Combinatorics and representation theory of diagram algebras. Zajj Daugherty The City College of New York & The CUNY Graduate Center February 3, 2020 Slides available at https://zdaugherty.ccnysites.cuny.edu/research/ Combinatorial representation theory Representation theory: Given an algebra A... • What are the A-modules/representations? (Actions A ˝ V and homomorphisms ' : A Ñ EndpV q) • What are the simple/indecomposable A-modules/reps? • What are their dimensions? • What is the action of the center of A? • How can I combine modules to make new ones, and what are they in terms of the simple modules? In combinatorial representation theory, we use combinatorial objects to index (construct a bijection to) modules and representations, and to encode information about them.
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Combinatorics and representation theoryof diagram algebras.
Zajj Daugherty
The City College of New York& The CUNY Graduate Center
February 3, 2020
Slides available at https://zdaugherty.ccnysites.cuny.edu/research/
Combinatorial representation theory
Representation theory: Given an algebra A. . .
• What are the A-modules/representations?
(Actions A V and homomorphisms ' : A Ñ EndpV q)
• What are the simple/indecomposable A-modules/reps?
• What are their dimensions?
• What is the action of the center of A?
• How can I combine modules to make new ones, and what arethey in terms of the simple modules?
In combinatorial representation theory, we use combinatorialobjects to index (construct a bijection to) modules andrepresentations, and to encode information about them.
Motivating example: Schur-Weyl Duality
The symmetric group Sk (permutations) as diagrams:
1
1
2
2
3
3
4
4
5
5
1 2 3 4 5
“
1
1
2
2
3
3
4
4
5
5
(with multiplication given by concatenation)
Motivating example: Schur-Weyl Duality
GLnpCq acts on Cnb Cn
b ¨ ¨ ¨ b Cn“ pCn
qbk diagonally.
g ¨ pv1 b v2 b ¨ ¨ ¨ b vkq “ gv1 b gv2 b ¨ ¨ ¨ b gvk.
Sk also acts on pCnq
bk by place permutation.
v1 v2 v3 v4 v5
b
b
b
b
b
b
b
b
v2 v4 v1 v5 v3
These actions commute!
gv1 gv2 gv3 gv4 gv5
b
b
b
b
b
b
b
b
gv2 gv4 gv1 gv5 gv3
vs.
v1 v2 v3 v4 v5
b
b
b
b
b
b
b
b
gv2 gv4 gv1 gv5 gv3
Motivating example: Schur-Weyl DualitySchur (1901): Sk and GLn have commuting actions on pCn
qbk.
Even better,
EndGLn
´pCn
qbk
¯
loooooooooomoooooooooon(all linear maps thatcommute with GLn)
� are distinct irreducible GLn-modulesS� are distinct irreducible Sk-modules
For example,
Cnb Cn
b Cn“
´G b S
¯‘
´G b S
¯‘
´G b S
¯
Representation theory of V bk
V “ C “ Lp q, Lp q b Lp q b Lp q b Lp q b Lp q ¨ ¨ ¨
H
......
...
H
More centralizer algebras
Brauer (1937)Orthogonal and symplectic groups(and Lie algebras) acting onpCn
qbk diagonally centralize
the Brauer algebra:
va vb vc vd ve
vi vi va vd vdb
b
b
b
b
b
b
b
�b,c
nÿ
i“1
with “ n
Temperley-Lieb (1971)GL2 and SL2 (and gl2 and sl2) act-ing on pC2
qbk diagonally centralize
the Temperley-Lieb algebra:
va vb vc vd ve
va vi vi vb veb
b
b
b
b
b
b
b
�c,d
2ÿ
i“1
with “ 2
Diagrams encode maps Vbk
Ñ Vbk that commute with the
action of some classical algebra.
More centralizer algebras
Representation theory of Vbk, orthogonal and symplectic:
V “ C “ Lp q, Lp q b Lp q b Lp q b Lp q ¨ ¨ ¨
H
H
H
H
......
...
More diagram algebras: braids
The braid group:
1
1
1
“
(with multiplication given by concatenation)
More diagram algebras: braids
The a�ne (one-pole) braid group:
1
1
1
“
(with multiplication given by concatenation)
Quantum groups and braidsFix q P C, and let U “ Uqg be the Drinfeld-Jimbo quantum groupassociated to Lie algebra g.U b U has an invertible element R “
∞R R1 b R2 that yields a map
RVW : V b W ݄ W b V
W b V
V b W
that (1) satisfies braid relations, and(2) commutes with the action on V b W
for any U -module V .
The two-pole braid group shares a commuting actionwith U on M b V
bkb N :
V
V
b
b
V
V
b
b
V
V
b
b
V
V
b
b
V
V
Mb
Mb
bN
bN
Around the pole:
MbV
MbV
“ RMV RVM
Orthogonaland
symplectic(types B, C, D)
V bk M b V bk M b V bk b N
Qu. grps:BMW algebra A�ne BMW 2-bdry BMW
Lie algs:Brauer algebra Deg. a↵. BMW Deg. 2-bdry BMW
Nazarov (95): Introduced degenerate a�ne Birman-Murakami-Wenzl(BMW) algebras, built from Brauer algebras and their Jucys-Murphyelements.
Haring-Oldenburg (98) and Orellana-Ram (04): Introduced thea�ne BMW algebras. [OR04] gave the action on M b V
bk commutingwith the action of the quantum groups of types B, C, D.
D.-Ram-Virk: Used these centralizer relationships to study these twoalgebras simultaneously. Results include computing the centers, handlingthe parameters associated to the algebras, computing powerfulintertwiner operators, etc.
Two boundary algebras (type A)Nienhuis, de Gier, Batchelor (2004): Studying the six-vertex modelwith additional integrable boundary terms, introduced the two-boundaryTemperley-Lieb algebra TLk:
k dots
even
#dot
s
non-crossing diagrams
de Gier, Nichols (2008): Explored representation theory of TLk usingdiagrams and established a connection to the a�ne Hecke algebras oftype A and C.D. (2010): The centralizer of gln acting on tensor space M b V
bkb N
displays type C combinatorics for good choices of M , N , and V .
The two-boundary (two-pole) braid group Bk is generated by
Tk “ , T0 “ and Ti “
i
i
i+1
i+1
for 1 § i § k ´ 1,
subject to relationsT0 T1 T2 Tk´2 Tk´1 Tk
i.e.
TiTi`1Ti “ “ “ Ti`1TiTi`1,
T1T0T1T0 “ “ “ T0T1T0T1,
and, similarly, Tk´1TkTk´1Tk “ TkTk´1TkTk´1.
(1) The two-boundary (two-pole) braid group Bk is generated by
Tk “ , T0 “ and Ti “
i
i
i+1
i+1
for 1 § i § k ´ 1,
subject to relations T0 T1 T2 Tk´2 Tk´1 Tk .
(2) Fix constants t0, tk, t P C.The a�ne type C Hecke algebra Hk is the quotient of CBk by therelations
pT0 ´ t1{20 qpT0 ` t
´1{20 q “ 0, pTk ´ t
1{2k qpTk ` t
´1{2k q “ 0
and pTi ´ t1{2
qpTi ` t´1{2
q “ 0 for i “ 1, . . . , k ´ 1.
(3) Set
“ t1{20 ´ pe0 “ t
1{20 ´ T0q
“ t1{2k ´ pek “ t
1{2k ´ Tkq
“ t1{2
´ pei “ t1{2
´ Tiq
so that e2j “ zjej (for good zj).
The two-boundary Temperley-Lieb algebra is the quotient of Hk by therelations eiei˘1ei “ ei for i “ 1, . . . , k ´ 1.
(1) The two-boundary (two-pole) braid group Bk is generated by
Tk “ , T0 “ and Ti “
i
i
i+1
i+1
for 1 § i § k ´ 1,
subject to relations T0 T1 T2 Tk´2 Tk´1 Tk .
(2) Fix constants t0, tk, t “ t1 “ t2 “ ¨ ¨ ¨ “ tk´1 P C.The a�ne type C Hecke algebra Hk is the quotient of CBk by the
relations pTi ´ t1{2i qpTi ` t
´1{2i q “ 0.
(3) Set
“ t1{20 ´ pe0 “ t
1{20 ´ T0q
“ t1{2k ´ pek “ t
1{2k ´ Tkq
“ t1{2
´ pei “ t1{2
´ Tiq
so that e2j “ zjej (for good zj).
The two-boundary Temperley-Lieb algebra is the quotient of Hk by therelations eiei˘1ei “ ei for i “ 1, . . . , k ´ 1.
Theorem (D.-Ram)
(1) Let U “ Uqg for any complex reductive Lie algebras g.Let M , N , and V be finite-dimensional modules.
The two-boundary braid group Bk acts on M b pV qbk
b N and this
action commutes with the action of U .
(2) If g “ gln, then (for correct choices of M , N , and V ),
the a�ne Hecke algebra of type C, Hk, acts on M b pV qbk
b N
and this action commutes with the action of U .
(3) If g “ gl2, then the action of the two-boundary Temperley-Lieb
algebra factors through the T.L. quotient of Hk.
Some results:
(a) A diagrammatic intuition for Hk.
(b) A combinatorial classification and construction of irreduciblerepresentations of Hk (type C with distinct parameters) via centralcharacters and generalizations of Young tableaux.
(c) A classification of the representations of TLk in [dGN08] via centralcharacters, including answers to open questions and conjecturesregarding their irreducibility and isomorphism classes.
V
V b
b V
V b
b V
V b
b V
V b
b V
VMb
Mb
bN
bN
Move both polesto the left
Ó
V
V b
b V
V b
b V
V b
b V
V b
b V
VMb
Mb
Nb
Nb
Jucys-Murphy elements:
Yi “
i
i
§ Pairwise commute
§ ZpHkq is (type-C) symmetricLaurent polynomials in Zi’s
§ Central characters indexed byc P Ck (modulo signed permutations)
Back to tensor space operators propertiesThe eigenvalues of the Ti’s must coincide with the eigenvalues ofthe corresponding R-matrices, which can be computedcombinatorially.
0 “ pT0 ´ t0qpT0 ´ t´10 q “ pTk ´ tkqpTk ´ t
´1k q “ pTi ´ t
1{2qpTi ` t
´1{2q
T0 “ 9 RVM RMV Tk “ 9 RNV RV N Ti “
i
i
i+1
i+1
9 RV V
a0
´b0
0 M
ak
´bk
0 N
1
´1
0 V
t0 “ ´q2pa0`b0q
tk “ ´q2pak`bkq
t “ q2
Exploring M b N b Lp qbk
Products of rectangles:
Lppab00 qq b Lppak
bkqq “
à
�P⇤
Lp�q (multiplicity one!)
where ⇤ is the following set of partitions. . .
pab00 q b “ ‘ ‘
‘ ‘ ‘
Exploring M b N b Lp qbk
k “ 0
k “ 1
k “ 2
b0
a0
L
ˆ ˙b L
´ ¯b L
` ˘b L
` ˘b L
` ˘b L
` ˘b L
` ˘
Y1 fiÑ t5.5
Y2 fiÑ t3.5
Y3 fiÑ t´4.5
Y4 fiÑ t´5.5
Y5 fiÑ t´2.5
0 1 2 3 4 5 6
-1
-2
-3
-4
-5
Shift by 12 pa0 ´ b0 ` ak ´ bkq
0 1 2 3 4 5 6-1
-2
-3
-4
-5
-6
1
2
3
4
5
5
4
1
3
2
Y1 fiÑ t5.5
Y2 fiÑ t3.5
Y3 fiÑ t´4.5
Y4 fiÑ t´5.5
Y5 fiÑ t´2.5
Y1 fiÑ t´5.5
Y2 fiÑ t2.5
Y3 fiÑ t4.5
Y4 fiÑ t3.5
Y5 fiÑ t5.5
-5
5
4
1
3
2
5
4
1
3
2
Y1 fiÑ t5.5
Y2 fiÑ t3.5
Y3 fiÑ t´4.5
Y4 fiÑ t´5.5
Y5 fiÑ t´2.5
(˚) Hk representations in tensor space are labeled by certain partitions �.(˚) Basis labeled by tableaux from some partition µ in pa
cq b pb
dq to �.
(˚) Calibrated (Yi’s are diagonalized): Yi acts by t to the shifted diagonalnumber of boxi. (Think: signed permutations.)