.2. The representations pA and pt t. We define a representationpA\ Dr -+ XR{Ar))of Dr in <£*C<4r) of right ideals of A = Ar in the followingway. We set

where %{A is the right ideal generated by £/; and ^- is a generator ofDr (i = 1, . . . , r). If x and j ; are lattice polynomials in Dr', then, bydefinition, p^ (x O;;) = p^ (x) O p^ (y) and pA(x +y) = pA (x) + pA(y), wherepA(x) and p^Cv) are right ideals of A.

Let ;4£r be the left ideal generated by £ r For every / > 0 and^ = 0, 1, . . . , r, we set

(8.1) Vtil = (Att)[]Vh

It is immediately clear from this formula that Vt 0 = 0 for z ^ 0 andVOtO = Kô. It is also easy to see that all the subspaces Vt l (i ^ 0) areone-dimensional: Vt x = K%f. Any space Vt /} where (t, /) = (0, 1) or/ > 1, is the sum 2 ^ a ? f of all subspaces K%a%t such that d( f a^ ) = /.

We define a linear representation pt f. Dr -* % (K, Vt> i) in Vt t by setting

(8.2) Pt.i(ej) = V t t l n ( t j A ) ( / = 1 , . . . , r ) ,

where J;- 4 is the right ideal generated by £;-.We also define the representation p e in the one-dimensional space Ke

so that p e (^) = 0 for all / = 1, . . . , r.EXAMPLE 1. pUl\ Dr -* X{K, Vi^lO ¥= 1) is the representation in the

space Vitl = K%t such that pitl{et) = K%t and P/fl(ey-) = 0 if / =£ /.EXAMPLE 2. p0 j : Dr -• ^ (/iT, Fo, I) is the representation in Fo>1 that is

r

generated by the vectors ^^0, . . . , £ £0 ( 2 £/£o = 0)- Clearly,/ = 0

dim V01 = r- 1. By definition, for this representation pOtl(ej) = K$j%0.PROPOSITION 8.1. For every algebra Ar (r > 4)

(8 .3 ) P A = Pe © Po, o © ( © ( 0 Pt, i))>1=1 t=0

The proof reduces to a verification of two elementary assertions.1) For every / > 1 the space Vl is isomorphic to a direct sum:


Vt s* Vo, i © Vu l © . . . 0 Vr, ,.

2) Clearly, every ideal %tA = p(e,-) is homogeneous with respect to ther

grading A = Vo © Vx © . . . , that is, £z-.4 = © (F, n £z-,4). To completei= l

the proof of the proposition it remains to check that every subspaceVt O ^ splits into a direct sum

(vt n itA) ~ © Pit z (eo = © (Ff, z n 5i^) - © {Ait n ^ n 5^). •<o <o <o

REMARK. (8.3) also holds for the algebras A1, A2, and A3. True, inthese cases the sum on the right contains only finitely many terms. For

example , A3 = p © p © ( © ( © pt / ) ) .1=1 t=o

8.3. The connection between the representations ptj and p\j. Bydefinition, the atomic representation p] l in the one-dimensional spaceV\x = K is such that: a) p+

0A{et) = 0 for every / = 1, . . . , r\b) pl.iej) = Vjtl and pjîe-) = 0 if i ± j . Thus,

Po. i^Po.o and Pi, i = Pi,i (i¥=0).

The representations p* z, / > 1, were constructed from the atomic repres-def

entations by means of the functors $ , namely, pt l = ( $ ) pt x forevery ^ = 0, 1, . . . , r.

PROPOSITION 8.2. For every I > 1,

Po, z-i = po, i, pt,i = P+tj, f = l , . . . , r .

For / = 1, as we have already mentioned, the proposition is evident. For/ > 1 the proof is by induction. We assume that the isomorphismPj l = pjl (or Poj-l — Po/) n a s already been proved and we claim thatsimilar isomorphisms hold for I ¥= 1. To see this, obviously, it is enoughto prove that

(8.4) <D~pj, i = Pj, i+u

(8.5) ^"Po, i-i = Po, j .

Let us prove (8.5), say. We denote by £;-L the linear map of A intoitself defined by the formula £/L : x -• fyx for every x G ^4. The map

0) a c ts on the standard monomial %a G Kf z_x by the formula

° if / = ii,

Hence /jL Kf>/_j_ C Vtl for / # 0; moreover, it is easy to see that


rLet fx denote the map Vo /_ x -> © p0 /(e7-) defined by the formula

/= l

lix^-fax, . . . , lrx).r r

Let 0 denote the map 0: e p0 z(g.) -+ Fo z such that 0(JC15. . . , xr) = 2 xb/=i ' ' i=i

where xt E p0 /(^). Thus, we have a sequence

(8-6) Fo, ;_i A J> p0, i (ej) \ Vo> t

for which dfx = 0. It is clear from the definition of <£~ that to establishthe isomorphism &~poi_1 = pol we have to prove the following:a) Po,Md — ô,/-i 'Poj-iiejYi b) the sequence (8.6) is exact at the middleterm, that is, Ker 6 = Im /x; c) Vo t = Coker JU (however, if Ker 0 = Im /x,then Voj = Coker ju, because 0 is an epimorphism).

We omit the simple proof of these assertions."COROLLARY 8.1. The algebras Ar, r > 4, are infinite-dimensional, and

the numbers nx = dim Vl form an increasing sequence n0 <n1 <n2 < . . .The proof follows from the isomorphism pt t = Q~pt /_x and the formulae

of Proposition (6.1), in which the dimension of <£~p can be found fromthat of p."

8.4. The lattice polynomials et z and the representation pA. The latticepolynomials ea = et t in Dr were defined as follows. Let A(r, I) be theset whose elements are sequences a = (il9 . . . , zz) of integers 1 < ij < rsuch that ij ^ iJ+1 for every 1 < / < / - 1. We set A(r, 1) = {1, . . . , r}.The elements ea are defined by induction on /: if / = 1 and (a) = (/j),then ea = et ; if / > 1 and a = (ix, . . . , i{), then ea = e{ ( D e»), where

1 (jer(a)

y4(r, / - l)and

We denote by At(r, /), / E {0, 1, . . . , r}, the set whose elements aresequences j3 = (/j, . . . , / j _ l f /) such that ij G { 1, . . . , r}, z';- ¥= ij+i,//_! =£ /. For every j3 E ^40(r, /), / > 1, we define

et = eii..-il_io = ei S ^Y»^ J 4761X3) Y

where F(j8) C A(r, I - 1) and F(j3) is constructed from the sequence0 = (/l5 . . . , //_!, 0) in the following way:

r (P) == {Y= ( i» • • •' ^-i) I ki *£ ii ^


For example,eiio=eil 2 * * i eiii2o = e i i ( 21 *ftifcS).

fîi feiîn'2ft 2=^ 12

The next theorem establishes a remarkable connection between the mono-mials ea and the representation pA: Dr -> #fl (-4) (p^ (e,) = %jA).

THEOREM 8.1. Let pA be the representation of Dr defined above in thelattice of right ideals of A. Then for every a = (i1, . . ., ij_lt t) G At(r, /):PA (ea) = %aA> where %aA is the right ideal generated by the element

This theorem is based on the following proposition, which is proved in [7].Note that this proposition refines Proposition 7.1.

PROPOSITION 8.3. Let p be an arbitrary representation of Dr (r > 4),and p1 = 3>+p- Let <pj\ V1 -> V be the standard map from the representationspace V1 of p1 into the representation space V of p, defined by the

formula <pftlt . . . , * , , . . . , * , ) = inhere fa, . . . , *r) G V1 *=> S £f. = 0.

= P (*j, o), (piiP1 (e»4.. .i^t) = 0,

cPiP1K...i/_1O=Pfei1...iz_1t), if i¥=i. •

We have proved that the representations ptj satisfy &+ptj = Pf,/_i, andthat the map J/Z : Vtl_l -• Fr>/ is the one we have denoted by «p;-: F

1 -> F.Consequently, Proposition 8.3 can be restated for s G {0, 1, . . . , r) in thefollowing way.

LEMMA 8.1 . For every m > 1 and every a = (il9 . . . , il_l, t) Ât{r, /)

Ps, m(*iio)=SiiV8f m- i ,

Ps, m^tiia.-.i^^) =SiiPs, m-1 (ei2. . .tM<) i

where £z- p (^ ) w f/ze /mage of the subspace p(e^ ) C Vs m_l under f ^ .•PROOF OF THEOREM 8.1. Since p^ splits into a direct sum of repres-

entations p5 m , we can write

(8.7) P A ( ' « ) = 2 S P..m(«a).

Let us find the subspaces ps m{e0L)-ps (et t t) for various s and m.

For this we have to analyze the cases t =£ 0 and /• = 0. We first consider t=£0.I. a) Let /• ¥= 0. We determine ps m {et A t) when m <l. Applying Lemma

8.1 repeatedly, we obtain


(8.8) pSt m {eit... i^t) = ^ tP«, m-1 (*i2. .. i^t) ~

= 111 (Ii2ps, m-2 (^3. . .*/_!*)) =

By definition ei t t Eet (I — m + I) C hp(I — m + 1) for any p ^ t.

Consequently, ps l(ei t t)^Ps i(hv(l-m + 1)). As we know, the ele-' TYl '" I— 1 ' "

ment hp(l - m + 1) is perfect, and Ps \ — P*s \ if s =£ 0, and p 0 t = p+0 2 .

Therefore, if / > m, then psA(hv(l-m + 1)) = 0, and a fortioriPf.e ( % ...//_! r) = °- T h u s ' i f m < l> t h e n Ps.m <<ei1 ...//M r) = °-

b) We now determine ps m(ei t t) for m = /. By analogy to the chain

of equalities (8.8), we find

If s = 0, then, by definition, POti(et) = K£t£0 = £t(K%0), and if s = t,then ps i(et) = ATJf = £rA" and p^ i(e r) = 0 for s ¥= t. Thus, we can write

{ lii...il_itK%o if 5 = 0 ,

0 if s^t.

c) For m > / we obtain

v" ' y / P«» m(g i1 . . . i /_1«)= =Si1 . • . i^Pa, m-Z+l(^*)«

By definition, for n > 1 the subspaces ps n(et) satisfyPs nêt^ = %t^s n — \' Consequently, ps m((?f / t)

= %j f- ? ^ s m-/# ^ e

can therefore rewrite (8.9) as follows:

Ps, m fax) = Ps, m (eit. . .tz_t<) ==&!•. . i ^ j ^ . m-l = \aVs, m-l (fn > Z).

d) Now we insert all the relevant expressions for ps m(ea) in (8.7).This gives

= 2 Ps, I {ea) + 2 P«, l+l (e*) + • • • + 2 Ps, Z+n (<?a) + • • • =

2 , 2 .iB5=0 3=0

2 2n=l s=0


The proof that ea = ei{. . . eij_i 0 proceeds similarly, with the help oft h e f o r m u l a p o m ( e i 0 ) = ^ Vo>m_1 .m

§9. Complete irreducibility of the representations p+t t and pj t

In this section all vector spaces are finite-dimensional over a field K ofcharacteristic zero.

9.1. Systems of vectors. Let R C V be a finite set of elements of a finite-dimensional space V. Then i? is called a system of vectors in V if all ele-ments a E i? are non-zero and i? generates V. A subset i?' Ci? is called asubsystem of R if R' generates V.

EXAMPLE 1. A root system (for the definition, see [4]) is a system ofvectors.

EXAMPLE 2. Let Vt! C Ar be the representation space of pt /. Then theset of £a = £z- ,- f (monomials of degree /) is a system of vectors.

A subset B C R is called a Z?as/s o/ i? if B is a basis of V.A system 7? is called indecomposable if for every subset Rx C R the

intersection of the subspaces Vx = X Ka and V2 = 2 Koc is non-

empty. (Ka denotes the one-dimensional subspace generated by a.)We introduce several concepts, which allow us to give a convenient

criterion for the indecomposability of a system R. Let B = {al9 . . . ,an}be a fixed basis of R. We associate with each vector /3 E R a subset(chamber) C C B in the following way:

a n d ^=7^0).

Two vectors o- and ay in B are called simply-connected if they belongto the same chamber (that is, if there exists a 0 £ R such thatP = biai + bjOLj + 2 £*;%, where &,- =£ 0 and £.- =£ 0). Two basis vectors

at and ak are called connected, in symbols c - ~ ak, if there is a sequenceof vectors af = ot°\ a^l\ . . . , a(m) = ak such that every pair ct^, oSJ+lîs simply-connected. If we take every vector a E B to be connected toitself, then it is not difficult to check that connectedness is an equivalencerelation. A basis B is called connected if every pair of vectors at, oy GBis connected.

PROPOSITION 9.1. The following assertions are equivalent:(i) Every basis B of R is connected.(ii) The system R is indecomposable.We claim that a basis is not connected if and only if the system is

decomposable.1) Let B be a disconnected basis of R and Bl C B be a non-trivial

connected component. Then for every vector 0 E R with chamber CB


either C, CBt or C, C^-B^. We set Vx = X KocmdV1= 2 Ka. ItaE5, a^B-Bl

follows easily from the definition of a chamber that j3 E 2 Aa. Thus,

for every vector j3 E i?, either j3 E Fx or j3 E F2. This means that R is decomposable.2) If R is decomposable, then there exist two non-empty complementary sub-

s e t s ^ andi?2 (7?j DR2 =0,Rt + R2 = R) such that the subspacesF- = 2 A$ (z = 1, 2) are disjoint. Let B be any basis of R. Clearly, each basis

fieR

vector belongs to one of the subspaces Vx or F2. Thus, the set B splits into twocomplementary subsets Bx and B2, where a E Bj <=» a: E F;-. Here i?;- is a basis ofF O' = 1,2). Let /3 E/? be an arbitrary vector. Then the condition of decompos-ability implies that either jft E Fi orjSE F2. Now j3 G Fz- clearly implies thatCp C .. Consequently, B is not connected."

An indecomposable system R is called minimal if for every ]3 E 7? thesubsystem i?' = /? -{]3}is decomposable.

PROPOSITION 9.2. Let R be a minimal indecomposable system andB = {ax, . . . , an } a basis of R. Then:

a) the number m of elements in R - B satisfies 1 < m < n;b) for each (3 E R - B the chamber Cp C B contains at least two

elements;c) the set R - B can be numbered in such a way that

R - B = { ft, . . . , Pm} and for each j > 1 the chamber C; = Cfi./-I /-I

satisfies 0 C C H ( 2 C) C £ Q (where C denotes strict inclusion) and the1 i= 1 z= 1

itsubsets 2 Q /orm a strictly increasing chain:

i 1i= 1

^ . . . c r S

We omit the simple proof of this proposition."Let R be a system in V = K71. The lattice generated by the one-

dimensional subspace Ka, a E R, is denoted by >//(i?). A system /? in Vis called completely irreducible if c#(i?) ^ <£(Qn), (where ^(Qn) is thelattice of linear subspaces of Q").

A system R is called rational if there is a basis B = { ax, . . . , an } C /?such that each vector j3 G 7? can be written as a sum

j3 = 2 A,-o:z- with rational coefficients %i./= l

It is easy to show that if R is rational, then every vector /3 E /? for any

basis 5 = {aJ}Ci? can be written as a sum ]3 = 2 X/a:z- with Xj G Q.l


THEOREM 9.1. Let R be an indecomposable rational system in V = Kn

over a field K of characteristic 0. / / dim V = n > 3, then R is completelyirreducible, that is o£ (R)s*X (Qn).

The proof requires several Lemmas.MAIN LEMMA 9.1. Let V = Kn, where K is a field of characteristic 0,

n

and let R = {<x0, c^ , . . . , an } be a system in V such that 2 OL = 0. / /

n > 3, then R is completely irreducible, that is, o/H (R) = X (Qn)).We do not give the rather tedious proof of this lemma. We only remark

that it is very similar to the standard procedure for introducing coordinatesin a projective space.

REMARK It is easily seen that any n vectors {at} of the system R inLemma 9.1. form a basis of V.

LEMMA 9.2. Let R be a system as in Lemma 9.1 {that is,

R C Kn, R = {a0, . . . , o^}, 2 at = 0), and let o/f (R) be the latticei=0

generated by the subspaces Kat. Let xQ, xi9 . . . , xn be non-zero elements

of Q/IL (R) such that any n of them are linearly independent {that is

V i i\i*jxi ^ xk ~ 0)- Then x0, xlt . . . , xn are generators of S {R).k * i,j

PROOF. It follows easily from the lemma that thexz- are one-dimensionalsubspaces of V. Therefore, by applying the fundamental lemma, we see that each

n

element xt can be represented in the form xt = Kfb where ft - 2 X o:,- and

Since dim V = n and all ft =£ 0, clearly, the n + 1 vectorsn

fo> fi> • - • >fn a r e u n e a r ly dependent, consequently,/0 = S jjfj with yt E Q.

By assumption, any n of the vectors f0, fl , . . . , / „ are linearly independent.Starting from this it is not hard to show that all the coefficients yt in the

n

expansion f0 = X ytft are non-zero./ = i

We set /o = - / 0 and f( = yifi if i ^ 0. Then the systemR' ~ { /o» fu • ••>/«} satisfies the conditions of Lemma 9.1, hence,S (Rr) ^ X (Qn). Since Kf\ = x{ and xt G Jl (R), it follows thatJl (Rf)<=<>$ (R).

It follows from the isomorphism S (Rr) ^ X (Qn) that S (Rf) containsn

every one-dimensional space of the form Ky, where y - 2 [ijl withn i= l

li: G Q. By definition, f'( = yt 2 X/.-a.-. Hence it is easy to show that the/=i


OLj can be expressed rationally in terms of the fj. This means thatKoij G G#(i?') and so nM{R') so^(f l ) . We have shown earlier thatdl{R')<^J({R). Therefore, S{R') = oM(R). m

LEMMA 9.3. Let R = {a0, ai, • • • > an + i} be a system of n + 2 vectors

m n + lin V = Kn such that 2 OL = 0, 2 OL = 0, where 1 < k < m < n. If

i=o j=k '

n > 3, ^^« /? is completely irreducible.PROOF. By assumption, the vectors a7- generate V. The conditions of the

lemma then imply that B = {a l 5 . . . , c^} is a basis of R. We considerseparately the cases m = 1, k = n, and 1 < m, k < n.

n + l1) Let m - \. This means that a0 4- c^ = 0 and 2 a, = 0. Since

/=i 7

Ab:0 = Koii, t h e s u b s p a c e s âz - ( / = 1, 2 , . . . , « + 1) a re g e n e r a t o r s o fo/^(jR),Thus, t h i s case r e d u c e s t o t h e m a i n l e m m a .

n2) Let k = n. This means that 2 <x- = 0 and a 4- a: + 1 = 0 . This is

clearly another application of the main lemma.3) Let 1 < m and k < n. We claim that <M (R) contains the element Kf,

n m nwhere / = 2 a.-. We write Vo = 2 KOL, VX - 2 Ka, (that is,

/ = i i = i /=*

mdim Fo = m, dim F! =n~ k + 1). The condition 2 o . = 0 implies that the

»=osubspace y0 = a 0 is defined on the basis { ax, . . . , a.m } of Vo by thesystem of equations

Here it is clear that each individual equation xt = xi+1(l < / < m) definesin VQ an (m - l)-dimensional subspace, which we denote by Wt. We write yjfor the line Kotj. It is easy to check thatWj = (y0 + • • • +J>/_i +î + ^ / + i +J^/+2 + - • • + J ; mM 1 <i<m), where thehats over yt and > /+1 denote that the corresponding summands are omitted.

Thus, Wt e QM(R) because yt e o4t(R).Similarly, the line yn + i - K&n + \ is defined in the basis {ak, . . . , an]

of Vx by the system of equations

Here each individual equation Xj = x/+1(A: < / < n) is that of the sub-space

268 I- M. Gel'fand and V. A. Ponomarev

f (*/o+i/*-i+£* + */

We define elements Lt in Jl (R) as follows:

.. .+J/n) for

•. + yn+i) for

It is not difficult to verify that in the coordinate system {ax, . . . , <xn}the hyperplane Lt(\ < / < n) is defined by the equation xf = xi+l. Conse-quently, the system of equations

defines a one-dimensional subspace Kf, where / has the coordinatesw - l

(1, 1, . . . , 1) in the basis {o^ , . . . , an ). It is also clear that Kf = n Lt. Sincei= 1

the Lt are elements of S (R) we see that Kf E cd(R). We denote the system{oclf . . . , an, - / } by R. Obviously, this system satisfies the conditions ofthe main lemma 9.1, hence Jt (R) s* X (Qn). It follows from Kf e Jl (R)that GM{R)^QM (R).

Owing to the isomorphism S (R) ^ X (Qn) the one-dimensional subspacesm n

y0 = Ka0 = K ( 2 a,-) and yn + l = Kan + 1 = K( 2 a ) belong to

Consequently, oM {R) <=oM(R). We have proved above that e/ /(S)e c//(i?).Therefore &ft(R) = <JZ (R) s «^(Qn). •

PROOF OF THEOREM 9.1. We choose in i? any minimal indecomposablesubsystem Rf and claim that R' is completely irreducible, that isS(R') - £(Qn)).

Let B = { « ! , . . . , an } be a basis of /?'. We denote by m the numberof elements in R' - B. It follows from Proposition 9.2 that 1 < m < «.We recall that with each element /3 £ /?' we associate a chamber C (a sub-set of B) by the following rule: (ai G Cp) ^=^ (|3 = \-az- + 2 X;-a;., with

;>/Xz- # 0). It follows from Proposition 9.2 that if 0 G / * ' - £ , then the num-ber of elements d(j3) in Cp is at least 2 (d(0) > 2).

We break the proof into several steps and use induction on the numberm of elements of R' - B.

Step 1. Let m = 1, that is, R - B = {j3}. Since Rf is indecomposable itn

evidently follows that Cfi = B, that is, |3 = 2 X,-^, where all Xz- ^ 0. Wez = l

set a[) = -0 and ot- = X.- o:.- for / > 1. Since 2 a- = 0, the system


^1 = (ao> • • • > a'n } cleai*ly satisfies the conditions of the main Lemma 9.1,and so Jl (R!) = Jf(Ri) s* X (Qn).

Step 2. Let Rf - B = {ft, j32}. It follows from Proposition 9.2 that thechambers Cx and C2 of ft and ft satisfy Q U C2 = 5, Cj n C 2 ^ 0 , CHence it is easy to see that the basis B can be numbered in the following

*, nway: B = {OL1, . . . 9an}, where ft = 2 X^. and ft = 2 /i/°9> with

1 /

1 < s0 < sx < n, and all of the coefficients Xt and JU;- are non-zero. (Notethat sx = d(Cx) and n -s0 4- 1 = d(C2), where d{Ct) is the number of ele-ments in Cz. Here, sx ~s0 4- 1 = d(Cx O C2).) We consider separately the casesd(Cx O C2) = 1 and d ( d n C 2 ) > l .

Step 2a). Let d(Cx n C2) = 1. This means in other words that

sx =s0 =s, that is, ft = 2 X -a,- and ft = 2 j -ay, where 1 <s <n. We define

vectors a' in the following way. We set a'o = - r— ft; a/ = ^ a.-, if 1 < / < s;/*. _1 ^ «+ i

C6 = -^- a;-, if s < / < w; aw + 1 = — ft. Then 2 ^ = 0 and 2 a' = 0, andVs Vs / = 0 j = s

{a\, . . . , ocn } is a basis of F. Let i?x* denote the system(oco, . . . , cx + 1 ) . Clearly, ^ x satisfies the conditions of Lemma 9.3, and soo/fl (Ri) = X (Qn). From the construction it is clear that QM(R')=^ Qff (R{)thus, G/ / / ( i r)^#(Qn) .

Step 2b). Let d(Cx n C2) > 1. In other words, we can write

13J = 2 XjCX;, ]32 = 2 jHyO , where 1 < s0 < sx < n. We use the notation

Vx = 2 ATc and F2 = 2 AToy. From 1 < s0 < sx < n it follows that1-1 /= s0

dim Fj = sx > 3 and dim F2 = n - s0 + 1 > 3. Let^1 = {î5 • • • > as? i i} and /?2 = {c*v . . . , an9 ft). Thus, 7?- is a systemin Fz- (/ = 1, 2). The same arguments as in step 1 show that Rx and R2

are completely irreducible, that is,

o# (i?0 ^ ^ (QSI), &tf (R2) ^ X (Qn"so+1).

Since J/(Rt) ^ X (QS1), the lattice Jl (Ri) contains every one-dimensional

subspace of the form Kx, where x = 2 7 ^ with a,- G Q. In particular, if1=1

a0 = — 2 OLh then Ka0 G G///(/?!).

It follows from Lemma 9.2 that the subspaces Ka0, Kax,. . . , Kas can


be chosen as new generators of o#(i?i) that is, GM{Ri) = j(i

(Ka0, Kax,. . . , Kas ), where &tft (Ka0, . . . , Kas ) denotes thelattice generated by the one-dimensional subspaces Ka0, . . . , Kas .

The same statements can be made about the lattice (J((R2), generated bythe system R2 = {OLS , . . . , an, j3 } in V2. Namely, o/ft (R2) contains

def "

K a n + l , w h e r e an + 1 = - 2 ay a n d J l ( R 2 ) = Jf2 (Kas , . . . ,K<xn, K a n + l ) .

We write <d(R') = oM (Kf$lf Kau . . . , Kan, ATj32). Now it is easy to see that<M(R') = Q/K (Ka0, Kax, . . . , Kan, Kocn + 1). New generators for dl (R!)

*i n + lcan be chosen so that 2 az- = 0 and 2 a.- = 0. Consequently, the

*-o /=s0

conditions of Lemma 9.1 are satisfied and &ft (Rr) = ^ (Qn)-Step 3. We now turn to the general case R' — B ={13!, . . . , 0m + 1 } .

From Proposition 9.2 it follows that we can renumber B so that in the newnumbering the vectors j3y- G Rf — B can be written in the following form:

S2

Pm+1— S ma i ^ m + i n L m J= sm+1

where C;- is the chamber of the element fy, and Lk = {l, 2, . . . , sk) =A:

= 2 Cy. From Proposition 9.2 it also follows that

1 < Si < s2 < . . . < sm < sm + l = n. Since m > 1, we have sm > 3,and so by induction we may assume that the proposition is proved for the

def sm

system Rm = { otx,. . . , as , ft, . . . , j3m } in Vm = 2 Kat. This means

that the lattice oflm = &ff {Kax, . . . , Kas , ^j!?!, . . . , Kf$m ) is isomorphic to#(Qsm). Consequently, there is an elemenT>> = Ka0 in oMm where

def sm&0 = - 2 a,-. Here, the elements Ka0, Kax, . . . , A a, are generators of

7 = 1

c//m. Hence we conclude that Q4( (R) = <#(KOLU . . . , Kan, Kpl9 . . . , A/3OT+1) =, ATQ:O, ^T]3W + 1 ) . Consequently, the arguments of step 2 apply


') and so

Step 4. Now we return to the case of an arbitrary (non-minimal) systemR. Let R' = {<*!, . . . , art, &, . . . , ft } be a minimal subsystem. It is clearthat a basis B = { a{, . . . , <*„ } of Rf is also a basis of # . LetR - R' = { 7 1 , . . . , 7fr}. Since R is rational, each vector 7 can be repres-ented in the form yf = 2 A^-, where X.;. G Q. Since ^(iT) £g #(Q"), we

see that A ^ e oM(R'). Consequently, <M(R) = (#(#') o* #(Qn). •THEOREM 9.2. All but a finite number of the representations

ptl\ Dr -* X(K, Vtti) over a field K of characteristic 0 are completelyirreducible.

Only the following representations are not completely irreducible:a) p0 0 and pt x(i ^ 0) for any r > 4,b) p/f'2(i 9^0)'/or r = 4.PROOF. The representations p 0 0 and pz- x are not completely irreducible,

since dim Vo 0 = dim F M = 1. It is also easy to find that dim Vi2 = 2for r = 4 and z ^ 0, so that the systems pf 2 (/ = 0, r = 4) are not com-pletely irreducible.

For all other representations pt / we can show that dim Vt / > 3.To prove the complete irreducibility of the other representations pt /, we

have to verify that: 1) the system of vectors Rt l = = {^}, aEAt (r, /), isrational, and that 2) the system Rt / is indecomposable.

The rationality of Rtj follows easily from the fact that the completesystem of equations satisfied by the vectors £a consists of2 ti , „/. / t ~ ^' where the summation is over all vectors £a ins ! "" j— 1 i + 1 ' " l— 1

which the indices ik (k ¥= /) are fixed.The indecomposability of Rt t clearly follows from that of the

representations pt /.

References

[1] M. Auslander, Representation theory of Artin algebras. I, II. Comm. Algebra 1 (1974),177-268; 269-310. MR 50 #2240. - and I. Reiten, III, Comm. Algebra 3 (1975),239-294. MR 52 # 504.

[2] I. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev, Coxeter functors and Gabriel'stheorem, Uspekhi Mat. Nauk 28:2 (1973), 19-33.= Russian Math. Surveys 28:2 (1973), 17-32.

[3] G. Birkhoff, Lattice Theory, Amer. Math. Soc, New York 1948.Translation: Teoriya struktur, Izdat. Inost. Lit., Moscow 1952.

[4] N. Bourbaki, Elements de mathe'matique, XXVI, Groupes et algebres de Lie, Hermannet Cie., Paris 1960. MR 24 # A2641.Translation: Gruppy i algebry Li, Izdat. Mir, Moscow 1972.

[5] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103.


[6] I. M. Gel'fand and V. A. Ponomarev, Problems of linear algebra and classification ofquadruples of subspaces in a finite-dimensional vector space, Coll. Math. Soc. IanosBolyai 5, Hilbert space operators, Tihany (Hungary) 1970, 163-237 (in English).(For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765.= Soviet Math. Dokl. 12 (1971), 535-539.

[7] I. M. Gel'fand and V. A. Ponomarev, Free modular lattices and their representations,Uspekhi Mat. Nauk 29:6 (1974), 3-58.= Russian Math. Surveys 29:6 (1974), 1-56.

[8] V. Dlab and C. M. Ringel, Representations of graphs and algebras, Carleton Math. Lect.Notes No. 8 (1974).

[9] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in the coll."Investigations in the theory of representations", Izdat. Nauka, Leningrad 1972,5-31.

Received by the Editors 9 April 1976

Representation Theory: Selected Papers

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