From Schur-Weyl duality to quantum symmetric pairs

Transcription

1 .. From Schur-Weyl duality to quantum symmetric pairs Chun-Ju Lai Max Planck Institute for Mathematics in Bonn Dec 8, 2016

2 Outline...1 Schur-Weyl duality

3 Background.. GL n := GL n (C) = general linear group of C n Schur s 1901 dissertation on the polynomial representations of GL n : Each polynomial representation V is semi-simple (i.e. V = irreducible modules) Each polynomial representation is homogeneous for some degree d N There is a correspondence { } { } polynomial representation 1:1 representation of of GL n of degree d symmetric group S d {irreducibles} {irreducibles} {partitions λ = (λ 1,..., λ n ) d}

4 Background.. Weyl s 1926 research on representation of semi-simple Lie group based on repn theory of Lie algebra, integration over compact form has no counterpart to Sd in Schur s method In 1927, Schur re-derived his 1901 dissertation by showing the double centralizer property for (GL n, S d ), which was publicized by Weyl in his 1939 book The Classic Groups. The methods used in the classical Schur-Weyl are still important in representation theory today

5 Dual actions.. gl n := gl n (C): general linear Lie algebra V := C n natural representation of gl n action on V d by the Leibniz rule, e.g., g (v 1 v 2 ) = (g v 1 ) v 2 + v 1 (g v 2 ) so that its exponential e tg acts group-like e tg (v 1 v 2 ) = e tg v 1 e tg v 2 S d has a (right) action on V d by permuting tensor factors, e.g., (v 1 v 2 ) (12) = v 2 v 1 gl n ψ V d S d general linear Lie algebra tensor space symmetric group φ

6 Example: commutivity.. Here s an example showing ( ) ( ) g(v 1 v 2 ) (12) = g (v 1 v 2 )(12) V V (12) V V v 1 v 2 v 2 v 1 g g gv 1 v 2 + v 1 gv 2 gv 2 v 1 + v 2 gv 1 V V (12) V V

7 .. Double Centralizer Property We have ψ : gl n End(V d ), φ : S d End(V d ) and gl n ψ V d S d general linear Lie algebra tensor space symmetric group φ Schur-Weyl duality (1927) 1. The actions of gl n and S d on the tensor space V d commute 2. The algebras generated by the actions of gl n and S d in End(V d ) are centralizing algebras of each other, i.e., End φ(a) (V d ) = ψ(b), End ψ(b) (V d ) = φ(a), where A = C[S d ] = group algebra, B = U(gl n ) = univ env algebra

8 Double centralizer property.. The double centralizer property leads to Corollary There is a decomposition V d = λ d V λ L λ, where {V λ } = Irrep(S d ); {L λ } are distinct irreducibles of gl n or 0.

9 .. quantization U(gl n ) (C n ) d C[S d ] univ env algebra group algebra of sym group. U q (gl n ) quantum group. (Q(v) n ) d H(S d ) Hecke algebra quantization Here the quantized algebras are over Q(v) with v = q 1/2 : indeterminate

10 Hecke algebra.. Recall that the symmetric group S d is generated by simple reflections s 1, s 2,..., s d 1 subject to the braid relations and s 2 i = 1. Hecke algebra H(S d ) is a Q(v)-algebra generated by T 1, T 2,..., T d 1 subject to braid relations and the Hecke relations (T i + 1)(T i q) = 0. (1) Specializing q 1, (1) recovers s 2 i = 1

11 Hecke algebra.. H(S d ) has a linear basis (called standard basis) {T w w S d } H(S d ) has a involution : H(S d ) H(S d ) sending v v 1, T w T 1 w 1 H(S d ) has a bar-invariant basis {C w } w Sd called the Kazhdan-Lusztig(KL) basis The famous Kazhdan-Lusztig theory ( 79) offered a solution the the difficult problem of determining irreducible character problem for category O of semisimple Lie algebras using the KL basis

12 Quantum groups.. Recall that the universal enveloping algebra U(gl n ) is an associative algebra generated by {e i, f i, d j, d 1 j }, subject to Chevalley/Serre relations (e.g. e 2 1e 2 + e 2 e 2 1 = 2e 1 e 2 e 1 ) Around 1985, Drinfeld and Jimbo introduced the quantum group U = U q (gl n ) as a Q(v)-algebra generated by {E i, F i, D j, D 1 j }, subject to q-chevellay/serre relations (e.g. E 2 1E 2 + E 2 E 2 1 = (v + v 1 )E 1 E 2 E 1 )

13 Quantum groups.. U provides solutions to the Yang-Baxter equation U is a Hopf algebra. Particularly it has a comultiplication : U U U There is a triangular decomposition U = U U 0 U + There is a modified (i.e. idempotented) quantum group U: U quantum group { replacing 1 with idempotents take certain infinite sum U modified quantum group U-modules} 1:1 {weight modules of U} U is viewed as a preadditive category in categorification

14 Canonical basis.. U has a sheaf theoretic interpretation bar involution : U U from the Verdier duality U admits canonical basis, i.e., U has a standard basis {a i } i I that is unitriangular and integral : a i a i + j<i Z[v, v 1 ]a j, and a (unique) canonical basis {b i } i I that is bar-invariant and positive : b i = b i a i + v 1 N[v 1 ]a j j<i This is analogous to (dual) Kazhdan-Lusztig basis for Hecke algebra

15 Canonical basis.. Canonical basis = Kashiwara s global crystal basis, i.e., for any weight λ, {b i v λ } is a basis of irreducible integrable U-module L(λ) := U v λ Canonical basis {b i } has positivity, i.e. b i b j k N[v, v 1 ]b k U does not admit canonical basis, while U does Existence of canonical basis has connection/application in Categorification Algebraic combinatorics Category O Cluster algebra Geometric representation theory

16 Outline...1 Schur-Weyl duality

17 of type A.. Our goal here is to describe the quantized Schur-Weyl duality below in examples: U q (gl n ) quantum group ψ V d φ H(Sd ) Hecke algebra We start with describing first a tensor space V d admitting natural actions from both sides

18 .. Natural representation V of U q (gl n ) V := n Q(v)v i : natural representation of U q (gl n ), e.g. n = 3 i=1 F 1 F 2 v 1 v 2 v 3 E 1 E 2 D q D2 1 1 q 1 1 U q (gl n ) acts on V d via comultiplication, e.g. (F 2 ) = D 1 2 D 3 F 2 + F 2 1 Hence F 2 (v 3 v 2 ) = D2 1 D 3v 3 F 2 v 2 + F 2 v 3 v 2 = qv 3 v 3 F 2 (v 2 v 3 ) = D2 1 D 3v 2 F 2 v 3 + F 2 v 2 v 3 = v 3 v 3

19 Hecke algebra action.. For H(S 2 ), we have the following equalities in one-line notation (1 = [12], s 1 = [21]) T [12] T 1 = T [21] T [21] T 1 = qt [12] + (q 1)T [21] This leads to a natural Hecke algebra action on V 2 respecting the multiplication rule, e.g. v b v a if b > a (v a v b )T 1 = qv b v a + (q 1)v a v b if b < a qv a v a if b = a

20 Example: commutivity.. Here s an example showing ( ) F 2 (v 3 v 2 ) )T 1 = F 2 ((v 3 v 2 )T 1 V V T 1 V V v 3 v 2 (q 1)v 3 v 2 + qv 2 v 3 F 2 V V qv 3 v 3 q 2 v 3 v 3 = (q 2 q)v 3 v 3 + qv 3 v 3 T 1 F 2 V V

21 .. U q (gl n ) quantum group ψ V d φ H(Sd ) Hecke algebra of type A (Jimbo 86) The algebras U q (gl n ) and H(S d ) satisfy the double centralizer property: End φ(a) (V d ) = ψ(b), End ψ(b) (V d ) = φ(a), where A = H(S d ), B = U q (gl n )

22 q-schur algebra.. In other words, there is a surjection, for each d, from U q (gl n ) to the q-schur algebra S n,d = End H(Sd )(V d ), and hence we have U(gl n ) quantum group S n,d q-schur algebra V d tensor space H(S d ) Hecke algebra

23 .. for other types It is known that U q (gl n ) is a type A quantum group. There are (q-)schur duality for other types, e.g., O n orthogonal group S o n,d orthogonal Schur algebra V d B d Brauer algebra U q (so n ) quantum group S o n,d orthogonal q-schur algebra V d BMW d Birman-Murakami-Wenzl algebra In these pictures the types are associated to the classical/quantum groups

24 .. for other types If we associate the types to the Hecke algebra instead, we obtain another family of (q-)schur duality: S X n,d := End H X d (V d ) q-schur algebra of type X V d H X d type X Hecke algebra Question: Is there a bottom-top construction that recovers the quantum group from the Schur algebras?

25 .. The answer is Yes, the construction for type A is provided in 1990 by Beilinson-Lusztig-MacPherson (BLM) geometrically using partial flags and a so-called stabilization procedure. We can paraphrase it algebraically as below: K A n stabilization algebra of type A S A n,d := End H A d (V d ) q-schur algebra of type A V d H A d := H(S d) type A Hecke algebra Stabilization algebra K A n inverse limit lim S A n,d d N

26 .. K A n stabilization algebra of type A S A n,d := End H A d (V d ) q-schur algebra of type A V d H A d := H(S d) type A Hecke algebra Canonical bases for S A n,d (d N) lift compatibly to canonical basis of K A n K A n U(gl n ) A concrete realization of canonical basis of U(gl n )

27 .. A (for type X) produces from a family of q-schur algebra {S X n,d d N} an Q(v)-algebra K X n that enjoys similar properties K X n stabilization algebra of type X S X n,d := End H X d (V d ) q-schur algebra of type X V d H X d type X Hecke algebra

28 New quantum groups.. In type B/C, there are two types of s due to Bao-Kujawa-Li-Wang ( 14) associated to involutions (type ı, ȷ) of the Dynkin diagram of type A n n n n n 1 n n n n+3 2 type ȷ : n even type ı : n odd K BC n is NOT the modified Drinfeld quantum group of type B/C; it is a modified quantization of certain product gl sl n+1

29 .. Applications of new quantum groups Depending on parity of n, K BC n is isomorphic to involutive quantum groups U ı n or U ȷ n introduced by Bao-Wang ( 13). The canonical basis theory for U ı n, U ȷ n gives a new formulation of KL theory for Lie algebra of type B/C establishes for the first time KL theory for Lie superalgebra osp(2m + 1 2n) (i.e. super type B/C) reformulation of KL theory for Lie algebra of type D and for Li superalgebra osp(2m 2n) (i.e. super type D) by Bao ( 16) We expect similar applications for other types (e.g. affine classical)

30 BLM-type constructions.. Here s a summary on what are known Geometric Algebraic (method) (dimension counting on flags) (combinatorics) Affine A Ginzburg-Vasserot ( 93) Du-Fu ( 14) Lusztig ( 99) Finite B/C BKLW ( 14) Finite D Fan-Li ( 14) Affine C Fan-Lai-Li-Luo-Wang1 ( 16) FLLLW2 ( 16)

31 Coideal subalgebras.. It is worth mentioning that U ı n, U ȷ n are coideal subalgebras of U(gl n ), i.e., the comultiplication : U(gl n ) U(gl n ) U(gl n ) satisfies that (U ı n) U ı n U(gl n ) (U ȷ n) U ȷ n U(gl n ) A coideal subalgebra is different from a Hopf subalgebra B of U s.t. (B) B B

32 Outline...1 Schur-Weyl duality

33 Symmetric pairs.. A symmetric pair (g, g θ ) consists of a Lie algebra g and its fixed-point subalgebra g θ with respect to an involution θ : g g It plays an important role in the study of real reductive groups Classification of symmetric pairs of finite type = Classification of real simple Lie algebras Satake diagrams (of finite type)

34 .. The quantum symmetric pair(qsp) is introduced by Letzter ( 02) for finite type; and by Kolb ( 14) for symmetrizable Kac-Moody as a quantization of the symmetric pair (g, g θ ) The pair (U, B) is a QSP if U = U(g) and B = B(θ) is certain coideal subalgebra of U, i.e., the comultiplication : U U U satisfies that (B) B U

35 Examples.. The following are examples of QSP: (U q (ĝl n), twisted Yangian) from Yang-Baxter equation (U q (g), B) where g is of classical type, and B is constructed based on solutions of the reflection equation RKRK = KRKR (U q (ŝl 2), q-onsager algebra) from Ising model Quantum algebras from. In other words, quantum algebras arising from Schur-Weyl duality

36 .. QSP from Recall that stabilization take inf sum {S X n,d d N} ========= K X n ========= K X n The resulting quantum algebras are: type Finite A quantum algebra K X n U q (gl n ) Finite B/C K BC n U ı n, U ȷ n Affine A U q (ĝl n) Affine C KĈn (four variants) Proposition The pairs (U(gl n ), U ı n), (U(gl n ), U ȷ n) and all four variants of (U(ĝl n), KĈn) are quantum symmetric pairs.

37 .. Applications and future work A canonical basis theory for QSP of Satake diagrams finite type is developed recently by Bao-Wang ( 16). It may lead to similar application as in (U(gl n ), K X n) with further study? canonical basis theory for affine QSP? from {S X n,d } when X is of affine B/D, finite and affine exceptional type? BLM-type construction for other Schur-type dualities (e.g., for Brauer algebras, partition algebras)? BLM-type construction for Howe duality

38 .. Thank you for your attention