Algebraic codes and geometry of some classical generalized ...statmath/eprints/2013/20.pdf · classical generalized polygons N.S.Narasimha Sastry Division of Theoretical Statistics

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Algebraic codes and geometry of someclassical generalized polygons

N.S.Narasimha Sastry

Indian Statistical Institute, Bangalore Centre8th Mile Mysore Road, Bangalore, 560059 India

Algebraic codes and geometry of someclassical generalized polygons

N.S.Narasimha SastryDivision of Theoretical Statistics and Mathematics

Indian Statistical Institute8th Mile, Mysore Road,

R.V. College PostBangalore - 560059 India

Abstract

Some results about the geometry of, and the q- ary codes associatedwith, finite generalized polygons W (q), q = 2n; H(q), q = 3n; and O(q),q = 22m+1, are surveyed and a few simple observations and related questionsare mentioned (in italics). Using the polarity in the cases W (q), q = 22m+1,and H(q), q = 32m+1, we define a nondegenerate symmetric form on, anda polarity of, the q− ary code associated with these geometries which isstabilized by the centralizer of the polarity in the automorphism group ofthe geometry.

1 Introduction

(i) In this article, we consider the q - ary codes, q = pn, p a prime andn ≥ 1, related to the finite generalized polygons X = (P,L): mostly theregular (q, q)- generalized quadrangle (GQ, for short) W (q), q = 2n anda few remarks about the regular (q, q)-generalized hexagon (GH, for short)H (q), q = 3n, and the (q, q2) - generalized octagon (GO, for short) O (q),q = 22m+1. The automorphism groups of these geometries contain subgroupsG isomorphic to PSP (4, 2n) if X = W (2n), to G2 (3n) if X = H(3n) andto the Ree group G = 2F4 (22m+1) if X = O(22m+1). In each case, G actstransitively on the point - set P , the line-set L and the set F ={(x, l) ∈P × L : x ∈ l} of all flags of X. Recall that a generalized n− gon is abipartite graph (undirected, with no loops and multiple edges) of diameter nand girth 2n. To all unexplained terminology and results, see the standardreferences [25], [37] and [50]. These codes are of interest because of their closeconnection to the rich geometry defining these groups. See [3],[5],[6] for anapplication to study ovoids in the projective 3- space over F2n . In this note,

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we record a few simple observations and put together some related questions.These codes are also of interest because they are interesting submodules ofsome permutation kH- modules with large indecomposable sections, wherek is a field and H is a finite group of Lie type defined over a field of the samecharacteristic as k:

• the k - row span of the incidence matrix of a Lie incidence system ofH whose ‘point-set’ and the ‘line-set’ are objects of two given distincttypes of the Lie geometry for H with respect to a H- invariant incidencerelation; more generally

• the image and the kernel of a kH- homomorphism between permutationkH- modules.

(ii) Let k be an algebraically closed field of characteristic p. For M ∈{P,L}, let kM denote the kG- module of all functions from M to k with theaction of G defined by (g · f)(x) = f(g−1x) for g ∈ G, f ∈ kM and x ∈ M .For A ⊆M , we denote by 1A ∈ kM the function taking x ∈M to one if x ∈ Aand zero otherwise. For a collection ∆ of subsets of M , by abuse notation,we denote by 〈∆〉 the k- subspace of kM generated by {1A : A ∈ ∆}. Wewrite C for 〈L〉 ≤ kP . Since (|M | , p) = 1, the map c from kM to k takingf ∈ kM to its content Σm∈Mf(M) has a splitting taking λ ∈ k to λ1A ∈ kM .So, we have a kG− module decomposition kM = k1M ⊕ Y , where Y = ker c.For any subset A of kP , we write A ∩ Y as A0. Since the number of lines inX containing any given point of X is 1(mod p), 1P ∈ C and C = k1P ⊕ C0.Being a permutation module kP , and so Y , are self-dual. For any subset Aof P , we denote by A⊥ the orthogonal of A with respect to the inner productf on kP for which the characteristic functions of singleton subsets of P forman orthonormal basis.

(iii) In the generalized n− gons X above, each pair of points distance napart is contained in, and determines, a unique (1, t)- subgon of X ( wheret = q2 if X = O(q) and t = q if X = W (q) or H(q)). There is a naturalpartition A ∪ B of the point-set P1 of a (1, q) - subgon of X, where A andB are the equivalence classes in P1 with respect to the equivalence relation: x ∼ y if, and only if, d(x, y) = 0 or 4 (mod n). The incidence system(A,B), with a ∈ A incident with b ∈ B if a and b are collinear in X, is acomplete bipartite graph with |A| = |B| = q + 1 and A and B are lines, notof W (q), forming a dual grid described in (§3.B.iv) if X = W (q); A and B

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are Desarguesian Projective planes of order q if X = H(q) (see [41], §.6, forthis in terms of the “ ideal line condition”) and the GQ W (q) if X = O(q).Further, each word of minimum weight in C⊥ appears as a nonzero scalarmultiple of 1A − 1B, where A ∪ B is a partition as above of a point set ofa (1, t)-subgon of X (see, for example, [42]). Our first question asks for thekG-module structures of : (i) the subspace of kp generated by {1A}, where thesets A ⊂ P are as above; and (ii) the subcode of C⊥ generated by its wordsof minimum weight. This is a proper subcode of C⊥ if X = W (q), q > 2(§3.D. first para of i). Finally, I dedicate this to the memory of saintly andmaternal figures in my life Kanakammagaru and Venkatalakshammagaru.

2 The GQ W (q), q = 2n

(A) Geometry of W (q): A duality of PG(3, q) is an inclusion reversingbijection of the set of all subspaces of PG(3, q). It interchanges the set P ofpoints and the set Π of planes of PG(3, q), and fixes the set L0 of all lines ofPG(3, q). A duality of order 2 is a polarity. Given a polarity τ of PG(3, q),a subspace A of PG(3, q) is said to be absolute (relative to τ) if it is incidentwith τ (A). We say that τ is a null polarity if each point (and so, eachplane) is absolute. The set of absolute lines of PG(3, q) with respect to anull polarity is called a general linear complex. We denote by L the set of allgeneral linear complexes in PG(3, q). A polarity of PG(3, q) is induced by anondegenerate sesquilinear form f on F4q. It is a null polarity if f (x, x) = 0for each x ∈ F4q ([25],p.42-43). We note that the natural action of PGL(4, q)on the set of all null polarities on PG(3, q), and so on L, is transitive. Further,the stabilizer of a member of L is isomorphic to PSP (4, q). If τ is a nullpolarity of PG(3, q) and L ∈ L is the set of absolute lines with respect toτ , then the incidence system X = (P,L) is a (q, q) - GQ (see [37], 3.1.1),denoted by WL (q) (or W (q) if L in question is clear). Recall that an (s, t) -generalized quadrangle is an incidence system X1 = (P1, L1), where P1 is thepoint set and L1 is the line set with any two points incident with at mostone line, s + 1 points incident with each line, t + 1 lines incident with eachpoint and, for each pair of nonincident point - line pair (p, l), there is exactlyone line containing p and sharing a point with l. Then, |P | = (s+ 1) (st+ 1)and |L| = (t+ 1) (st+ 1). In the GQ W (q) = (P,L) defined by a polarity ofPG(3, q) as above, for x ∈ P , πx = ∪x∈l∈Ll = τ (x) ∈ Π.

A duality of an incidence system X1 = (P1, L1) is an incidence preserving

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bijection of P1 ∪ L1 which interchanges P1 and L1, thus defining an isomor-phism between X1 and its dual incidence system X∗1 = (L1, {Lp : p ∈ P}),where Lp = {l : p ∈ l ∈ L}. A polarity of X1 is a duality of order 2. For apolarity τ , an element of P1∪L1 incident with its image under τ is said to beabsolute. For each n, W (2n) admits a duality (see [37], 3.2.1). It admits apolarity if, and only if, n ≥ 1 is odd (Tits, [49]).To understand this geometry,Klein correspondence (see [30], Chap.15) and the fact that the tangent linesof a nondegenerate quadric in PG (2n, q) for even q and n ≥ 1 meet at apoint (called the nucleus of the quadric) ([31], Corollary 2, p.10) are veryuseful. By Klein correspondence, we have a useful

(α) isomorphism from the dual of W (q) (and so, from W (q)) to the (q, q)-GQ Q (4, q) of points and lines contained in a nondegenerate quadric E inPG(4, q); and

(β) an embedding of Q (4, q) (and so, of W (q)) as a subGQ of the (q, q2)−GQ Q− (5, q) of points and lines of an elliptic quadric E in PG(5, q) by

realizing E as E ∩ π for a hyperplane π of PG (5, q). We note that each lineof Q− (5, q) is either a line of Q (4, q) or contains exactly one point of Q (4, q).

We briefly explain. Under Klein correspondence, lines of PG (3, q) aremapped bijectively on to the points of a hyperbolic quadric K of PG (5, q);for L ∈ L, members of L to the points of a nondegenerate quadric E = π∩Kof π, where π is a hyperplane in PG (5, q); and elements of L through a pointof PG(3, q) to points of a line contained in K∩π. This gives an isomorphismfrom the dual of WL (q) to the GQ Q (4, q) mentioned in (α). For (β), we

choose an elliptic quadric E in PG (5, q) with E∩K = E.

To briefly describe the polarity of W (q), we fix an embedding of PG(3, q)as a subspace A of PG (5, q). Ensuring that the nucleus n of Q is not in A, weproject from n the points and lines of E to PG (3, q) yielding points and linesof a GQ isomorphic to W (q) embedded in PG (3, q). Applying a collineationof PG(3, q), we obtain a duality of W (q). If ϕ is a polarity of W (q), thenϕ ◦ψ−1 is a collineation δ of PG(3, q). Requiring that δ ◦ψ be an involution,we get a polarity of W (q).

(B) Ovoids and spreads in PG(3, q) and in W (q): (i) An ovoid ofPG(3, q), q > 2, is a set of q2 + 1 points no three collinear. An ovoid ofPG(3, 2) is a set of five points, no four coplanar ([25]). By definition, anelliptic quadric in PG(3, q) is an ovoid of PG(3, q). For q odd (a case notdiscussed further here), these are the only ovoids in PG(3, q) (independently

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due to Barlotti and Panella, see [10] and [36]). We denote by O (respectively,E) the set of all ovoids (respectively, all elliptic ovoids) of PG(3, q). An ovoidof W (q) is a set of its q2 + 1 pairwise noncollinear points. An ovoid of W (q)is also an ovoid of PG(3, q) ([37],1.8.2).

By a fundamental Theorem due to Segre (see [33], 25.10, p.127), if q iseven, the set L of all tangents to an element θ of O is a general linear complexand θ is an ovoid of the GQ WL (q). So, classification of ovoids of PG(3, q)and ovoids of W (q) are equivalent. We denote by OL the set of all ovoids inPG(3, q) with L as the set of its tangent lines. Write E ∩OL as EL. Clearly,the sets {OL}L∈L of ovoids of PG(3, q) partition O.

A collection of q2 +1 lines of PG(3, q) partitioning P is called a spread ofPG(3, q). It is called a symplectic spread of PG(3, q) or a spread of WL(q) ifall these lines belong to a general linear complex L. We recall the importantconcept of a regular spread. Given any three pair-wise skew-lines `1, `2, `3 inPG(3, q), there is a unique transversal to `1, `2, `3 (that is, a line of PG(3, q)meeting each `i) through any point on any one of the lines li. If R′ is the setof the q+ 1 transversals of `1, `2, `3, then a transversal to any three membersof R′ is a transversal to all members of R′. The set R of all transversals ofmembers of R′ is called the regulus in PG(3, q) determined by `1, `2, `3. Aspread S in PG(3, q), q > 2, is said to be regular if it contains the regulusin PG(3, q) determined by each triple of its members ([25], 5.1); equivalently[40], if, for some `1, `2 ∈ S, `1 6= `2, S contains the regulus determined by`1, `2, `3 for each `3 ∈ S\{`1, `2}. Any regular spread is symplectic. Anovoid in W (q) is elliptic if, and only if, its image under a duality of W (q)is a regular spread of W (q). Under Klein correspondence, a regular spreadS of PG(3, q) is mapped on to an elliptic quadric K ∩ `⊥ ' O− (3, q) inthe 3-subspace `⊥ of PG (5, q), where K is the Klein quadric and ` is aline of PG (5, q) disjoint from K. Each of the q + 1 hyperplanes of PG(5, q)containing `⊥ intersects K in a nondegenerate quadric isomorphic to Q (4, q).Under Klein correspondence, these are precisely the hyperplane sections ofK corresponding to the q + 1 members of L containing S.

(ii) Classical ovoids: (a) Elliptic ovoids: An elliptic quadric in PG(3, q)is an ovoid of WL (q) , L ∈ L, if, and only if, its defining quadratic formpolarizes to a nondegenerate symplectic bilinear form on PG(3, q) such thatL is the set of its isotropic lines. Each elliptic ovoid of WL (q) can be viewedin the following useful ways:

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(α) as (exactly) one of the q + 1 orbits for the semi-regular action on Pof a cyclic subgroup of Aut(WL (q)) of order q2 + 1 (see [26], (B.vi.a))

(β) as the set Ex ⊂ E of all points of Q(4, q) (isomorphic to W (q), see

(A.β) and the notation there) collinear to a given point x ∈ E of the (q, q2)- GQ Q− (5, q) containing Q(4, q) as a hyperplane intersection, but x not apoint of Q(4, q).

An elliptic quadric θ of Q(4, q) is Ex for some x ∈ E\E if, and only

if, x belongs to the secant line θ⊥ of E (Metz, see [24]). Thus, there is a

2− to −1 map from E\E on to the set of all elliptic ovoids in Q(4, q) and

|EL| = |E\E|/2 = q2 (q2 − 1) /2. By Klein correspondence, any two ellipticovoids of Q(4, q) intersect either in a point or in a conic of E.

(γ) If l is a line of Q− (5, q) meeting E at a point p, then the set El ={Ex : p 6= x ∈ `} of q elliptic ovoids of Q (4, q), together with the lines ofQ (4, q) incident with p, define a partition of P\{p}, a fact used below andin the last para of § 3.D.i. Further, El ⊂ EL is a maximal in EL with respectto pairwise intersection at p. Also, for lines l and m of E meeting E at apoint, El = Em if, and only if l,m and the nucleus of E in E are coplanar.

Let y ∈ E\(E ∪ l). If y is collinear to p ∈ Q (4, q), then either y ∈ θ⊥

for some θ ∈ El and so Ey = θ ; or θ ∩ Ey is a conic in Q (4, q) containingp for each θ ∈ El. If y is not collinear to p, then p /∈ Ey, q − 2 members θof El intersect Ey in mutually disjoint conics, remaining two members of Elintersect Ey in at distinct points and the remaining q+1 points of Ey are thepoints of Q(4, q) collinear to both y and p. Thus, given an elliptic ovoid θ(= Ex) in Q(4, q) and p ∈ θ, θ is in a unique set Cθ,p(namely, Epx) of q ellipticovoids of Q (4, q) which is maximal with respect to pairwise intersection atp. Further, ∪{Cθ,p\{θ}}p∈θ is the set (having (q − 1) (q2 + 1) elements) of allelliptic ovoids of Q(4, q) which intersect θ at a point.

Let θ1 and θ2 be elliptic ovoids of Q(4, q). If θ1 ∩ θ2 = {p}, then Cθ1,p =Cθ2,p and each of the q−2 members of Cθ1,p\{θ1, θ2} intersects both θ1 and θ2at p. Any elliptic ovoid θ of Q (4, q) intersecting θ1 at a point x 6= p belongsto Cθ1,x\{θ1}. Restricting the partition of P\{x} described in (γ) to θ2, wesee that there is a unique member θ of Cθ1,x\{θ1} intersecting θ2 at a pointdifferent from p (and each of the remaining q − 2 members of Cθ1,x\{θ, θ1}intersect θ2 in a conic and each line m of Q (4, q) through x intersects θ2 at apoint). Thus, there are q2 +(q−2) elliptic ovoids of Q(4, q) intersecting each

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of θ1 and θ2 at only one point. If θ1 ∩ θ2 is a conic, then, by the uniquenessstated above, Cθ1,p and Cθ2,p are disjoint. Further, if x ∈ θ1\θ2, again by(γ), there are exactly two members of Cθ1,x intersecting (θ1 at x and) θ2 atdistinct points of θ2\θ1. Thus, there are 2 (q2 − q) elliptic ovoids of Q(4, q)intersecting each θi, i ∈ {1, 2}, at a point.

(b)Tits ovoids: If q = 22m+1, m ≥ 1, the set of absolute points (respec-tively, all absolute lines) of a polarity of W (q) is an ovoid (respectively,spread) of W (q) (Payne, see[37],1.8.2), called a Tits ovoid (respectively,Luneburg spread, see [33]). Since composition of two polarities of W (q)is an automorphism of W (q), Aut (W (q)) w PΓSP (4, q) acts transitively onthe set TL of all Tits ovoids of W (q) = WL (q). Its action on the set EL isalso transitive. Tits ovoids and elliptic ovoids in PG(3, q) are distinct forq ≥ 8 because their stabilizers in PSP (4, q), being respectively isomorphicto 2B2 (q) ([49]) and PSL2(q

2) · 2 ([30]), are nonisomorphic.

Elliptic ovoids (which exist for all q) and Tits ovoids (which exist onlyfor all odd powers of 2) are the only known ovoids of PG(3, q). They arecollectively called classical ovoids. An ovoid of W (q) admits a transitivegroup of automorphisms if, and only if, it is classical ([3],Theorem 1, see also[23]). For q ∈ {2, 4, 16}, each ovoid in PG(3, q) is elliptic. For q ∈ {8, 32},each ovoid in PG(3, q) is either elliptic or of Tits type (see [12]). A recentremarkable development in this direction due to Pentilla is that, if q is apower of 4, then any ovoid in W (q) is elliptic ([38]). I thank Pentilla for thiscommunication.

(iii) In view of the connection of ovoids of PG(3, q) to several other combi-natorial structures (for example, inversive planes, see [33], 25.6, p.126; trans-lation planes [48]; maximal arcs [48]; unitals [16], [35]; association schemes;group divisible designs [15]; semi-biplanes [4]; Tits (q, q2)- generalized quad-rangles [37], etc), study of various structures on O and their embeddings inPG(3, q) may be fruitful. Questions like classification of ovoids, intersectionpattern of members of O and structure of association schemes on O and onits suitable subsets; partitions of the point set P of PG(3, q) by ovoids or byq − 1 hyperbolic quadrics and 2 lines, etc. could be of interest. Identifyingmore differences between elliptic and Tits ovoids (like, for example, theirsecant plane sections ([12]), existence of special tangent lines ([25], §1.4.58,p.53), the subcodes of kP generated by their characteristic functions [43], etc)may yield pointers towards the nonexistence (conjecturally) of nonclassical

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ovoids.

(iv) Dual grids in W (q): We denote byH the set of all hyperbolic quadricsin PG(3, q). For L ∈ L, let HL denote the set of the members H of H suchthat each of the 2 (q + 1) lines in H is in L. For each H ∈ HL, the dualsin WL (q) of the two parallel classes of lines in H (containing q + 1 elementseach) are the lines m,m

⊥of PG(3, q) not in L. Here, for any subset A of P ,

A⊥ ⊆ P denotes the set of elements of P collinear in WL (q) to each element

of A. The subset m ∪ m⊥of P is called a dual grid of W (q). It has the

following equivalent descriptions: (α) the set of all points of a (1, q) - subGQof WL (q); (β) m

⊥= x

⊥∩y⊥and m = ∩z∈tr{x,y}z

⊥for any two distinct points

x, y of m; and (γ) if θ is any ovoid of WL (q), then one of m,m⊥ is a secantto θ meeting θ at, say, {x, y} and the other is x

⊥ ∩ y⊥, the intersection of the

tangent planes to θ at x and y.

(v) Ovoidal partitions of P : For a dual grid m ∪m⊥, by (iv.γ), m andm⊥ meet each ovoid of WL (q) in 0 or 2 points. Since q is even, P cannot bepartitioned by ovoids of WL (q).

(a) Existence of ovoidal partitions: Subgroups of A, A ∈ {PSL(4, q), G},PSP (4, q) ' G < PSL(4, q), of order q2 + 1 form a conjugacy class in A and

the centralizer in PSL(4, q) of such a subgroup T is a Singer cycle T = TK,where K is a subgroup of order q + 1. So, T and K act on P as well ason the set of lines of PG(3, q) semi-regularly. The orbits θ0, · · · , θq of T inP are elliptic quadrics in PG(3, q), defining a partition of P ([26], Theorem3, p.1167). Any two such subgroups of PSL(4, q) have at most one orbit incommon ([1], Lemma 4.1). If T ≤ Aut (WL (q)) for some L ∈ L, then onlyone of the θi’s , say θ0, is an ovoid of WL (q). One of the line orbits of Tis a regular spread S and is the set of all lines of PG(3, q) tangent to eachθi, i ∈ {0, · · · , q}. The orbits of K in P are the elements of S.

(b) Let {θi}0≤i≤q be a set of ovoids of PG(3, q) partitioning P and Li ∈ Ldenote the set of tangent lines to θi.Then, a line l of PG(3, q) is tangentto each θi or it is tangent to a unique ovoid θl, secant to q/2 of the otherovoids and passant to q/2 remaining ovoids. Klein correspondence impliesthat there is a regular spread S of lines of PG(3, q) such that Li ∩ Lj = Sfor i 6= j and {L0, · · · , Lq} are the only general linear complexes in PG(3, q)containing S (see [17], § 2 and Lemmas 2.4 and 2.5). On the other hand, anyregular spread S of PG (3, q) is contained in precisely q + 1 general linearcomplexes and the cyclic group K of collineations of PG (3, q) fixing each

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member of S acts regularly on these general linear complexes. Further, ifθ is an ovoid of PG(3, q) such that each member of S is tangent to θ, then{g (θ) : g ∈ K} is a partition of P (see [17], Lemma 2.5 and Theorem 3.1).In particular, there is a partition of P by Tits ovoids (see [22], Theorem 7).It is not clear if all partitions of P appear in this way (equivalently, whethera cyclic group of SP (4, q) of order q + 1 acts regularly on the ovoids of apartition). Existence of partitions of P by ovoids of PG(3, q) of differenttypes is not known (see [22], 2.3).

(vi) We present a few facts about O. For θ ∈ O, let θ∗ denote the set ofall planes tangent to θ and L (θ) ∈ L denote the set of all lines tangent to θ.

Proposition 1 Let θ1, θ2 ∈ O. Then,

(α) (Butler [18],Theorem 12) |θ1 ∩ θ2| ≤ 12q2 + 1;

(β) (Bruen and Hirschfeld [13],Theorem 5.1) |θ∗1 ∩ θ∗2| = |θ1 ∩ θ2|; and

(γ) (Butler [18], Lemma 2.2) if L(θ1) 6= L(θ2), then L(θ1)∩L(θ2) is eithera regular spread in PG(3, q) or a set of q2 + q + 1 lines meeting l (includingl itself) or a set of (q + 1)2 lines of a common (1, q) -sub GQ (equivalently,lines of a common dual grid).�

Pentilla has found two Tits ovoids of PG(3, 8) with 33 points in common(private communication). However, the bound in (α) seems to be excessivefor general q. Possible intersections of classical ovoids with different sets oftangent lines in PG(3, q) is not known. Analysis in ([27], III.C) using theclassification of pencils of quadrics in [14] shows that any two elliptic ovoidsof PG(3, q) meet in at most 2 (q + 1) points.

In (β), the points of tangency in θ1 and θ2 of an element of θ∗1∩θ∗2 need notbe in θ1∩θ2 (see [7] for a discussion). It would be interesting to see if there isa ‘natural” bijection between θ1∩ θ2 and θ∗1 ∩ θ∗2. More generally, the numberof points common to two nonsingular quadrics of the same type in a finiteprojective space of odd dimension is equal to the number of their commontangent hyperplanes. This is not true for finite projective spaces of evendimension, even for projective planes (there are disjoint ovals with commontangent lines (see [13], p.218 and [7], Lemma 2.1). Though there exist ovalsin a projective plane intersecting at a point with distinct tangent lines at thepoint of intersection, I do not know if a pair of ovoids in PG(3, q) intersectingat a point with distinct tangent planes at the point of intersection can exist.

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To see (γ), we note that the Klein correspondence τ maps L(θ1)∩L(θ2)on to the intersection of a projective 3-subspace B = 〈L(θ1)〉 ∩ 〈L(θ2)〉 ofthe projective 4−space 〈L(θ1)〉 with the nondegnerate quadric L(θ2). Thepossibilities for the later are : an elliptic quadric in B, a hyperbolic quadricin B and a quadratic cone ([30], chap.15). They correspond under τ to thepossibilities in (γ).

(Vii) Intersection of classical ovoids in WL (q) w W (q), q = 22m+1 ≥ 8,L ∈ L: We first describe the subsets of a Tits ovoid θ of WL (q) , each ofwhich appears as the intersection of θ with some Tits ovoid 6= θ of WL (q).If H w 2B2 (q) is the stabilizer of θ in PSL(4, q), then H < Aut (WL (q))and |H| = q2 (q2 + 1) (q + 1). The stabilizer in H of a point x of θ is PK,where P ∈ Syl2 (H), K ' Cq−1 and the action of P on θ \ {x} is regular (see[34], p. 90). Union of {x} and an orbit of the centre of P in θ is an ovaland is called a pseudo-circle in θ. Union of two distinct pseudo-circles in θthrough x is called a figure of eight in θ. H contains a unique conjugacy classA+ (respectively, A−) of cyclic subgroups of orders q + r + 1 (respectively,q − r + 1), where r2 = 2q. The normalizer in H of each of these groupsis maximal in H and is of order 4 (q + r + 1) (respectively, 4(q − r + 1)).Each member of A+ ∪ A− is self-centralizing in H and its centralizer inPSL (4, q) (respectively, in Aut (WL (q))) is a cyclic subgroup T of order |P |(respectively, a subgroup T of the Singer cycle T of order q2 + 1) (). see [34],Theorem 3.10, p.190). As noted in (ii.α), the T− orbits in P are ellipticquadrics and only one of them is an ovoid of WL (q). Since this accounts for|H|/4 (q + r + 1)+|H|/4 (q − r + 1) = q2(q2−1)/2 elliptic ovoids in W (q) (≡subgroups of order q2 +1 of Aut (WL (q))), the map T 7→ T ∩H is a bijectionfrom the set of subgroups of Aut (WL (q)) of order q2 + 1 and A+ ∪ A−.Thus, H has two orbits in the set EL of all elliptic ovoids of WL (q) (and twoorbits on the set of regular spreads in L, in view of the polarity of WL (q)centralized by H) with 4 (q + r + 1) and 4 (q − r + 1) elements. An A- orbitin θ is called a cap or a cup according as A ∈ A+ or A−. Each A- orbit inθ is the intersection of θ with the T - orbit E in P containing it. For use in(§3.C), we note that 1E = Σx∈T1x(θ) ([43], Lemma 14).

Theorem 2 Let L be a general linear complex in PG(3, q) and θ1, θ2 ∈ OLbe distinct. Then, the following hold:

(i) (Glynn [28]) |θ1 ∩ θ2| ≤ q(q − 1)/2.

(ii) (Butler [17]) |θ1 ∩ θ2| is odd. Further, |θ1 ∩ θ2| = 1 (mod 4) if θ1 is

10

an elliptic ovoid.

(iii) (Glynn [29]) If θ1 and θ2 are both elliptic, then θ1 ∩ θ2 is either apoint or a conic (that is, the intersection of θi with a nontangent plane).

(iv) (Bagchi and Sastry [5]) If θ1 is elliptic and θ2 is a Tits ovoid, thenθ1 ∩ θ2 is either a cup or a cap.

(v) (Bagchi and Sastry [5]) If θ1 and θ2 are both Tits ovoids, then θ1 ∩ θ2is one of the following: a point, a pseudo-circle, a figure of eight, a cup or acap. �

(See also [8]; [50], p.341 for (iv) and (v)).Thus any two ovoids (dually,any two spreads) of W (q) have at least one element in common. If an ovoidof W (q) intersects each elliptic ovoid of W (q) in a point or a conic (respec-tively, a cup or a cap), is it necessarily an elliptic ovoid (respectively, Titsovoid) of W (q)?

Other possibilities for intersection of ovoids exist if the sets of tangentlines are different.

Example: Let `∪m be a dual grid in the GQ W (q) = (P,L).The groupAut (W (q)) contains a cyclic subgroup L of order q − 1 which fixes eachpoint of `∪m and acts regularly on the (q + 1)2 sets xy \{x, y}, where x ∈ `,y ∈ m and the line xy ∈ L. Fix x0 ∈ `, y0 ∈ m and an ovoid θ of PG(3, q)containing x0 and y0. Then, aθ is an ovoid of PG(3, q) for each a ∈ L andthe q − 1 ovoids {aθ : a ∈ L} intersect pairwise at {x0, y0}.

(C) The graph ΓO: Let ΓO denote the graph with vertex set O andθ1, θ2 ∈ O defined to be adjacent if |θ1 ∩ θ2| = 1. For A ⊆ O, we denoteby ΓA the subgraph of ΓO induced on A. Properties like connectedness,regularity, diameter etc. of ΓA for A ∈ {O, E ,J , EL,JL for L ∈ L} would beinteresting. We make some remarks about their cliques.

Lemma 3 If q ≥ 4, a clique in ΓO has at most q + 1 vertices.

Proof. If there were q+ 2 pairwise adjacent vertices in ΓO, then their unionwould have at least (q + 2) (q2 + 1)−

(q+22

)points. This exceeds |P | by q(q−3)

2,

a positive number if q ≥ 4.W (2) has 6 ovoids (all elliptic) pairwise adjacent. Let x ∈ P , π be a

plane in PG(3, q) containing x and Ox,π ⊂ O denote the set of all ovoidsof PG(3, q) containing x with π as their common tangent plane at x. Each

11

clique in ΓOx,π of size q (in particular, the clique El in ΓEL described in(B.ii.a.γ) is maximal in ΓO. We now show that there is an elementary abeliansubgroup of PSL(4, q) of order q acting semi-regularly on Ox,π whose orbitsthus partition Ox,π by maximal cliques of size q.

Proposition 4 Let x, π and Ox,π be as above and L ∈ L. Assume that Lcontains each line in π incident with x. Let G < Aut(WL (q)) with G 'SP (4, q) and M be the subgroup of G consisting of all elements of G fixingeach point of π. Then, the following hold:

(i) M is isomorphic to the additive group of Fq.

(ii) If m ∈M fixes an element of P \ π, then m is trivial.

(iii) M stabilizes each line ` * π of PG(3, q) containing x and acts reg-ularly on ` \ {x}.

(iv) For θ ∈ Ox,π and m ∈M , m (θ) ∈ Ox,π and {m (θ)}m∈M is a maximalclique in ΓO.

Proof. With an appropriate choice of the symplectic basis for F4q for thestandard representation of SP (4, q) on F4q, M is seen to be conjugate to

{I4 + λE1,4 : λ ∈ Fq}, where E1,4 is the 4× 4- matrix over Fq whose (1, 4)th -entry is 1 and the rest are zero. So, (i) follows. If m ∈ M fixes z ∈ P \ πand l is a line incident with x, then m fixes ` ∩ π. Since order of m is even,m fixes a third point of ` not in {z, ` ∩ π} and so is identity on `. Thusm is identity on each line through z and so on P , proving (ii). If ` is as

in (iii), then `⊥ ⊂ z⊥ = π and, so ` =(`⊥)⊥

is fixed by each m ∈ M .Here, the perpendicularity ‘⊥′ is in WL (q). Now, (ii) completes the proof of(iii). For θ and m as in (iv), π is tangent to m (θ) at x. So, m (θ) ∈ Ox,π.Each of the q2 lines l * π incident with x is secant to m (θ) and, by (iii),l ∩m (θ) 6= l ∩m′ (θ) for distinct m,m′ ∈M . So, the ovoids {m (θ)}m∈M aredistinct and pairwise intersecting at {x}. Maximality of the clique followsbecause the complement of their union is π\{x}.

Proposition 5 Let L ∈ L and q > 2.

(i) (Hubaut, Metz [32]) ΓEL is a strongly regular graph with parameters

v =(q2

2

), k = (q − 1) (q2 + 1) , λ = (q − 1) (q + 2) = q2 + (q − 2) and µ =

2 (q2 − q).

12

(ii)Any clique in ΓEL containing three members intersecting mutually ata common point is a subset of a maximal clique of the type El described in(B.ii.a.γ).

(iii) (Van Maldeghem and Sastry) Let A be a clique in ΓEL whose membersmeet mutually at distinct points. Then, A has at most 6 elements. Any cliqueof three elements can be extended to a clique of size 5 if q = 4 and to a cliqueof size 6 if q > 4.

Proof. (i) is well-known. (ii) follows from the partition described in(B.ii.a.γ). For (iii), we use the set up outlined in the last two paragraphs of(B.ii.a.γ). It also contains a proof of (i). Let A be a clique in ΓEL with at

least 3 elements θ1, θ2, θ3. Let θ⊥i ∩ E = {ai, bi} ⊂ E\E, i = 1, 2, 3. Then,ai � bi and θi = Eai = Ebi . With proper indexing, we can take {a1, a2, a3}and {b1, b2, b3} to be triads in Q−(5, q) and their traces are conics C and D,respectively. If θ4 ∈ A and θ4 6= θi for i = 1, 2, 3, then θ4 = Ea4 = Eb4 for non

collinear points a4, b4 in E\E and, each of them is collinear to ai or bi (and notboth) for i = 1, 2, 3. Let b4 ∼ a1, say. Then, as a1 ∼ b2, b4 � b2. So, b4 ∼ a2.Similarly, b4 ∼ a3 and b4 ∈ C. If A has a fifth element θ5 = Ea5 = Eb5 , thenC contains an element b5 collinear to a4. Since the plane π containing C isnot contained in a⊥4 (since b4 ∈ a⊥4 ) and a⊥4 ∩ π is a line, there are at mosttwo possibilities for b5 and (iii) follows.

3 The code CLet X = (P,L) be the GQ W (q) and the notation be as in §1.i.

(A) Socle of kP and of C: Since q and the characteristic of k are even andsince the number of lines as well as the number of planes of W (q) incidentwith each x ∈ P is odd, 1πx ∈ C and 1P = Σx∈P1πx ∈ 〈Π〉 ⊆ C. We note that〈Π〉0 is a simple kG - module; in fact, the Steinberg module for G. If θ is anovoid of W (q), then the characteristic functions of its tangent planes {πx:x ∈ θ} form a basis for 〈Π〉 ([3], Lemma 6, p.144): in fact, for a secant planeπ of θ, each p ∈ P is on q + 1, 1, 2 or 0 tangent planes to θ at points of theoval π∩ θ according as p is the nucleus n of π∩ θ; n 6= p ∈ π; p /∈ π and p lieson, or does not lie on, the tangent planes considered. So, 1π = Σx∈θ∩π1πx .For any subset A of θ, the restriction of Σx∈Aλx1πx to θ is Σx∈Aλxx. This iszero if, and only if, λx = 0 for each x ∈ A. Thus, dimension of 〈Π〉 is q2 + 1.Since 1P ∈ 〈Π〉, 〈Π〉0 is of codimension one in 〈Π〉.

13

Proposition 6 (i) Socle (kP ) = 〈Π〉 w k1P ⊕〈Π〉0 w kP/rad (kP ) and is ofdimension q2 + 1.

(ii) Socle (Y ) = 〈Π〉0 w Y/rad (Y ). In particular, Y is indecomposable.

Proof. Let S be a Sylow 2- subgroup of G. Then, S fixes a flag (p, `) inW (q) and so, also πp. Further, it acts transitively on the sets ` \ {p}, πp \ `and P \ πp. So, 1{p},1`,1πp and 1P generate the space of S- fixed points inkP . Hence, k1P and 〈1P − 1πp : p ∈ P 〉 = 〈Π〉0 are the only simple kG-submodules of kP and socle of kP is 〈Π〉. From the preceeding remarks andthe self-duality of kP and Y , rest follows.

In particular, socle of C is 〈Π〉, that of C⊥ is 〈Π〉0 and that of any kG−submodule of kP is either k1P , 〈Π〉0 or 〈Π〉. Given a duality of δ of W (q),

the map from P to C taking x ∈ P to 1δ(x) defines a linear map δ from kP

onto C with kernel C⊥. Further, for ` ∈ L, δ(1`) = 1πδ(`) ∈ 〈Π〉 and δ is a

bijection between P and L. So, C ' kP/C⊥and C/C ∩ C⊥ ' 〈Π〉.For the decomposition of the permutation module kP = k1 ⊕ Y (as in

§1.A) for the permutation action of a finite group G with a split (B,N)- pairdefined over a field of the same characteristic p as that of k on the cosets ofa maximal parabolic subgroup, Y and Y/rad(Y ) are simple kG− modules.For details, see §.11 and Remark 11.4 of ([2]).

(B) Structure of C: The dimension of C is

1 + 2−2n[(1 +√

17)2n + (1−√

17)2n] = ΣI∈N4|I|, (*)

where N is the collection of all subsets of Z/2nZ which do not have consecu-tive elements ([43], Theorem 1, p.485; for the last equality, see [21], p.34-35).This formula is obtained by identifying the composition factors appearing ineach socle layer of C0. Note that (*) is not a rational function of q. The lat-tice of submodules of C is also known ([43], p.491). As each kG- compositionfactor of C0

appears precisely once, each submodule of C0is determined by

the isomorphism type of its quotient by its radical. Further, the number ofsubmodules of C0

is finite and is independent of k. The composition factorsof C0

and of kP are the same ([46], Theorem 1, 238). ‘Canonical‘ gener-ating sets for the socle layers of C2 = 〈1l : l ∈ L〉 ≤ kP , considered as aF2G−module, may be interesting from a geometric point of view.

In a significant development ([20] for even characteristic and [?, ?, ?] forodd characteristic), Chandler, Sin and Xiang have obtained a formula for

14

the dimension of the q− ary code generated by the characteristic functionsof isotropic subspaces of a fixed dimension l in a symplectic space over Fqof projective dimension n. They also identify the composition factors of themodule for the corresponding symplectic group. Their method is very sim-ilar to the representation theoretic approach in [9] for the famous Hamadaformula for the dimension of the q− ary code generated by the characteristicfunctions of the subspaces of fixed dimension l in a finite projective space overFq. If n = 3 and q = pn, p any prime, Fq-rank of point × isotropic line inci-

dence matrix reduces to 1 +αn1 +αn2 , where α1, α2 = p(p+1)2

4± p(p+1)(p−1)

12

√17.

It agrees with the expression above for p = 2 . However, the methods in [20]and in [43] are different.

The words of minimum nonzero weight in C are nonzero scalar multiplesof 1`, ` ∈ L.

(C) Subcodes of C generated by ovoids in W (q): For a subgroup A ofG, let σA denote the k− endomorphism of kP taking 1{x} ∈ kP , x ∈ P , toΣa∈A1{a(x)} ∈ kP . If T is a subgroup of G ' PSP (4, q) of order q2 + 1 (seeB.via), then σT (l) = 1P or 1θi according as l ∈ S or l is tangent to θi forsome i, 0 ≤ i ≤ q (see § 2.B.ii α and v.b). Thus, σT

(kP)

= 〈1θ0 , · · · ,1θq〉and σT (C) = 〈1θ0 ,1P 〉 ⊂ C. Since G is transitive on EL and C is a kG-module, 〈EL〉 ⊆ C. The dimension of 〈EL〉 is 5n . The composition factorsof each of its socle layers is known ([43],Theorem 13, p.493). Further, 〈EL〉= 〈HL〉 ([44],Theorem 3.1, p.5). However, the words of the minimum weightof 〈EL〉 are not known.

If W (q) admits a polarity τ , then C contains the characteristic function1θ of the set θ of all absolute points of τ , a Tits ovoid of W (q). Thisfollows because 1θ is the diagonal of the incidence matrix of W (q) (writtenas (ax,y)x,y∈P with ax,y = 1 or 0 according as x is, or is not, in yτ ) and the F2-row span of a symmetric (0, 1)- matrix contains the diagonal ([?], Theorem3, p.143; see also [11]). Since G is transitive on TL and C is a kG- module,〈JL〉 ⊆ C. As noted at the end of first para of (B.vii), the characteristicfunction of an elliptic ovoid of W (q) is a sum of the characteristic functionsof some Tits ovoids of W (q). So, 〈EL〉 ⊆ 〈JL〉 ⊆ C. For a proof of 1P /∈ 〈JL〉and more on 〈JL〉 / 〈EL〉, see [43], Theorem 15,p.495. Neither the dimensionnor the words of minimum weight of 〈JL〉 is known.

(D) On C⊥: (i) The subcode 〈D〉 of C⊥: Let D denote the set of all dualgrids in W (q) (see §2. B.iv). Each word of C⊥of minimum weight is of theform 1

m∪m⊥ , where m ∪m⊥ ∈ D and m is a line of PG(3, q) not in L ([42]).

15

An element of D meets each element of L∪D in either 0, 2 or 2(q+ 1) points.

So, 〈D〉 ⊆ C⊥∩D⊥

. Let T and θ0, · · · , θq be as in (§ 2.B.v.a) and m∪m⊥ ∈ D.Then, by a result with R.P.Shukla (unpublished), if θm, θm⊥ ∈ {θ1, · · · , θq}are the unique ovoids the lines m and m

⊥are tangent to, then θm 6= θ

m⊥ .

Consequently, for σT defined in (C.iii), σT

(m ∪m⊥

)= 1θm + 1θ

m⊥ which is

not in σT (C) if q > 2. If 〈D〉 = C⊥, then C = 〈D〉⊥ ⊇ 〈D〉, a contradiction ifq > 2. Thus, 〈D〉 6= C⊥ if q > 2. This is as in the case of the code orthogonalto the q− ary code generated by the lines of a projective 3− space over Fq.However we do not know the dimension of 〈D〉. Since 〈D〉 has only wordsof even weight, 〈D〉 6= 〈EL〉. Does 〈D〉 contains 〈1P\θ : θ ∈ EL〉?

For each θ ∈ WL (q), the set of lines of PG(3, q) not in L is partitionedinto the set S(θ) of secants of θ and the set E(θ) of external lines of θ. Themap l ←→ l⊥ is a bijection between S(θ) and E(θ) (which does not extendlinearly!) and D = {l ∪ l⊥ : l ∈ S(θ)}. Structure of the kH - submodules〈S(θ)〉 and 〈E(θ)〉 of kP , where H is the stabilizer of θ in Aut(WL (q)),for both elliptic and Tits ovoids of WL (q). See ([47]) for a study of similarcodes from ovals of Desarguesian projective planes of odd order.

By (γ) in (§2.B.ii.a), C⊥

is a one-step-completely orthogonalizable code.This means that, for each coordinate position x, its dual (that is, C) has2 (q + 1) − 1 = q + (q + 1) vectors whose supports intersect pairwise at {x}[4].

(ii) The subcode M of C⊥: For a subgroup T of G of order q2 + 1, letMT denote the subspace of kP of dimension q spanned by the characteristicfunctions of unions of even number of the T− orbits θ1, · · · , θq in P (see §2.B.v.a). Then, M = Σ{MT : T a subgroup of G of order q2 + 1} ≤ kP is akG− submodule of C⊥

(see §2.B.v.b). We do not know the dimension andwords of minimum weight of M . Is 〈D〉 ⊆ M?

(iii) The support of words of maximum weight in C⊥is the compliment of

an ovoid of W (q) ([3], [39]). Thus, determination of the weight enumeratorof C⊥

(more to the point, the number of words of C⊥of maximum weight,

only in the case q is an odd power of two, in view of the fact that ovoidsin W (4t) are all elliptic ([38])) settles the question of the existence of ovoidsin W (q) other than the classical ovoids. We mention that there are no codewords of C⊥

whose weight is in the interval (q3 + (5q − 4) /6, q3 + q) ([39],Theorem 16, p.3137). Study of the codes 〈OL〉 and 〈P\θ : θ ∈ OL〉 may be

16

instructive. From the structure of C in ([43],Theorem 1, p.485), C ∩ C⊥ is theradical of C and its radical series is known. If k = F2 and q > 2, then theweight of each element of C ∩ C⊥ is a multiple of 4, because if w ∈ C ∩ C⊥has weight congruent to 2 (mod4), then so does σT (w), but σT (C) has nosuch element. Considering the image under σT of sums of even number ofmembers of {θ1, · · · , θq}, we conclude that dim

(C⊥/C ∩ C⊥

)≥ q − 2. I do

not know the simple factors of C⊥/C ∩ C⊥.

4 A bilinear form and a polarity on CThe structure of the code C in §.3 as a k[H]− module, where H is thestabilizer in SP (4, q) of a quadratic form (either hyperbolic or elliptic) po-larizing to a symplectic form on PG(3, q) defining SP (4, q), is determined in[45],Theorem 1. We hope that the bilinear form and a polarity on C intro-duced in the next section (which exist when the incidence system admits apolarity) will be helpful in understanding C, particularly its structure as amodule for the stabilizer of a Tits ovoid in W (q), as well as the code over afield k of characteristic 3 associated with the (q, q)- GH H(32n+1) as a modulefor the Ree group 2G2 (32n+1).

Let X = (P,L) be a finite connected partial linear space with s + 1points on each line and s + 1 lines through each point and G = Aut(X).Assume that X admits a polarity τ , H = {g ∈ G : g ◦ τ = τ ◦ g on P ∪ L},O = {x ∈ P : x ∈ xτ} and S = {` ∈ L : `τ ∈ `}. The sets O and Sare H- invariant. The important examples here are W (q) and H(q) withq = p2m+1 (= s), p = 2 if X = W (q) and p = 3 if X = H(q).

(A) A nondegenerate bilinear form on C: Let k be a field of char-acteristic p; kP , kL and C be as in §1.ii and η : kL → C be the surjectivekG-morphism taking ` ∈ L to 1` = ` ∈ C. Let B′ be the symmeteic k−bilinear form on kL defined, for ` and m ∈ L, B′(`,m) to be one if `τ ∈ mand zero otherwise.

Proposition 7 (i) B′ is H- invariant, (ii) Rad B′ = ker η and (iii) B′

induces a symmetric nondegenerate H- invariant bilinear form B on C.

Proof. (i) follows from the equivalence of the following statements for `,m ∈L and h ∈ H: B′(h`, hm) = 1; (h`)τ ∈ hm;h(`τ ) ∈ hm; `τ ∈ m; B′(`,m) = 1.

17

(ii) follows from the equivalence of the following statements for α =Σl∈Lλ``, λ` ∈ k: α ∈ Rad(B′);B′(α,m) = 0 for each m ∈ L; Σmτ∈`∈Lλ` = 0for each m ∈ L; Σp∈`∈Lλ` = 0 for each p ∈ P ; α ∈ ker η.

(iii) If v, v′, w, w′ ∈ kL are such that η (v) = η(v′) and η (w) = η(w′), thenx = v − v′ and y = w − w′ are in ker η and B′(v, w) = B′(v′ + x,w′ + y) =B′(v′, w′), by (ii). So, B′ induces a symmetric H- invariant bilinear form Bon C. Further, if v = Σ`∈Lλ`` ∈ kL and w = η (v) ∈ C, then, for each m ∈ L,

B(1m, w) = B′(m,Σλ``) = Σλ`B′(m, `) = Σmτ∈`λ` = wmτ (**)

where, for p ∈ P , wp denotes the ‘pth-coordinate of w’. As τ is a bijectionbetween P and L, this implies that B(1m, w) = 0 for each m ∈ L if, and onlyif, w = 0. Thus, B is nondengererate.

Note that the radical of the restriction of f (defined in § 1.i) to C isC ∩ C⊥, which unlike radical of B, may be nonzero. For any subspace D ofC, we denote by D~ the orthogonal of D with respect to B. If (p, s+ 1) = 1,then 1P = Σ`∈L1` ∈ C and, for w ∈ C, by (**), B(w,1P ) = B(w,Σ`∈L1`) =Σ`∈Lw`τ

= Σp∈Pwp = f(w,1P ). So, 1~P = 1⊥P = C0. For x ∈ P , let πx =Σx∈m∈Lm ∈ kL and πx = η (πx) ∈ C. Then, for w ∈ C, B(πx, w) = Σp∈xτwp(by (**)) = f (w, xτ ). So, for Π = 〈πx : x ∈ P 〉 ≤ C, Π~ = C ∩ C⊥.

The subspaces M = {v ∈ C : B(v,1m) = f(v,1m) for each m ∈ L}, U =〈1` : ` ∈ S〉 and W = 〈1` : ` ∈ L \ S〉 are kH- submodules of C. An elementw ∈ C is inM if, and only if, wmτ = Σp∈mτwp for each m ∈ L. So, 1P ∈M.

Let p = 2 and write w ∈ C as w = Σ`∈Sλ`1l + Σ`∈L\Sµ`1l, λ`, µ` ∈ k.

Then, B(w,w) = Σ`∈Sλ2` = (Σ`∈Sλ`)

2. So, the set of all isotropic elementsof C with respect to B is U0 + W . Use of ([3], Theorem 3, p.143) againyields 1O ∈ C. Since x ∈ O if, and only if, xτ ∈ S, by (**), B(1m,1O) = 1

if mτ ∈ O and zero otherwise. So, 1O ∈ (U0 +W )~. Since U0 + W is of

codimension zero in C if O is empty and one otherwise, (U0 +W )~

= k1O.(B) A polarity of C: Let θ : kP −→ C be the surjective kH− module

homomorphism taking x ∈ P to τ(x). Recall A⊥ defined for A ⊆ kP .

Proposition 8 (i) 0 → C⊥ → kPθ→ C → 0 is an exact sequence of kH−

modules.(ii) For any subspace A of C, dim A+ dim θ(A⊥) = dim C.

Proof. (i) By definition, θ is onto. (i) follows from the equivalence of thefollowing statements for v = Σx∈Pvxx ∈ kP : v ∈ ker θ; Σx∈Pvx1xτ = 0;

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Σ{vx : x ∈ P such that p ∈ xτ} = 0 for each p ∈ P ; Σx∈pτvx = 0 for eachp ∈ P ; Σx∈`vx = 0 for each ` ∈ L; Σx∈Pvxx ∈ C⊥.

(ii) dim A+ dim A⊥ = |P | = dim C+ dim C⊥. Therefore, dim A+ dimA⊥− dim C⊥ = dim C. But C⊥ ⊆ A⊥ and A⊥/C⊥ ' θ(A⊥). So, (ii) holds.

Let P (V ) denote the set of all subspaces of a vector space V , partiallyordered by inclusion. Consider the maps

P(kP )⊥→ P(kP )

θ→ P(C),

where, for A ⊆ kP , ‘⊥’ takes A to its orthogonal compliment A⊥ with respectto f and, if A ∈ P(kP ), θ (A) = θ (A). We write A♦ = θ

(A⊥). Clearly, if A

is a kH− submodule of kP , then so is A♦.

Proposition 9 Let A ∈ P(kP ).

(a) C ⊆ A if, and only if, A♦ = 0.

(b) If A ∈ P(C), then

(i) dim A+ dim A♦ = dim C; and (ii) A = A♦♦.

(c) The map A → A♦, A ∈ P(C), is an inclusion reversing involutorypermutation of P(C).

(d) If (p, s+ 1) = 1, then 1P∈C, (k1P )♦ = C0 and (A♦)0 = (A+k1P )♦ foreach subspace A of C. In particular, the map in (c) is an inclusion reversingbijection between the set of all subspaces of C containing 1P and the set ofall subspaces of C0.

Proof. Since a subspace A contains C if, and only if, A⊥ ≤ C⊥, (a) followsfrom Proposition 8 (i). (b. i) is a restatement of Proposition 8 (ii).

Since the dimensions of A and A♦♦ are equal (by (b.i)), we need only toshow that A ⊆ A♦♦. Note that

A♦ ={

Σp∈P (Σx∈pτβx) p : Σp∈Pβpp ∈ A⊥}

.

Consider w = η(Σl∈Lλ``) ∈ C. Then, w = Σx∈P (Σx∈`∈Lλ`)x. Now, Σl∈Lλ``τ ∈(

A♦)⊥

if, and only if, for each w′ = Σβxx ∈ A⊥,

0 = Σ`τ∈Pλ` (Σ`τ∈xτβx) = Σ`τ∈Pλ` (Σx∈`βx) = Σx∈P (Σx∈`∈Lλ`) βx = f(w,w′).

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So, Σl∈Lλ``τ ∈

(A♦)⊥

if, and only if, w ∈ A⊥⊥ = A. Since w = θ (Σl∈Lλ``τ ),

it follows that A = A♦♦. So, (b.ii) follows. (c) is now follows from (b).

For w = Σx∈Pλxx ∈ kP , θ (w) = η(Σx∈Pλxxτ ) and c(θ(w)) = (s+ 1)c(w).

Since Σl∈L1l = (s+ 1)1P ∈ C and (p, s+ 1) = 1, 1P ∈ C. For any subspaceA of kP , θ(A ∩ 1⊥P ) = θ (A) ∩ 1⊥P ⊆ C0. So, (k1P )♦ = θ(k1⊥P ) ⊆ C0. SinceC0 and (k1P )♦ are both of codimension one in C (see Proposition 9 (b.i)),(k1P )♦ = C0. For A ∈ P(C), A⊥ ∩ 1⊥P = (A+ k1P )⊥. So,

(A♦)◦ = θ(A⊥)∩ 1⊥P = θ(A⊥ ∩ 1⊥P ) = θ((A+ k1P )⊥) = (A+ k1P )♦.

So, a subspace A of kP contains 1P if, and only if, A♦ ⊆ C0. So, (d) follows.

Let l ∈ L. If w = Σp∈Pβpp ∈ kP , then w ∈ l⊥ if, and only if, Σp∈lβp = 0.In this case, lτ -th coordinate of θ (w) is zero. So, θ(k1⊥l ) ≤ {w ∈ C : WL (q) =0}. Since both subspaces are of codimension one in C, equality holds. Asnoted above, (k1P )♦ = C0.

When (p, s+ 1) = 1, the polarity of P (C) taking A ∈ P (C) to A♦ is notsymplectic, because 1P /∈(k1P )♦ = C0.

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