Higher Schur-Weyl duality for cyclotomic walled Brauer algebras

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1 Higher Schur-Weyl duality for cyclotomic walled Brauer algebras (joint with Hebing Rui) Shanghai Normal University December 7, 2015

2 Outline 1 Backgroud 2 Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras 3 Decomposition numbers of cyclotomic walled Brauer algebras

3 Schur-Weyl duality GL n : general linear group V r : tensor product of natural representation of GL n S r : symmetric group on r letters [Schur,1901]: ϕ : CS r End GLn (V r ) op ϕ is injective if n r. GL n V r S r

4 Schur-Weyl duality GL n : general linear group V r : tensor product of natural representation of GL n S r : symmetric group on r letters [Schur,1901]: ϕ : CS r End GLn (V r ) op ϕ is injective if n r. GL n V r S r

5 Schur-Weyl duality GL n : general linear group V r : tensor product of natural representation of GL n S r : symmetric group on r letters [Schur,1901]: ϕ : CS r End GLn (V r ) op ϕ is injective if n r. GL n V r S r

6 Brauer algebra so N : orthogonal algebra sp 2n : symplectic algebra V r : tensor product of natural representation of so N or sp 2n B r (δ): Brauer algebra with parameter δ [Brauer,1937] so N V r B r (N ) sp 2n V r B r ( 2n)

7 walled Brauer algebra V : dual representation of the natural representation of gl n B r,t (δ): walled Brauer algebra with parameter δ [Koike,1989] and [Turaev,1989] gl n V r (V ) t B r,t (n) φ : B r,t End U (gln )(V r W t ) op, and φ is injective if and only if n r + t If we allow t = 0, then B r,t should be replaced by S r

8 affinization of S r : degenerate affine Hecke algebra Hr aff : degenerate affine Hecke algebra [Arakawa-Suzuki,1998] gl n M (λ) V r H aff r M (λ): Verma module of gl n [Brundan-Stroppel,2012] gl m n K (λ) V r H aff r φ : H 2,r End U (glm n )(K (λ p,q ) V r ) op, and φ is injective if and only if n r K (λ): Kac module of general linear Lie superalgebra gl m n

9 affinization of S r : degenerate affine Hecke algebra Hr aff : degenerate affine Hecke algebra [Arakawa-Suzuki,1998] gl n M (λ) V r H aff r M (λ): Verma module of gl n [Brundan-Stroppel,2012] gl m n K (λ) V r H aff r φ : H 2,r End U (glm n )(K (λ p,q ) V r ) op, and φ is injective if and only if n r K (λ): Kac module of general linear Lie superalgebra gl m n

10 affinization of walled Brauer algebra B aff r,t(ω): (degenerate) affine walled Brauer algebra with parameters ω = (ω i ) i Z 0 [Rui-Su,2013] gl m n V r K (λ) (V ) t B aff r,t

11 structure and representations of cyclotomic walled Brauer algebra B k,r,t := B aff r,t/i, level k (cyclotomic) walled Brauer algebra I : generated by two cyclotomic polynomials of degree k [Rui-Su,2013]: freeness, cellular basis, classification of simple modules of B k,r,t [Rui-Su,2013]: gl m n V r K (λ p,q ) (V ) t B 2,r,t If r + t min{m, n}, then B 2,r,t = Endg (V r K (λ p,q ) (V ) t ) op, decomposition numbers of B 2,r,t

12 structure and representations of cyclotomic walled Brauer algebra B k,r,t := B aff r,t/i, level k (cyclotomic) walled Brauer algebra I : generated by two cyclotomic polynomials of degree k [Rui-Su,2013]: freeness, cellular basis, classification of simple modules of B k,r,t [Rui-Su,2013]: gl m n V r K (λ p,q ) (V ) t B 2,r,t If r + t min{m, n}, then B 2,r,t = Endg (V r K (λ p,q ) (V ) t ) op, decomposition numbers of B 2,r,t

13 structure and representations of cyclotomic walled Brauer algebra B k,r,t := B aff r,t/i, level k (cyclotomic) walled Brauer algebra I : generated by two cyclotomic polynomials of degree k [Rui-Su,2013]: freeness, cellular basis, classification of simple modules of B k,r,t [Rui-Su,2013]: gl m n V r K (λ p,q ) (V ) t B 2,r,t If r + t min{m, n}, then B 2,r,t = Endg (V r K (λ p,q ) (V ) t ) op, decomposition numbers of B 2,r,t

14 structure and representations of cyclotomic walled Brauer algebra B k,r,t := B aff r,t/i, level k (cyclotomic) walled Brauer algebra I : generated by two cyclotomic polynomials of degree k [Rui-Su,2013]: freeness, cellular basis, classification of simple modules of B k,r,t [Rui-Su,2013]: gl m n V r K (λ p,q ) (V ) t B 2,r,t If r + t min{m, n}, then B 2,r,t = Endg (V r K (λ p,q ) (V ) t ) op, decomposition numbers of B 2,r,t

15 [Sartori,2014]: gl n M p (δ) V r (V ) t B 2,r,t [Sartori-Stroppel,2014]: graded representations of B 2,r,t [Brundan-Comes-Nash-Reynolds,2014]: basis of level k walled Brauer B k,r,t

16 [Sartori,2014]: gl n M p (δ) V r (V ) t B 2,r,t [Sartori-Stroppel,2014]: graded representations of B 2,r,t [Brundan-Comes-Nash-Reynolds,2014]: basis of level k walled Brauer B k,r,t

17 [Sartori,2014]: gl n M p (δ) V r (V ) t B 2,r,t [Sartori-Stroppel,2014]: graded representations of B 2,r,t [Brundan-Comes-Nash-Reynolds,2014]: basis of level k walled Brauer B k,r,t

18 Aim 1. Schur-Weyl duality: gl n M p (λ) V r (V ) t B k,r,t 2. decomposition numbers of B k,r,t

19 Aim 1. Schur-Weyl duality: gl n M p (λ) V r (V ) t B k,r,t 2. decomposition numbers of B k,r,t

20 1 Backgroud 2 Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras 3 Decomposition numbers of cyclotomic walled Brauer algebras

21 definition of affine walled Brauer algebra Definition (Rui-Su,2013) Let R be a commutative ring containing 1 and ω a, ω a a N. The affine walled Brauer algebra B aff r,t is the associative R-algebra generated by e 1, x 1, x 1, s i, s j (1 i < r), (1 j < t), such that 1 s i s, s j s, e 1 satisfy defining relations for B r,t, 2 s i s and x 1 satisfy defining relations for H aff r, 3 s i s and x 1 satisfy defining relations for H aff t, 4 e 1 (x 1 + x 1 ) = (x 1 + x 1 )e 1 = 0, s 1 e 1 s 1 x 1 = x 1 s 1 e 1 s 1, s 1 e 1 s 1 x 1 = x 1 s 1 e 1 s 1, 5 s i x 1 = x 1 s i, s i x 1 = x 1 s i, x 1 (e 1 + x 1 ) = (e 1 + x 1 )x 1, 6 e 1 x k 1 e 1 = ω k e 1, e 1 x k 1 e 1 = ω k e 1, k Z 0.

22 cyclotomic (or level k) walled Brauer algebra B k,r,t u i, ū i R, i k I : two-sided ideal of B aff r,t generated by f (x 1 ) and g( x 1 ) f (x 1 ) = k i=1 (x 1 u i ) g( x 1 ) = k i=1 ( x 1 ū i ) e 1 f (x 1 ) = ( 1) k e 1 g( x 1 ) cyclotomic (or level k) walled Brauer algebra: B k,r,t := B aff r,t/i

23 admissible condition f (x 1 ) = x k 1 + k 1 i=0 a k ix i B k,r,t is admissible: ω l = (a 1 ω l a k ω l k ), l k Theorem (Rui-Su,2013) B k,r,t is free over R with rank k r+t (r + t)! if and only if B k,r,t is admissible.

24 Cellular basis of B k,r,t Theorem (Rui-Su,2013) C = {C (s,κc,c)(t,κ d,d) (s, κ c, c), (t, κ d, d) δ(f, λ), (f, λ) Λ k,r,t } is a weakly cellular basis of B k,r,t over C in the sense of J.Graham and G.Lehrer.

25 decomposition numbers of B k,r,t C (f, λ): right cell module w.r.t the cellular basis C φ f,λ : an invariant form on C (f, λ), D f,λ = C (f, λ)/radφ f,λ, is either zero or absolutely irreducible. [C (f, λ) : D l,µ ]: the multiplicity of D l,µ in C (f, λ), Such a multiplicity is called a decomposition number of B k,r,t

26 parabolic category O of general linear Lie algebra g := gl n = n h n + p = n + + l l = gl q1 gl q2 gl qk, k i=1 q i = n O p : parabolic category O w.r.t p Λ p : the set of p-dominant weights M p (λ): parabolic Verma module w.r.t λ Λ p

27 certain tensor product module in O p d = (d 1, d 2,, d k ) C k such that d i d j Z if and only if d i = d j. p i = i j =1 q j, 1 i k, p 0 = 0 c i = d i + p i q 1, for all 1 i k δ c = k i=1 c i(ɛ pi ɛ pi ɛ pi ) Λ p M c := M p (δ c ): irreducible (projective, injective, tilting) M r,t c = M c V r (V ) t O p

28 actions of B aff r,t on M r,t c U := U(gl n ): universal enveloping algebra of gl n Casimir element in U U: Ω = 1 i,j n e i,j e j,i π a,b : U 2 U (r+t+1) such that π a,b (x y) = 1 1 ath x 1 1 bth y 1 1 s i = π i+1,i (Ω) M r,t, s c j = π j,j +1 (Ω) Mc r,t x 1 = π 01 (Ω) M r,t, x c 1 = π 0 1(Ω) M r,t, c e 1 = π 1 1(Ω) M r,t c

29 Isomorphism theorem Theorem (1) There is an algebra homomorphism ϕ : B aff r,t End O (M r,t c ) op, where B aff r,t is the affine walled Brauer algebra with some special parameters ω 0 = n and ω i, i Z >0. (2) ϕ factors through B k,r,t. Moreover, B k,r,t is admissible. (3) If r + t min{q 1, q 2,, q k }, then ϕ is an isomorphism. Remark For any highest weight module M, there is an algebra homomorphism ϕ : B aff r,t End O (M V r (V ) t ) op.

30 computations of u i, ū i 1 B k,r,t := B aff r,t/ < f (x 1 ), g( x 1 ) > 2 f (x 1 ) = k i=1 (x 1 u i ), g( x 1 ) = k i=1 ( x 1 ū i ) 3 u i = c i + p i 1, ū i = c i + n p i, i k, Remark 1 Any parameters u i, ū i can be realized in this way, so the special parabolic Verma module M p (δ c ) is enough to study the representation of B k,r,t arising from Schur-Weyl duality.

31 computations of ω a s Lemma The generating function of parameters ω a s satisfies 1 + a=0 ω a u a+1 = k i=1 u + n u i u + ū i. Perelomov-Popov formulas: 1 + ( 1) k χ(tre k ) a=0 = n u+l i +1 (u n+1) a+1 i=1 u+l i. E: n n matrix with (i, j )-th entry e i,j, l i = e i,i i + 1 Remark Note that there exist some admissible cyclotomic walled Brauer algebra with parameters ω a s which do not satisfy above formula.

32 Remark Problem: ϕ : B k,r,t End O (Mc r,t ) op is always surjective? idea: U (g e ) V r (V ) t grb k,r,t U (g, e) V r (V ) t B k,r,t U (gl n ) M p (δ c ) V r (V ) t B k,r,t

33 graded algebra associated to B k,r,t degs i = deg s j = deg e 1 = 0, deg x 1 = deg x 1 = 1, B k,r,t (B k,r,t ) (1) (B k,r,t ) (0) (B k,r,t ) ( 1) = 0 (B k,r,t ) [i] =(B k,r,t ) (i) /(B k,r,t ) (i 1). gr(b k,r,t )= i 0(B k,r,t ) [i] : Z-graded algebra associated to B k,r,t

34 Schur-Weyl duality between U(g e ) and gr(b k,r,t ) q = (q 1, q 2,, q k ): a partition of n e = (i,j ) K e i,j g: a nilpotent matrix K = {(i, j ) 1 i, j n, row(i) = row(j ), col(i) = col(j ) 1} 1 Example: t = 2 5 if (q 1, q 2, q 3 ) = (4, 3, 2) e = e 2,5 + e 3,6 + e 4,7 + e 6,8 + e 7,9 g e : centralizer of e in g deg e i,j = col(j )-col(i)

35 Proposition There is a graded algebra homomorphism ϕ : gr(b k,r,t ) End U(ge )(V r (V ) t ) op such that (1) ϕ(e 1 ) = π 1, 1 (Ω), (2) ϕ( s j ) = π j,j +1 (Ω), (3) ϕ(s i ) = π i+1,i (Ω), (4) ϕ(x 1 ) = 1 r 1 e 1 t, (5) ϕ( x 1 ) = 1 r e 1 t 1. ϕ is surjective, and it is injective if r + t q k. if t = 0, this is proved in [Brundan-Kleshchev,2009]

36 Finite W -algebra U(g, e) m = i<0 g i χ : U(m) C, sending x to tr(xe) η(e i,j ) = e i,j + δ i,j (q 1 q col(j ) q col(j )+1 q k ) Finite W -algebra: U(g, e) = {u U(p) [x, η(u)] U(g)I χ for all x m}

37 Skryabin s equivalence C: the category of all g-modules on which x χ(x ) acts locally nilpotently, x m Q χ = U/UI χ : (U, U(g, e))-bimodule Q χ U(g,e)? : U(g, e)-mod C: equivalence of categories Wh(?) : C U(g, e)-mod M C, Wh(M ) = {v M xv = χ(x )v, x m} Wh(?) is the inverse of Q χ U(g,e)?

38 Skryabin s equivalence C: the category of all g-modules on which x χ(x ) acts locally nilpotently, x m Q χ = U/UI χ : (U, U(g, e))-bimodule Q χ U(g,e)? : U(g, e)-mod C: equivalence of categories Wh(?) : C U(g, e)-mod M C, Wh(M ) = {v M xv = χ(x )v, x m} Wh(?) is the inverse of Q χ U(g,e)?

39 Skryabin s equivalence C: the category of all g-modules on which x χ(x ) acts locally nilpotently, x m Q χ = U/UI χ : (U, U(g, e))-bimodule Q χ U(g,e)? : U(g, e)-mod C: equivalence of categories Wh(?) : C U(g, e)-mod M C, Wh(M ) = {v M xv = χ(x )v, x m} Wh(?) is the inverse of Q χ U(g,e)?

40 actions of B aff r,t on V r (V ) t? X : C C, X is finite dimensional g-module By Skryabin s equivalence,? X : U(g, e)-mod U(g, e)-mod M X = Wh((Q χ U(g,e) M ) X ), C d : 1-dimensional p-module, e i,j 1 d = δ i,j d col(i) 1 d e 1 : C d V V C d V V s 1 : C d V V C d V V s 1 : C d V V C d V V x 1 : C d V C d V x 1 : C d V C d V s i, s i End U(g,e) (C d V r (V ) t ) C d V r (V ) t = C d V r,t = C d V r,t = V r (V ) t

41 actions of B aff r,t on V r (V ) t? X : C C, X is finite dimensional g-module By Skryabin s equivalence,? X : U(g, e)-mod U(g, e)-mod M X = Wh((Q χ U(g,e) M ) X ), C d : 1-dimensional p-module, e i,j 1 d = δ i,j d col(i) 1 d e 1 : C d V V C d V V s 1 : C d V V C d V V s 1 : C d V V C d V V x 1 : C d V C d V x 1 : C d V C d V s i, s i End U(g,e) (C d V r (V ) t ) C d V r (V ) t = C d V r,t = C d V r,t = V r (V ) t

42 actions of B aff r,t on V r (V ) t? X : C C, X is finite dimensional g-module By Skryabin s equivalence,? X : U(g, e)-mod U(g, e)-mod M X = Wh((Q χ U(g,e) M ) X ), C d : 1-dimensional p-module, e i,j 1 d = δ i,j d col(i) 1 d e 1 : C d V V C d V V s 1 : C d V V C d V V s 1 : C d V V C d V V x 1 : C d V C d V x 1 : C d V C d V s i, s i End U(g,e) (C d V r (V ) t ) C d V r (V ) t = C d V r,t = C d V r,t = V r (V ) t

43 actions of B aff r,t on V r (V ) t? X : C C, X is finite dimensional g-module By Skryabin s equivalence,? X : U(g, e)-mod U(g, e)-mod M X = Wh((Q χ U(g,e) M ) X ), C d : 1-dimensional p-module, e i,j 1 d = δ i,j d col(i) 1 d e 1 : C d V V C d V V s 1 : C d V V C d V V s 1 : C d V V C d V V x 1 : C d V C d V x 1 : C d V C d V s i, s i End U(g,e) (C d V r (V ) t ) C d V r (V ) t = C d V r,t = C d V r,t = V r (V ) t

44 actions of B aff r,t on V r (V ) t? X : C C, X is finite dimensional g-module By Skryabin s equivalence,? X : U(g, e)-mod U(g, e)-mod M X = Wh((Q χ U(g,e) M ) X ), C d : 1-dimensional p-module, e i,j 1 d = δ i,j d col(i) 1 d e 1 : C d V V C d V V s 1 : C d V V C d V V s 1 : C d V V C d V V x 1 : C d V C d V x 1 : C d V C d V s i, s i End U(g,e) (C d V r (V ) t ) C d V r (V ) t = C d V r,t = C d V r,t = V r (V ) t

45 Schur-Weyl duality between U(g, e) and B k,r,t gr(u(g, e)) = U(g e ) Theorem 1 There is an algebra homomorphism Φ : B k,r,t End U(g,e) (V r (V ) t ) op. 2 Φ is always surjective. It is injective if r + t q k.

46 Whittaker functor O d : Serre subcategory of O p generated by L(µ), µ Λ d R d (λ):category of rational representations of U(g, e) Whittaker functor ([Brundan-Kleshchev,2009]): V : O d R d (λ) as (U(g, e), B aff r,t)-bimodules: V(Mc r,t ) = V r (V ) t U(g, e) V r (V ) t B aff r,t U M r,t c B aff r,t

47 Schur-Weyl duality: surjective Φ : B k,r,t End U(g,e) (V r (V ) t ) op is always surjective ϕ : B k,r,t End O (Mc r,t ) op is always surjective Theorem 1 There is a B aff r,t-isomorphism k r,t : End O (M r,t c ) End U(g,e) (V r (V ) t ) 2 Φ = k r,t ϕ. So, ϕ is always surjective, and it is injective if r + t q k.

48 1 Backgroud 2 Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras 3 Decomposition numbers of cyclotomic walled Brauer algebras

49 construction of highest weight vectors of M r,t c idea: cellular basis of B k,r,t + Schur-Weyl duality construction of highest weight vectors cell module C (f, µ, ν): {e f x µ x ν w µ w ν y µ y ν d(t)dx κ d ( mod B f +1 k,r,t ) (t, d, κ d) δ(f, µ, ν )} v t,d,κd = v^λ µ,ν e f w µ,ν y µ y (ν o ) d(t)dx k d

50 classification of highest weight vectors Theorem Suppose r + t min{q 1, q 2,, q k }. (1) There is a bijection between Λ k,r,t and the set of p-dominant weights λ such that Mc r,t contains at least a highest weight vector with highest weight λ. Moreover, the map sends (f, µ, ν) to ^λ µ,ν. (2) The C-space V^λ µ,ν of g-highest weight vectors of Mc r,t highest weight ^λ µ,ν has a basis S = {v t,d,kd t T s (µ ) T s ((ν o ) ), d D f r,t, k d N f }. with

51 Decomposition numbers of B k,r,t Suppose r + t min{q 1, q 2,, q k }. Theorem 1 For any (f, µ, ν) Λ k,r,t, as right B k,r,t -modules Hom O (M p (^λ µ,ν ), M r,t c ) = C (f, µ, (ν o ) ). 2 For any (f, µ, ν) Λ k,r,t, (l, α, β) Λ k,r,t, (T (^λ α,β ) : M p (^λ µ,ν )) = [C (f, µ, (ν o ) ) : D (l,α,(β o ) ) ]. T (^λ α,β ):indecomposable tilting module w.r.t ^λ α,β

52 idea of the proof f := Hom O (Mc r,t,?), g = Mc r,t Bk,r,t? N p (^λ µ,ν ): dual parabolic Verma module f(n p (^λ µ,ν )) = C (f, µ, (ν o ) ) f(t p ((^λ α,β )) = P(f, α, (β o ) ) isomorphism theorem gf(t (^λ α,β )) = T (^λ α,β ) Hom O (T (^λ α,β ), N p (^λ µ,ν )) = Hom O (f(t (^λ α,β )), f(n p (^λ µ,ν ))) =Hom Bk,r,t (P(f, α, (β o ) ), C (l, µ, (ν o ) )).

53 idea of the proof f := Hom O (Mc r,t,?), g = Mc r,t Bk,r,t? N p (^λ µ,ν ): dual parabolic Verma module f(n p (^λ µ,ν )) = C (f, µ, (ν o ) ) f(t p ((^λ α,β )) = P(f, α, (β o ) ) isomorphism theorem gf(t (^λ α,β )) = T (^λ α,β ) Hom O (T (^λ α,β ), N p (^λ µ,ν )) = Hom O (f(t (^λ α,β )), f(n p (^λ µ,ν ))) =Hom Bk,r,t (P(f, α, (β o ) ), C (l, µ, (ν o ) )).

54 Reference Hebing Rui,, Mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer algebras", 2015, preprint.

55 Thanks!

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that