Representation theory and combinatorics of diagram algebras.

Transcription

1 Representation theory and combinatorics of diagram algebras. Zajj Daugherty May 15, 2016

2 Combinatorial representation theory

3 Combinatorial representation theory Representation theory: Given an algebra A... What are the A-modules/representations? What are the simple/indecomposable A-modules/reps? What is the action of the center of A? What are their dimensions? How can I combine modules to make new ones, and what are they in terms of the simple modules?

4 Combinatorial representation theory Representation theory: Given an algebra A... What are the A-modules/representations? What are the simple/indecomposable A-modules/reps? What is the action of the center of A? What are their dimensions? 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.

5 Motivating example: Schur-Weyl Duality The symmetric group S k (permutations) as diagrams:

6 Motivating example: Schur-Weyl Duality The symmetric group S k (permutations) as diagrams: (with multiplication given by concatenation)

7 Motivating example: Schur-Weyl Duality The symmetric group S k (permutations) as diagrams: (with multiplication given by concatenation)

8 Motivating example: Schur-Weyl Duality The symmetric group S k (permutations) as diagrams: = (with multiplication given by concatenation)

9 Motivating example: Schur-Weyl Duality

10 Motivating example: Schur-Weyl Duality GL n (C) acts on C n C n C n = (C n ) k diagonally. g (v 1 v 2 v k ) = gv 1 gv 2 gv k.

11 Motivating example: Schur-Weyl Duality GL n (C) acts on C n C n C n = (C n ) k diagonally. g (v 1 v 2 v k ) = gv 1 gv 2 gv k. S k also acts on (C n ) k by place permutation. v 2 v 4 v 1 v 5 v 3 v 1 v 2 v 3 v 4 v 5

12 Motivating example: Schur-Weyl Duality GL n (C) acts on C n C n C n = (C n ) k diagonally. g (v 1 v 2 v k ) = gv 1 gv 2 gv k. S k also acts on (C n ) k by place permutation. v 2 v 4 v 1 v 5 v 3 v 1 v 2 v 3 v 4 v 5 These actions commute! gv 2 gv 4 gv 1 gv 5 gv 3 gv 2 gv 4 gv 1 gv 5 gv 3 vs. gv 1 gv 2 gv 3 gv 4 gv 5 v 1 v 2 v 3 v 4 v 5

13 Motivating example: Schur-Weyl Duality Schur (1901): S k and GL n have commuting actions on (C n ) k. Even better, End GLn ( (C n ) k) }{{} (all linear maps that commute with GL n) = π(cs k ) }{{} (img of S k action) and End Sk ((C n ) k) = ρ(cgl n ). }{{} (img of GL n action)

14 Motivating example: Schur-Weyl Duality Schur (1901): S k and GL n have commuting actions on (C n ) k. Even better, End GLn ( (C n ) k) }{{} (all linear maps that commute with GL n) = π(cs k ) }{{} (img of S k action) and End Sk ((C n ) k) = ρ(cgl n ). }{{} (img of GL n action) Why this is exciting: The double-centralizer relationship produces (C n ) k = G λ S λ as a GL n -S k bimodule, λ k where Gλ are distinct irreducible GL n -modules S λ are distinct irreducible S k -modules

15 Motivating example: Schur-Weyl Duality Schur (1901): S k and GL n have commuting actions on (C n ) k. Even better, End GLn ( (C n ) k) }{{} (all linear maps that commute with GL n) = π(cs k ) }{{} (img of S k action) and End Sk ((C n ) k) = ρ(cgl n ). }{{} (img of GL n action) Why this is exciting: The double-centralizer relationship produces (C n ) k = G λ S λ as a GL n -S k bimodule, λ k where Gλ are distinct irreducible GL n -modules S λ are distinct irreducible S k -modules For example, C n C n C n = ( G S ) ( G S ) ( G S )

16 Representation theory of V k V = C = L( )

17 Representation theory of V k V = C = L( ), L( )

18 Representation theory of V k V = C = L( ), L( ) L( )

19 Representation theory of V k V = C = L( ), L( ) L( ) L( )

20 Representation theory of V k V = C = L( ), L( ) L( ) L( ) L( )

21 Representation theory of V k V = C = L( ), L( ) L( ) L( ) L( ) L( )...

22 Representation theory of V k V = C = L( ), L( ) L( ) L( ) L( ) L( )...

23 Representation theory of V k V = C = L( ), L( ) L( ) L( ) L( ) L( )...

24 More centralizer algebras Brauer (1937) Orthogonal and symplectic groups (and Lie algebras) acting on (C n ) k diagonally centralize the Brauer algebra: n δ b,c i=1 v i v i v a v d v d v a v b v c v d v e with = n

25 More centralizer algebras Rep theory of V k, orthogonal and symplectic: V = C = L( )

26 More centralizer algebras Rep theory of V k, orthogonal and symplectic: V = C = L( ), L( )

27 More centralizer algebras Rep theory of V k, orthogonal and symplectic: V = C = L( ), L( ) L( )

28 More centralizer algebras Rep theory of V k, orthogonal and symplectic: V = C = L( ), L( ) L( ) L( )

29 More centralizer algebras Rep theory of V k, orthogonal and symplectic: V = C = L( ), L( ) L( ) L( ) L( )...

30 More centralizer algebras Brauer (1937) Orthogonal and symplectic groups (and Lie algebras) acting on (C n ) k diagonally centralize the Brauer algebra: n δ b,c i=1 v i v i v a v d v d v a v b v c v d v e with = n Temperley-Lieb (1971) GL 2 and SL 2 (and gl 2 and sl 2 ) acting on (C 2 ) k diagonally centralize the Temperley-Lieb algebra: δ c,d i=1 2 v a v i v i v b v e v a v b v c v d v e with = 2

31 More centralizer algebras Brauer (1937) Orthogonal and symplectic groups (and Lie algebras) acting on (C n ) k diagonally centralize the Brauer algebra: n δ b,c i=1 v i v i v a v d v d v a v b v c v d v e with = n Temperley-Lieb (1971) GL 2 and SL 2 (and gl 2 and sl 2 ) acting on (C 2 ) k diagonally centralize the Temperley-Lieb algebra: δ c,d i=1 2 v a v i v i v b v e v a v b v c v d v e with = 2 Either way: Diagrams encoding maps V k V k that commute with the action of some classical algebra.

32 Quantum groups and braids Fix q C, and let U = U q g be the Drinfeld-Jimbo quantum group associated to Lie algebra g.

33 Quantum groups and braids Fix q C, and let U = U q g be the Drinfeld-Jimbo quantum group associated to Lie algebra g. U U has an invertible element R = R R 1 R 2 that yields a map Ř V W : V W W V that (1) satisfies braid relations, and (2) commutes with the action on V W for any U-module V. W V V W

34 Quantum groups and braids Fix q C, and let U = U q g be the Drinfeld-Jimbo quantum group associated to Lie algebra g. U U has an invertible element R = R R 1 R 2 that yields a map Ř V W : V W W V that (1) satisfies braid relations, and (2) commutes with the action on V W for any U-module V. The braid group shares a commuting action with U on V k : V V V V V W V V W V V V V V

35 Quantum groups and braids Fix q C, and let U = U q g be the Drinfeld-Jimbo quantum group associated to Lie algebra g. U U has an invertible element R = R R 1 R 2 that yields a map Ř V W : V W W V that (1) satisfies braid relations, and (2) commutes with the action on V W for any U-module V. W V V W The one-pole/affine braid group shares a commuting action with U on M V k : M V V V V V Around the pole: M V = ŘMV ŘV M M V V V V V M V

36 Quantum groups and braids Fix q C, and let U = U q g be the Drinfeld-Jimbo quantum group associated to Lie algebra g. U U has an invertible element R = R R 1 R 2 that yields a map Ř V W : V W W V that (1) satisfies braid relations, and (2) commutes with the action on V W for any U-module V. W V V W The two-pole braid group shares a commuting action with U on M V k N: M V V V V V N Around the pole: M V = ŘMV ŘV M M V V V V V N M V

37 Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Brauer algebra Sym. group Temperley-Lieb V = V V

38 Universal Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Brauer algebra Sym. group Braid group BMW algebra Hecke algebra = a + Temperley-Lieb V = V V Quantum groups

39 Universal Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Brauer algebra Sym. group Braid group BMW algebra Hecke algebra = a + Temperley-Lieb V = V V Quantum groups A ne braids A ne BMW A ne Hecke of type A (+twists) Two-pole braids Two-pole BMW A ne Hecke of type C (+twists) One-boundary TL Two-boundary TL M V k M V k N

40 Universal Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Braid group Brauer algebra BMW algebra Sym. group Hecke algebra = a + Temperley-Lieb V = V V Quantum groups A ne braids A ne BMW A ne Hecke of type A (+twists) Two-pole braids Two-pole BMW A ne Hecke of type C (+twists) One-boundary TL Two-boundary TL M V k M V k N

41 Type B, C, D (orthog. & sympl.) Lie grp/alg Brauer algebra BMW algebra V = V V Quantum groups A ne BMW Two-pole BMW M V k M V k N

42 Type B, C, D Nazarov (95): degenerate affine BMW algebras Lie grp/alg (orthog. & sympl.) Brauer algebra BMW algebra V = V V l = z l C act on M V k, commuting with the action of the Lie algebras of types B, C, D. Quantum groups A ne BMW Two-pole BMW M V k M V k N

43 Type B, C, D Nazarov (95): degenerate affine BMW algebras Lie grp/alg Quantum groups (orthog. & sympl.) Brauer algebra BMW algebra A ne BMW Two-pole BMW V = V V M V k M V k N l = z l C act on M V k, commuting with the action of the Lie algebras of types B, C, D. Häring-Oldenburg (98) and Orellana-Ram (04): affine BMW algebras act on M V k, commuting with the action of the quantum groups of types B, C, D.

44 Type B, C, D Nazarov (95): degenerate affine BMW algebras Lie grp/alg Quantum groups (orthog. & sympl.) Brauer algebra BMW algebra A ne BMW Two-pole BMW V = V V M V k M V k N l = z l C act on M V k, commuting with the action of the Lie algebras of types B, C, D. Häring-Oldenburg (98) and Orellana-Ram (04): affine BMW algebras act on M V k, commuting with the action of the quantum groups of types B, C, D. Centralizer perspective: (D.-Ram-Virk) Use centralizer relationships to study these the affine and degenerate affine algebras simultaneously (representation theory of the quantum groups and the Lie algebras are basically the same). Some results: (a) The center of each algebra. (b) Difficult admissibility conditions handled. (c) Powerful intertwiner operators.

45 Universal Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Braid group Brauer algebra BMW algebra Sym. group Hecke algebra = a + Temperley-Lieb V = V V Quantum groups A ne braids A ne BMW A ne Hecke of type A (+twists) Two-pole braids Two-pole BMW A ne Hecke of type C (+twists) One-boundary TL Two-boundary TL M V k M V k N

46 Universal Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Brauer algebra Sym. group Braid group BMW algebra Hecke algebra = a + Temperley-Lieb V = V V Quantum groups A ne braids A ne BMW A ne Hecke of type A (+twists) Two-pole braids Two-pole BMW A ne Hecke of type C (+twists) One-boundary TL Two-boundary TL M V k M V k N

47 Universal Type B, C, D Type A Small Type A Universal Type B, C, Type Small Type (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Qu Qu grp grp Two-pole braids Two-pole BMW Two-pole braids Two-pole BMW A ne Hecke of ne type Hecke C of type (+twists) (+twists) Two-boundary TL Two-boundary TL M V V k N N

48 Universal Type B, C, D Type A Small Type A Universal Type B, C, Type Small Type (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Qu Qu grp grp Two-pole braids Two-pole BMW Two-pole braids Two-pole BMW A ne Hecke of ne type Hecke C of type (+twists) (+twists) Two-boundary TL Two-boundary TL Two boundary algebras: Mitra, Nienhuis, De Gier, Batchelor (2004): Studying the six-vertex model with additional integrable boundary terms, introduced the two-boundary Temperley-Lieb algebra T L k : non-crossing diagrams M V V k N N even # dots k dots

49 Universal Type B, C, D Type A Small Type A Universal Type B, C, Type Small Type (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Qu Qu grp grp Two-pole braids Two-pole BMW Two-pole braids Two-pole BMW A ne Hecke of ne type Hecke C of type (+twists) (+twists) Two-boundary TL Two-boundary TL Two boundary algebras: Mitra, Nienhuis, De Gier, Batchelor (2004): Studying the six-vertex model with additional integrable boundary terms, introduced the two-boundary Temperley-Lieb algebra T L k : non-crossing diagrams M V V k N N even # dots k dots De Gier, Nichols (2008): Explored representation theory of T L k using diagrams and established a connection to the affine Hecke algebras of type A and C.

50 Affine type C Hecke algebra and two-boundary braids k 2 k 1 Fix constants t 0, t k, and t = t 1 = = t k 1. The affine Hecke algebra of type C, H k, is generated by T 0, T 1,..., T k with relations k 2 if i j T i T j... }{{} = T j T i... }{{} m i,j factors m i,j factors where m i,j = 3 if 4 if i i j j and T 2 i = (t 1/2 i t 1/2 i )T i + 1.

51 Affine type C Hecke algebra and two-boundary braids k 2 k 1 Fix constants t 0, t k, and t = t 1 = = t k 1. The affine Hecke algebra of type C, H k, is generated by T 0, T 1,..., T k with relations k T i T j... }{{} = T j T i... }{{} m i,j factors m i,j factors and T 2 i = (t 1/2 i t 1/2 i )T i + 1.

52 Affine type C Hecke algebra and two-boundary braids k 2 k 1 Fix constants t 0, t k, and t = t 1 = = t k 1. The affine Hecke algebra of type C, H k, is generated by T 0, T 1,..., T k with relations k T i T j... }{{} = T j T i... }{{} m i,j factors m i,j factors and T 2 i = (t 1/2 i t 1/2 i )T i + 1. The two-boundary (two-pole) braid group B k is generated by i i+1 T k = T 0 = and T i = for 1 i k 1. i i+1

53 Affine type C Hecke algebra and two-boundary braids k 2 k 1 Fix constants t 0, t k, and t = t 1 = = t k 1. The affine Hecke algebra of type C, H k, is generated by T 0, T 1,..., T k with relations k T i T j... }{{} = T j T i... }{{} m i,j factors m i,j factors and T 2 i = (t 1/2 i t 1/2 i )T i + 1. The two-boundary (two-pole) braid group B k is generated by i i+1 T k = T 0 = and T i = for 1 i k 1. i i+1 Relations: T i T i+1 T i = = = T i+1 T i T i+1

54 Affine type C Hecke algebra and two-boundary braids k 2 k 1 Fix constants t 0, t k, and t = t 1 = = t k 1. The affine Hecke algebra of type C, H k, is generated by T 0, T 1,..., T k with relations k T i T j... }{{} = T j T i... }{{} m i,j factors m i,j factors and T 2 i = (t 1/2 i t 1/2 i )T i + 1. The two-boundary (two-pole) braid group B k is generated by i i+1 T k = T 0 = and T i = for 1 i k 1. i i+1 Relations: T i T i+1 T i = = = T i+1 T i T i+1 T 1 T 0 T 1 T 0 = = = T 0 T 1 T 0 T 1

55 Affine type C Hecke algebra and two-boundary braids Punchline: For any for any complex reductive Lie algebras g, the quantum group U q g and the two-boundary braid group B k have commuting actions on M (V ) k N. When g = gl n, for good choices of M, N, and V, the action of the two-boundary braid group factors to an action of the affine Hecke algebra of type C.

56 Affine type C Hecke algebra and two-boundary braids Punchline: For any for any complex reductive Lie algebras g, the quantum group U q g and the two-boundary braid group B k have commuting actions on M (V ) k N. When g = gl n, for good choices of M, N, and V, the action of the two-boundary braid group factors to an action of the affine Hecke algebra of type C. Some consequences: (a) A combinatorial classification and construction of irreducible representations of H k (type C with distinct parameters). (b) A diagrammatic intuition for H k. (c) A classification of the representations of T L k via the action of its center.

57 Universal Type B, C, D Type A Small Type A (orthog. & sympl.) (gen. & sp. linear) (GL 2 &SL 2 ) Lie grp/alg Brauer algebra Sym. group Braid group BMW algebra Hecke algebra = a + Temperley-Lieb V = V V Quantum groups A ne braids A ne BMW A ne Hecke of type A (+twists) Two-pole braids Two-pole BMW A ne Hecke of type C (+twists) One-boundary TL Two-boundary TL M V k M V k N