TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 1, January 2008, Pages 189–213 S 0002-9947(07)04179-7 Article electronically published on August 16, 2007
BRAUER ALGEBRAS, SYMPLECTIC SCHUR ALGEBRAS AND SCHUR-WEYL DUALITY
RICHARD DIPPER, STEPHEN DOTY, AND JUN HU
Abstract. In this paper we prove the Schur-Weyl duality between the sym- plectic group and the Brauer algebra over an arbitrary infinite field K.We show that the natural homomorphism from the Brauer algebra Bn(−2m)to the endomorphism algebra of the tensor space (K2m)⊗n as a module over the symplectic similitude group GSp2m(K) (or equivalently, as a module over the symplectic group Sp2m(K)) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for GSp2m(K)totheen- 2m ⊗n domorphism algebra of (K ) as a module over Bn(−2m), is derived as an easy consequence of S. Oehms’s results [S. Oehms, J. Algebra (1) 244 (2001), 19–44].
1. Introduction Let K be an infinite field. Let m, n ∈ N.LetU be an m-dimensional K- vector space. The natural left action of the general linear group GL(U)onU ⊗n commutes with the right permutation action of the symmetric group Sn.Letϕ, ψ be the natural representations op ⊗n ⊗n ϕ :(KSn) → EndK U ,ψ: KGL(U) → EndK U , respectively. The well-known Schur-Weyl duality (see [Sc], [W], [CC], [CL]) says that ⊗n ≥ (a) ϕ KSn =EndKGL(U) U ,andif m n,thenϕ is injective, and hence ⊗n an isomorphism onto End KGL (U) U , ⊗n (b) ψ KGL(U) =EndKSn U , op (c) if char K = 0, then there is an irreducible (KGL(U), (KSn) )-bimodule decomposition ⊗n λ U = ∆λ ⊗ S ,
λ=(λ1,λ2,··· )n (λ)≤m λ where ∆λ (resp., S ) denotes the irreducible KGL(U)-module (resp., irre- ducible KSn-module) associated to λ,and(λ) denotes the largest integer i such that λi =0.
Received by the editors April 8, 2005 and, in revised form, September 27, 2005. 2000 Mathematics Subject Classification. Primary 16G99. The first author received support from DOD grant MDA904-03-1-00. The second author gratefully acknowledges support from DFG Project No. DI 531/5-2. The third author was supported by the Alexander von Humboldt Foundation and the National Natural Science Foundation of China (Project 10401005) and the Program NCET.
c 2007 American Mathematical Society 189
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 190 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU
Let τ be the automorphism of KSn which is defined on generators by τ(si)=−si for each 1 ≤ i ≤ n − 1. Then (by using this automorphism) it is easy to see that the same Schur-Weyl duality still holds if one replaces the right permutation action of Sn by the right sign permutation action, i.e., ⊗···⊗ − ⊗···⊗ ⊗ ⊗ ⊗ ⊗···⊗ (vi1 vin )sj := (vi1 vij−1 vij+1 vij vij+2 vin ), ≤ ≤ − ··· ∈ for any 1 j n 1andanyvi1 , ,vin U. In the case of K = C, there are also Schur-Weyl dualities for other classical groups—symplectic groups and orthogonal groups. In this paper, we shall consider only the symplectic case.1 Recall that symplectic groups are defined by certain bilinear forms ( , ) on vector spaces. Let V be a 2m-dimensional K-vector space equipped with a nondegenerate skew-symmetric bilinear form ( , ). Then (see [Gri], [Dt, Section 4]) the symplectic similitude group (resp., the symplectic group) rela- tive to ( , )is GSp(V ):= g ∈ GL(V ) ∃ 0 = d ∈ K, such that (gv, gw)=d(v, w), ∀ v, w ∈ V resp., Sp(V ):= g ∈ GL(V ) (gv, gw)=(v, w), ∀ v, w ∈ V . By restriction from GL(V ), we get natural left actions of GSp(V )andSp(V )on V ⊗n. Again we denote by ψ the natural K-algebra homomorphism ψ : KGSp(V ) → End V ⊗n , K ⊗n ψ : KSp(V ) → EndK V . Note √ that if√ 0 = d ∈ K is such that (gv, gw)=d(v, w) for any v, w ∈ V ,then −1 −1 ∈ ( d g)v,√( d g)w =(v, w) for any v, w V . Therefore, if K is large enough such that d ∈ K for any d ∈ K,theng ∈ GSp(V )impliesthat(a idV )g ∈ Sp(V ) for some 0 = a ∈ K.Inthatcase, ψ(g)=ψ (a−1 id )(a id )g = ψ a−1 id ψ (a id )g V V V V −n −n = a idV ⊗n ψ (a idV )g = a ψ (a idV )g . It follows that (1.1) ψ KSp(V ) = ψ KGSp(V ) provided K is large enough. In the setting of Schur-Weyl duality for the symplectic group, the symmetric group Sn should be replaced by Brauer algebras (introduced in [B]). Recall that the Brauer algebra Bn(x) over a Noetherian integral domain R (with parameter x ∈ R) is a unital R-algebra with generators s1, ··· ,sn−1,e1, ··· ,en−1 and relations (see [E]): 2 2 ∀ ≤ ≤ − si =1,ei = xei,eisi = ei = siei, 1 i n 1, sisj = sjsi,siej = ejsi,eiej = ejei, ∀ 1 ≤ i Note that Bn(x) was originally defined as the linear space with basis the set of all Brauer n-diagrams, graphs on 2n vertices and n edges with the property that 1In this paper, we will use the results by Oehms and also by Donkin on symplectic Schur algebras. To deal with the orthogonal case, one needs analogous results for orthogonal Schur algebras, which are not presently available. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 191 every vertex is incident to precisely one edge. One usually thinks of the vertices as arranged in two rows of n each, the top and bottom rows. Label the vertices in each row of an n-diagram by the indices 1, 2, ··· ,n from left to right. Then si corresponds to the n-diagram with edges connecting vertices i (resp., i +1)onthe top row with i +1(resp., i) on the bottom row, and all other edges are vertical, connecting vertex k on the top and bottom rows for all k = i, i +1. ei corresponds to the n-diagram with horizontal edges connecting vertices i, i + 1 on the top and bottom rows, and all other edges are vertical, connecting vertex k on the top and bottom rows for all k = i, i + 1. The multiplication is given by the linear extension of a product defined on diagrams. For more details, see [B], [GW]. There are right actions of Brauer algebras (with certain parameters) on tensor space. The definition of the actions depends on the choice of an orthogonal basis with respect to the defining bilinear form. Let δij denote the value of the usual Kronecker delta. For any 1 ≤ i ≤ 2m,seti := 2m +1− i. We fix an ordered basis v1,v2, ··· ,v2m of V such that (v ,v )=0=(v ,v ), (v ,v )=δ = −(v ,v ), ∀ 1 ≤ i, j ≤ m. i j i j i j ij j i For any i, j ∈ 1, 2, ··· , 2m ,let ⎧ ⎨⎪1ifj = i and i License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 192 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU The aim of this work is to remove the restriction on K in part 1) and part 2) of the above theorem. We shall see that the following holds for any infinite field K. ⊗n Proposition 1.3. For any infinite field K, ψ KGSp(V ) =EndBn(−2m) V . In fact, this is an easy consequence of [Oe, (6.1), (6.2), (6.3)] and [Dt, (3.2(b))]. The proof is given in Section 2. The main result of this paper is Theorem 1.4. Let K be an arbitrary infinite field. Then ⊗n ⊗n ϕ Bn(−2m) =EndKGSp(V ) V =EndKSp(V ) V , and if m ≥ n,thenϕ is also injective, and hence an isomorphism onto ⊗n EndKGSp(V ) V . Remark 1.5. 1) Note that when m 2. The algebra s AR(m) In this section, we shall show how Proposition 1.3 follows from results of [Oe, (6.1), (6.2), (6.3)] and [Dt, (3.2(b))]. Z s We shall first introduce (following [Oe, Section 6]) a -graded R-algebra AR(m) for any Noetherian integral domain R. Over an algebraically closed field, this algebra is isomorphic to the coordinate algebra of the symplectic monoid, and the dual of its n-th homogeneous summand is isomorphic to the symplectic Schur algebra introduced by S. Donkin ([Do2]). 2 Let R be a Noetherian integral domain. Let xi,j , 1 ≤ i, j ≤ 2m be 4m commut- ing indeterminates over R.LetAR(2m) be the free commutative R-algebra (i.e., polynomial algebra) in these xi,j , 1 ≤ i, j ≤ 2m.LetIR be the ideal of AR(2m) generated by elements of the form ⎧ 2m ⎪ x x , 1 ≤ i = j ≤ 2m; ⎨k=1 k i,k j,k 2m (2.1) x x , 1 ≤ i = j ≤ 2m; ⎪k=1 k k,i k ,j ⎩ 2m − ≤ ≤ k=1 k(xi,kxi ,k xk,jxk ,j ), 1 i, j m. s The R-algebra AR(2m)/IR will be denoted by AR(m). Write ci,j for the canon- s ≤ ≤ s ical image xi,j + IR of xi,j in AR(m)(1 i, j 2m). Then in AR(m)wehavethe relations ⎧ 2m ⎪ c c =0, 1 ≤ i = j ≤ 2m; ⎨k=1 k i,k j,k 2m (2.2) c c =0, 1 ≤ i = j ≤ 2m; ⎪k=1 k k,i k ,j ⎩ 2m − ≤ ≤ k=1 k(ci,kci ,k ck,jck ,j )=0, 1 i, j m. Note that AR(2m) is a graded algebra, AR(2m)= n≥0 AR(2m, n), where the AR(2m, n) is the subspace spanned by the monomials of the form xi,j for (i,j) ∈ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 193 I2(2m, n), where I(2m, n):= i =(i1, ··· ,in) 1 ≤ ij ≤ 2m, ∀ j , 2 I (2m, n)=I(2m, n) × I(2m, n),xi,j := xi ,j ···xi ,j . 1 1 nn s s s Since IR is a homogeneous ideal, AR(m)isgradedtooandAR(m)= n≥0 AR(m, n), s where AR(m, n) is the subspace spanned by the monomials of the form ci,j for (i,j) ∈ I2(2m, n), where ··· ci,j := ci1,j1 cin,jn . By convention, throughout this paper, we identify the symmetric group Sn with the set of maps acting on their arguments on the right. In other words, if σ ∈ Sn and a ∈{1,...,n} we write (a)σ for the value of a under σ. This convention carries the consequence that, when considering the composition of two symmetric group elements, the leftmost map is the first to act on its argument. For example, we have (1, 2, 3)(2, 3) = (1, 3) in the usual cycle notation. Note that the symmetric group Sn acts on the right on the set I(2m, n)bythe rule2 iσ := (i(1)σ−1 , ··· ,i(n)σ−1 ),σ∈ Sn. s ∼ It is clear (see [Dt]) that AR(m, n) = AR(2m, n)/IR(n), where IR(1) = 0, and for n ≥ 2, IR(n)istheR-submodule of AR(2m, n) generated by elements of the form ⎧ 2m ⎨⎪ x ··· ··· , k=1 k (i1, ,in),(k,k ,k3, ,kn) 2m (2.3) x ··· ··· , ⎪k=1 k (k,k ,i3, ,in),(j1, ,jn) ⎩ 2m − k=1 k(x(i,i ,i3,··· ,in),(k,k ,j3,··· ,jn) x(k,k ,i3,··· ,in),(j,j ,j3,··· ,jn)), ≤ ≤ ∈ where 1 i, j m, i,j I(2m, n) such that i1 = i2,j1 = j2. Furthermore, if one defines ∆(xi,j )= xi,k ⊗ xk,j ,ε(xi,j )=δi,j , ∀ i,j ∈ I(2m, n), ∀ n, k∈I(2m,n) then the algebra AR(2m) becomes a graded bialgebra, and each AR(2m, n)isa subcoalgebra of AR(2m). Its linear dual SR(2m, n):=HomR(AR(2m, n),R)isthe s s so-called Schur algebra over R (see [Gr]). Let SR(m, n):=HomR(AR(m, n),R). s By [Oe, Section 6], AR(m, n) is in fact a quotient coalgebra of AR(2m, n); hence s SR(m, n) is a subalgebra of SR(2m, n). We define (i,j) ∼ (u,v)ifthereexistssomeσ ∈ Sn with iσ = u,jσ = v. 2 2 Let I (2m, n)/∼ be the set of orbits for the action of Sn on I (2m, n). For each 2 (i,j) ∈ I (2m, n)/∼, we define ξi,j ∈ SR(2m, n)by ∼ 1, if (i,j) (u,v), 2 ξi,j (xu,v)= ∀ (u,v) ∈ I (2m, n)/∼. 0, otherwise, 2 The set ξi,j (i,j) ∈ I (2m, n)/∼ forms an R-basis of SR(2m, n). The natural ⊗n action of SR(2m, n)onV is given as follows: ξ : V ⊗n → V ⊗n i,j ⊗···⊗ → ∀ ··· ∈ va := va1 van vb, a := (a1, ,an) I(2m, n). b∈I(2m,n), (a,b)∼(i,j) 2This action is the so-called right place permutation action. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 194 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU ∈ Let ξ = (i,j)∈I2(2m,n)/∼ ai,j ξi,j SR(2m, n). By (2.3), it is easy to see that ξ ∈ Ss (m, n) if and only if R ⎧ 2m ⎨⎪ a ··· ··· =0, k=1 k (i1, ,in),(k,k ,k3, ,kn) 2m (2.4) a ··· ··· =0, ⎪k=1 k (k,k ,i3, ,in),(j1, ,jn) ⎩ 2m − k=1 k(a(i,i ,i3,··· ,in),(k,k ,j3,··· ,jn) a(k,k ,i3,··· ,in),(j,j ,j3,··· ,jn))=0, ≤ ≤ ∈ where 1 i, j m, i,j I(2m, n) such that i1 = i2,j1 = j2. Let R = K be an infinite field. Recall the ordered basis v1,v2, ··· ,v2m of V .Let(, ) be the unique (nondegenerate) skew-symmetric bilinear form on V such that (vi,vj)=0=(vi ,vj ), (vi,vj )=δij = −(vj ,vi), ∀ 1 ≤ i, j ≤ m. This form is given (relative to the above ordered basis) by the block matrix 0 J J := m , −Jm 0 where Jm is the unique anti-diagonal m × m permutation matrix. With respect to the above ordered basis of V , the group GSp(V ) may be identified with the group GSp (K)givenby 2m T GSp2m(K):= A ∈ GL2m(K) ∃ 0 = d(A) ∈ K, such that A JA = d(A)J . Let M2m(K) denote the affine algebraic monoid of n × n matrices over K.With respect to the above basis of V , the symplectic monoid SpM(V ), which by definition consists of the linear endomorphisms of V preserving the bilinear form up to any scalar (see [Dt, Section 4.2]), may be identified with T SpM2m(K):= A ∈ M2m(K) ∃ d(A) ∈ K, such that A JA = d(A)J . Let K be the algebraic closure of K. The coordinate algebra K[M2m(K)] is ⊗ isomorphic to AK (2m):=AK (2m) K. The coordinate algebra of GL2m(K)is −1 isomorphic to K[det (xi,j)2m×2m; xi,j ]1≤i,j≤2m. The embedding GSp2m(K) → GL2m(K) induces a surjective map K[GL2m(K)] K[GSp2m(K)]. Denote by Asy(m)(resp.,Asy(m, n)) the image of A (2m)(resp.,ofA (2m, n)) under this K K K K map. Then, by [Do2], (1) Asy(2m) is isomorphic to the coordinate algebra of SpM (K), K 2m (2) Asy(2m)= Asy(m, n), and the dimension of Asy(m, n) is indepen- K 0≤n∈Z K K dent of the field K, (3) the linear dual of Asy(m, n), say, Ssy(m, n) is a generalized Schur algebra K K in the sense of [Do1]. The algebra Ssy(m, n) is called by S. Donkin the symplectic Schur algebra. K sy sy We define AK (m)(resp.,AK (m, n)) to be the image of AK (2m)(resp.,of AK (2m, n)) under the surjective map K[GL2m(K)] K[GSp2m(K)]. It is clear that Asy(m) ⊗ K = Asy(m),Asy(m, n) ⊗ K = Asy(m, n), K K K K sy sy and hence AK (2m)= 0≤n∈Z AK (m, n). On the other hand, by definition of SpM2m(K), it is easy to check that the defin- ing relations (2.1) vanish on every matrix in SpM2m(K). It follows that there is an s sy epimorphism of graded bialgebras from AK (m)ontoAK (m). Note that for each License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 195 ≤ ∈ Z s sy 0 n , the dimensions of both AK (m, n) (see [Oe, (6.1)]) and AK (m, n)are s ∼ sy independent of the field K. By [Dt, (9.5)], AC(m, n) = AC (m, n). So the two coal- s ∼ sy gebras always have the same dimension. It follows that AK (m, n) = AK (m, n)and s ∼ sy s ∼ sy AK (m) = AK (m). In particular, we have that SK (m, n) = SK (m, n). Therefore we have Theorem 2.5 ([Oe, (6.2)]3). For any infinite field K, there is an isomorphism of s sy s ∼ sy graded bialgebras from AK (m) onto AK (m).Inparticular,AK (m, n) = AK (m, n) ∼ sy and Ss (m, n) = S (m, n) for each n ∈ N. K K Z ⊗n ⊗n Z As a -submodule of EndZ VZ , the algebra EndBn(−2m)Z VZ is a free - module of finite rank. Oehms proved in [Oe, (6.3)] that the symplectic Schur s ⊗n algebra S (m, n) is isomorphic to the centralizer algebra EndBn(−2m) V over any Noetherian integral domain. The following two results follow directly from the construction of his isomorphism. Theorem 2.6 ([Oe, (6.3)]) . For any field K, under the natural homomorphism → ⊗n s SK (2m, n) End V , the subalgebra SK (m, n) is mapped isomorphically onto ⊗n the subalgebra EndBn(−2m) V . Corollary 2.7 ([Oe, (6.3)]). For any field K, the map which sends f ⊗ a to af naturally extends to a K-algebra isomorphism ⊗n ⊗ ∼ ⊗n EndBn(−2m)Z VZ Z K = EndBn(−2m) V . Now we can prove Proposition 1.3. By Theorem 2.6 and the canonical iso- sy ∼ s morphism SK (m, n) = SK (m, n) from Theorem 2.5, we know that the natural sy ⊗n sy homomorphism from SK (m, n)toEndV maps SK (m, n) isomorphically onto ⊗n EndBn(−2m) V . Now the second author showed in [Dt, (3.2(b))] that the im- sy ages of KGSp (V )andofSK (m, n) (which is denoted by Sd(G) in [Dt, (3.2(b))]) in End V ⊗n are the same when K is infinite. It follows that for any infinite field K, ⊗n ψ(KGSp(V )) = EndBn(−2m) V . Note that this is also equivalent to the fact that the natural evaluation map → sy ∼ s (2.8) KGSp(V ) SK (m, n) = SK (m, n) is surjective. This completes the proof of Proposition 1.3. ⊗n 3. The action of Bn(−2m) on V for m ≥ n In this section, we shall give the proof of Theorem 1.4 in the case where m ≥ n. Let R be a Noetherian integral domain with q ∈ R a fixed invertible element. It is well known that the Hecke algebra HR,q(Sn) associated with the symmetric group Sn, and hence the group algebra of the symmetric group Sn itself, are cellular algebras. An important cellular basis of HR,q(Sn) is the Murphy basis, introduced in [Mu]. Another cellular basis is the Kazhdan-Lusztig basis [KL]. The latter one was extended by Graham and Lehrer to a cellular basis of the Brauer algebra. Xi extended this in [Xi] to the Birman-Murakami-Wenzl algebra, a quantization of the Brauer algebra; this algebra is also cellular. It is known that ([GL], [Xi], [E]) 3Note that though Oehms assumed in [Oe, (6.2)] that K is an algebraically closed field, the validity over an arbitrary infinite field is an immediate consequence (as shown in our previous discussion). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 196 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU any cellular basis of the Hecke algebra HR,q(Sk)(k ∈ N) can be extended to a cellular basis of the Birman-Murakami-Wenzl algebra. We shall follow Enyang’s formulation in [E], which describes the basis explicitly in terms of the generators. We will use the Murphy basis of RSk (k ∈ N), extended to a cellular basis of − Bn( 2m). We now describe this basis. For a composition λ =(λ1, ··· ,λs)ofk (i.e., λi ∈ Z≥0, λi = k), let Sλ = S{1,··· ,λ } × S{λ +1,··· ,λ +λ } ×··· 1 1 1 2 be the corresponding Young subgroup of S ,andsetx = w ∈ RS .The k λ w∈Sλ k Young diagram associated with λ consists of an array of nodes in the plane with λi many nodes in row i.Aλ-tableau t is such a diagram in which the nodes are replaced by the numbers 1, ··· ,k,insomeorder.Theinitial λ-tableau tλ is the one obtained by filling in the numbers 1, ··· ,k in order along successive rows. For example, 123 45 is the initial (3, 2)-tableau. The symmetric group Sk acts naturally on the set of λ-tableaux (on the right), and for any λ-tableau t we define d(t) to be the unique λ element of Sk with t d(t)=t.Aλ-tableau t is called row standard if the numbers increase along rows. If λ1 ≥ ··· ≥ λs, i.e., λ is a partition of k,thent is called column standard if the numbers increase down columns, and standard if it is both row and column standard. The set Dλ = d(t) t is a row standard λ-tableau is a set of right coset representatives of Sλ in Sk; its elements are known as distinguished coset representatives. For any standard λ-tableaux s, t, we define −1 mst = d(s) xλd(t). Murphy [Mu] showed Theorem 3.1 ([Mu]). mst λ k, s, t are standard λ-tableaux is a cellular basis of RSk for any Noetherian integral domain R. To describe Enyang’s cellular basis of the Brauer algebra Bn(x), we need some f more notation. First we fix certain bipartitions of n,namelyν = νf := ((2 ), − f ··· − (n 2f)), where (2 ):=(2, 2, , 2)and(n 2f) are considered as partitions of f copies 2f and n−2f respectively, and 0 ≤ f ≤ [n/2]. Here [n/2] is the largest non-negative integer not greater than n/2. In general, a bipartition of n is a pair (λ(1),λ(2))of partitions of numbers n1 and n2 with n1 + n2 = n. The notions of Young diagram, bitableaux, etc., carry over easily. Let tν be the standard ν-bitableau in which the numbers 1, 2, ··· ,n appear in order along successive rows of the first component tableau, and then in order along successive rows of the second component tableau. We define (t(1), t(2))=tν d is row standard and the first column of t(1) is D := d ∈ S . ν n an increasing sequence when read from top to bottom For each partition λ of n − 2f,wedenotebyStd(λ) the set of all the standard λ- tableaux with entries in {2f +1, ··· ,n}. The initial tableau tλ in this case has the numbers 2f +1, ··· ,n in order along successive rows. Again, for each t ∈ Std(λ), λ let d(t) be the unique element in S{2f+1,··· ,n} ⊆ Sn with t d(t)=t. For each integer f with 0 ≤ f ≤ [n/2], we denote by B(f) the two-sided ideal (f) of Bn(−2m) generated by e1e3 ···e2f−1.NotethatB is spanned by all Brauer License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 197 diagrams with at least 2f horizontal edges (f edgesineachofthetopandthe bottom rows in the diagram). Let f be an integer with 0 ≤ f ≤ [n/2]. Let σ ∈ S{2f+1,··· ,n} and d1,d2 ∈Dν , f − −1 ··· where again ν is the bipartition ((2 ), (n 2f)) of n.Thend1 e1e3 e2f−1σd2 corresponds to the Brauer diagram where the top horizontal edges connect (2i−1)d1 and (2i)d1, the bottom horizontal edges connect (2i − 1)d2 and (2i)d2,fori = ··· −1 1, 2, ,f, and the vertical edges are determined by d1 σd2. By [Xi, (3.5)], every Brauer diagram d canbewritteninthisway. Theorem 3.2 ([E]). Let R be a Noetherian integral domain with x ∈ R.Let B (x) be the Brauer algebra with parameter x over R. Then the set n R ≤ ≤ − ∈ −1 ··· 0 f [n/2], λ n 2f, s, t Std(λ), d1 e1e3 e2f−1mstd2 f d1,d2 ∈Dν ,whereν := ((2 ), (n − 2f)) is a cellular basis of the Brauer algebra Bn(x)R. As a consequence, by combining Theorems 3.1 and 3.2, we get that Corollary 3.3. With the above notation, the set −1 0 ≤ f ≤ [n/2], σ ∈ S{2f+1,··· ,n}, d1,d2 ∈Dν , d e e ···e − σd . 1 1 3 2f 1 2 where ν := ((2f ), (n − 2f)) is a basis of the Brauer algebra Bn(x)R, which coincides with the natural basis given by Brauer n-diagrams. We now specialize R to be a field K, assume m ≥ n, V = Km and consider the special Brauer algebra Bn(−2m)=Bn(−2m · 1K )K . As pointed out in Section 1, this algebra acts on the tensor space V ⊗n, centralizing the action of the symplectic similitude group GSp(V ) and hence that of the symplectic group Sp(V ) as well. The proof of the next result will be given at the end of the section, after a series of preparatory lemmas. Theorem 3.4. Let K be a field. If m ≥ n, then the natural homomorphism − → ⊗n ϕ : Bn( 2m) EndK V is injective; if furthermore K is infinite, then it is in ⊗n fact an isomorphism onto EndKSp(V ) V . Suppose that m ≥ n. Our first goal here is to show that the action of Bn(−2m) ⊗n ⊗n on V is faithful; that is, the annihilator annB (−2m)(V ) is (0). Note that n ⊗n annBn(−2m)(V )= annBn(−2m)(v). v∈V ⊗n ∈ ⊗n Thus it is enough to calculate annBn(−2m)(v) for some set of chosen vectors v V such that the intersection of annihilators is (0). We write ∈ − ann(v)=annBn(−2m)(v):= x Bn( 2m) vx =0 . Recall that (v1, ··· ,v2m) denotes an ordered basis of V ,and I(2m, n) denotes ··· ∈ ··· ··· the set of multi-indices i := (i1, ,in)withij 1, , 2m for j =1 , ,n.We ⊗···⊗ ··· ∈ ∈ write vi = vi1 vin for i := (i1, ,in) I(2m, n). Thus vi i I(2m, n) ⊗n is a K-basis of V . Consider the action of the symmetric group Sn on I(2m, n) given by iπ =(i(1)π−1 , ··· ,i(n)π−1 )fori := (i1, ··· ,in) ∈ I(2m, n)andπ ∈ Sn. (π) Thus, in particular, by definition, viπ =(−1) viπ.Fori ∈ I(2m, n), an ordered ≤ ≤ pair (s, t)(1 s License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 198 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU (s, t)and(u, v) are called disjoint if s, t ∩ u, v = ∅. We define the symplectic length s(vi) to be the maximal number of disjoint symplectic pairs (s, t)ini.For σ, π ∈ Sn and 1 ≤ j ≤ n − 1, it is easy to see that viσejπ is zero or a linear combination of tensors vj with s(vi)=s(vj). Moreover, for f>s(vi)wehave (f) (π) B ⊆ ann(vi). Note that π → (−1) π for π ∈ Sn defines an automorphism τ of the group algebra KSn, and that our action of Sn on tensor space is precisely the standard place-permutation action (see Section 2) twisted by this automorphism. ⊗n In particular, this shows that KSn acts faithfully on V for m ≥ n.Moreover, −1 for π ∈ Sn and i ∈ I(2m, n), ann(viπ) = ann(viπ)=π ann(vi). Now suppose again that m ≥ n. We shall prove by induction on f that B(f) ⊇ ⊗n (f) annBn(−2m) V for all f.SinceB =0forf>[n/2], this shows the main ⊗n result of this section, that is, Bn(−2m) acts faithfully on V if m ≥ n.Thestart of the induction is the following. ⊗n ⊆ (1) Lemma 3.5. annBn(−2m) V B . Proof. Since m ≥ n, the tensor v := v1 ⊗ v2 ⊗···⊗vn is defined. Then vπ = (π) (1) (−1) v(1)π−1 ⊗···⊗v(n)π−1 for π ∈ Sn.NowB is contained in the annihilator of vπ, hence is contained in the intersection of all annihilators of vπ,asπ ranges (1) over Sn. Hence B annihilates the subspace S spanned by the vπ,whereπ runs through Sn. Then the subspace S becomes a Bn(−2m)-submodule of tensor space (since B(1) acts as zero). On the other hand, since S, as a module for the symmetric group part, which − (1) is isomorphic with Bn( 2m) modulo the ideal B , is faithful, it follows that the (1) ⊗n ⊆ (1) annihilator of S must be in B . Hence annBn(−2m) V B . ⊗n (f) Suppose that we have already shown ann − V ⊆ B for some natural Bn ( 2m) ≥ ⊗n ⊆ (f+1) number f 1. We want to show annBn(−2m) V B .Iff>[n/2], we are done already. Thus we may assume f ≤ [n/2]. For i := (i1, ··· ,in) ∈ I(2m, n), we define the weight λ(vi)=λ to be the composition λ =(λ1, ··· ,λ2m)ofn into 2m parts, where λj is the number of times vj occurs as a tensor factor in vi, j =1, ··· , 2m. Note that the tensors of weight λ for a given composition λ of n span a KS -submodule M λ of V ⊗n;thus n V ⊗n = M λ λ∈Λ(2m,n) as a KSn-module, where Λ(2m, n) denotes the set of compositions of n into 2m parts. It is well known that M λ is isomorphic to the sign permutation representation of Sn on the cosets of the Young subgroup Sλ of Sn. As a consequence, each element v ∈ V ⊗n canbewrittenasasum v = vλ λ∈Λ(2m,n) λ for uniquely determined vλ ∈ M . ∈ ··· ≤ ≤ Fix an index c I(2m, 2f)oftheform(i1,i1,i2,i2, ,if ,if )with1 is 2m for 1 ≤ s ≤ f; for example, c =(1, 1 , 2, 2 , ··· ,f,f ). Since e1e3 ···e2f−1 acts only on the first 2f parts of any simple tensor vi, i ∈ I(2m, n), we may consider these operators as acting on V ⊗2f . f Let ν = νf := ((2 ), (n − 2f)). Consider the subgroup Π of S{1,··· ,2f} ≤ Sn (1) permuting the rows of tν but keeping the entries in the rows fixed. Obviously, Π License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 199 ν(1) normalizes the stabilizer S(2f ) of t in S2f ,whereS2f := S{1,2,··· ,2f}.Infact, it is well known that the semi-direct product Ψ := S(2f ) Π is the normalizer of S(2f ) in S2f . (1) Let λ ∈ Λ(2m, 2f)betheweightofvc with c =(f +1, (f +1) , ··· , 2f,(2f) ) ∈ I(2m, 2f). Note, if j =(j1, ··· ,jn−2f ) ∈ I(2m, n − 2f)satisfies2f +1≤ js ≤ m (2) for s =1, ··· ,n − 2f,andifλ ∈ Λ(2m, n − 2f) denotes the weight of vj ∈ ⊗n−2f (1) (2) V , then we obtain the weight λ ∈ Λ(2m, n)ofv ⊗ v by adding λ to λ c j (1) (2) componentwise. Note that s λs =0 ∩ s λs =0 = ∅.Wewriteforthis (1) (2) weight λ = λ ⊗ λ . We define Ef ∈ Bn(−2m)tobee1e3 ···e2f−1. (1) Lemma 3.6. The weight component of vce1e3 ···e2f−1 to weight λ is f f (ψ) − − − vcEf λ(1) =( 1) vcψ =( 1) ( 1) vcψ. ψ∈Ψ ψ∈Ψ Proof. By definition, m ⊗f m ⊗f f vcEf = vj ⊗ vj − vj ⊗ vj =(−1) vj ⊗ vj − vj ⊗ vj . j=1 j=1 (1) To obtain the components in the weight space (V ⊗2f )λ ,wehavetoconsider all occurring simple tensors which are obtained from vc = w1 ⊗ ··· ⊗ wf with wi = vf+i⊗v(f+i) by first permuting the tensors wi, which is done by a permutation π ∈ Π, and then replacing (for some i ∈{1, ··· ,f}) wi by wi = v(f+i) ⊗ vf+i, which amounts to applying a permutation σ ∈ S(2f ). On the other hand, each such tensor occurs exactly once, and the sign (−1)(ψ) is calculated taking in account f that if we factor out (−1) ,thewi carry a positive sign and the wi carry a negative ⊗n sign, the elements of Π all have even length and the action of Sn on V considered here carries a sign as well. This proves the lemma. f − (1) (2) D Recall ν = νf := ((2 ), (n 2f)) = (ν ,ν ) and the definition of the set νf in the beginning of this section. We set D D ∩ − ∈ f = νf Sµ where µ = ((2f), (n 2f)) Λ(2,n). ν(2) ∈D D ∈D Since t d is row standard for any d ν f ,thus f consists of all d νf which ··· D D ∩ fix every element in the set 2f +1, ,n .Thatis, f = νf S2f . Lemma 3.7. We have the equality S2f = Ψd, d∈Df where “ ” means a disjoint union. ν(1) (1) −1 Proof. Let t = t w,wherew ∈ S2f ,beaν -tableau. Then w S(2f )w is its row −1 stabilizer and w Πw is the subgroup of S2f permuting the rows of t. We therefore −1 find a ρ ∈ S(2f ) such that tw ρw is row standard, and then a π ∈ Π such that (1) tw−1ρww−1πw = tν ρπw is row standard and has increasing first column. Thus ν(1) ν(1) ∈D ∩ D t ρπw = t d for some d νf S2f = f .Thuswehaveshownψw = d with ψ := ρπ ∈ Ψ, and hence w ∈ Ψd. To show that the union is disjoint, let d1,d2 ∈Df ν(1) and suppose d1 = ψd2 for some ψ ∈ Ψ. Consider ti = t di, i =1, 2. We see from d1 = ψd2 that t1 and t2 have the same numbers in their rows, in fact up to License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 200 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU a permutation the same rows, since they are row standard. But the first column D has to be increasing, by definition of νf ; hence the orders of the rows in t1 and t2 have to be the same as well. This proves d1 = d2 and the union is disjoint. D ∈D (1) (2) We now turn to the full set νf .Fixd νf and let t =(t , t )bethe (2) corresponding νf -bitableau. Since t consists of a single row with increasing en- tries, it is completely determined by those entries. On the other hand, taking an arbitrary set partition {1, ··· ,n} = {i1, ··· ,i2f }{i2f+1, ··· ,i2n}, and inserting (1) the entries of the first set in increasing order along successive rows in tν ,andthe ν(2) numbers in the second set in increasing order into t ,weobtainaνf -bitableau (1) (2) ∈D t =(t , t ) such that obviously d(t) νf . Thus we may index those elements D P { ··· } ∈P of νf by the set f of subsets of 1, ,n of size 2f.WritingdJ for J f , ∈D νf (1) (2) { ··· } for an arbitrary d νf with t d = t =(t , t ), the subset J of 1, ,n of (1) entries of t is an element of Pf , and one sees by direct inspection that there is − ∈D D ∩ νf 1 an element d1 f = νf S2f such that t = t dJ (dJ d1dJ )=d1dJ .Thatis, d = d1dJ . Note also that each element dJ is a distinguished right coset representa- tive of S(2f,n−2f) in Sn.Thuswehaveshown Lemma 3.8. D D νf = f dJ . J∈Pf We define If to be the set of multi-indices (i2f+1, ··· ,in)oflengthn − 2f with 2f +1 ≤ iρ ≤ m for ρ =2f +1, ··· ,n, where we choose the position index ρ to run from 2f +1 to n in order to keep the notation straight, when we act by an element of Sn. Note that for 2f +1≤ i ≤ m,wehavei >m; hence s(vk) = 0 for all k ∈ If . ∈ ⊗n ⊗···⊗ For an arbitrary element v V ,wesaythesimpletensorvi = vi1 vin is involved in v if vi has a nonzero coefficient in writing v as a linear combination ∈ ⊗n j∈I(2m,n) kj vj of the basis vj j I(2m, n) of V . ⊗n Lemma 3.9. Let k ∈ If , v = vc ⊗ vk ∈ V .Let1 = d ∈ Sn.Ifeither ∈ ∈D D ∩ −1 ∈ d S(2f,n−2f) or d f = νf S(2f,n−2f),thend zEf ann(v) for any z ∈ Ψ. Proof. Write vc = w1 ⊗···⊗wf with wj = vj ⊗ vj ,j =1, ··· ,f.Ifd ∈ S(2f,n−2f), −1 then d is not contained in S(2f,n−2f) either. In particular, there is some j, ≤ ≤ ≤ −1 ≤ 2f +1 j n, such that 1 jd 2f, and hence the basis vector vkj with ≤ ≤ −1 −1 ≤ − 2f +1 kj m appears at position jd in vd . However, m We are now ready to prove the key lemma from which our main result in this section will follow easily. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 201 Lemma 3.10. Let S be the subset ∈D −1 d1,d2 νf , d1 =1, d1 Ef σd2 σ ∈ S{2f+1,··· ,n} of the basis (3.3) of Bn(−2m),andletU be the subspace spanned by S.Then (f) (f+1) B ∩ ann(vc ⊗ vk) = B ⊕ U. k∈If Proof. Since s(vk) = 0, by definition of If , hence s(vc ⊗ vk)=f, it follows that (f+1) B ⊆ ann(vc ⊗ vk). This, together with Lemma 3.9, shows that the right-hand side is contained in the left-hand side. (f) Now let x ∈ B ∩ ∈ ann(vc ⊗vk) . Using Lemma 3.9 and the basis (3.3) of k If B (−2m), we may assume that x = E z d,whereν = ν = ((2f ), (n−2f)) n f d∈Dν d f and the coefficients zd, d ∈Dν are taken from KS{2f+1,··· ,n} ⊆ KSn.Wethen have to show x =0. (1) Fix k ∈ If and write v = vc ⊗ vk. As in Lemma 3.6, choose the weight λ ∈ Λ(2m, 2f)tobetheweightofvc = w1 ⊗···⊗wf ,wherewi = vf+i ⊗ v(f+i) ,i = (2) (1) (2) 1, ··· ,f,andletλ be the weight of vk;thusλ = λ ⊗λ is the weight of vc ⊗vk. ⊗n λ Since V is the direct sum of its weight spaces M , we conclude (vx)µ =0forall µ ∈ Λ(2m, n). In particular, 0=(vx)λ = (vc ⊗ vk)x = vcEf ⊗ vk zdd λ λ d∈D ν ⊗ = (vcEf )λ(1) vk zdd. d∈Dν The latter equality holds, since the action of Snpreserves weight spaces. f − By Lemma 3.6 we have vcEf λ(1) =( 1) ψ∈Ψ vcψ = v, where again Ψ is the normalizer of the Young subgroup S(2f ) in S2f . Thuswehavetoinvestigate v ⊗ v z d = 0 for the unknown element z ∈ KS{ ··· }.Notethat d∈Dν k d d 2f+1, ,n (v ⊗ vk)zd = v ⊗ (vkzd). ∈D { ··· } We fix d νf .ByLemma3.8wefinda2f-element subset J of 1, ,n and d1 ∈Df ⊆ S2f such that d = d1dJ .Thus ⊗ ⊗ ⊗ ⊗ v vk zdd = v vkzd d = v vkzd d1dJ = vd1 vkzd1dJ dJ , since d1 ∈ S2f and zd ∈ KS{2f+1,··· ,n}. If J, L ∈Pf ,J = L,choose1≤ l ≤ n with l ∈ J but l ∈ L.Thusthereexists ∈{ ··· } −1 a j 1, 2, , 2f which is mapped by dJ to l, but (l)dL > 2f.Notethat for any d ∈Df all basis vectors vi occurring in vd as factors have index in the set {f +1,f +2, ··· , 2f,(2f) , ··· , (f +2), (f +1)},andallthosevi occurring in ⊗···⊗ vkzddJ , respectively in vkzddL , have index i between 2f +1 and m.Letvi1 vin ⊗ ⊗···⊗ beasimpletensorinvolvedin vd 1 vkzd1dJ dJ and vj1 vjn be a simple ⊗ ∈D tensor involved in vd2 vkzd2dL dL for d1,d2 f . Then, by the above, we have ≤ ≤ ≤ ≤ that 2f +1 jl m, and either vil = vk or vil = vk for some f +1 k 2f . ∈ ⊗ Consequently the simple tensors vi,i I(2m, n)involvedin (vd1 vkzd1dJ )dJ ⊗ and in (vd2 vkzd2dL )d L are disjoint; hence both sets are linearly independent. We conclude that vd ⊗v z d =0foreachJ ∈P ; hence vd ⊗ d∈Df k ddJ J f d1∈Df 1 vkzd1dJ =0,sincedJ is invertible. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 202 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU Lemma 3.7 says in particular that vd 1 is a linear combination of basis tensors vi = ⊗···⊗ ∈ D vi1 vi2f ,withi cΨd1, and that we obtain by varying d1 through f precisely the partition of S2f into Ψ-cosets. These are mutually disjoint. This is because S2f acts faithfully on the K-span of vc σ σ ∈ S2f , and hence all the basis vectors vi, 1 ≤ i ≤ n appearing as factors in vd 1 are pairwise distinct. Consequently the cosets of Ψd1,d1 ∈Df , partition the basis vectors in this set into mutually disjoint subsets, and we conclude that the basic tensors involved in vd1 are disjoint for different choices of d ∈D. Therefore, the equality vd ⊗ v z =0 1 f d1∈Df 1 k d1dJ ⊗ ∈D ∈ implies that vd1 vk zd1dJ =0foreachfixedd1 f .Nowwevaryk If .TheK- span of v k ∈ I is isomorphic to the tensor space V ⊗n−2f for the symmetric k f ∼ group S{2f+1,··· ,n} = Sn−2f .Sincem − 2f ≥ n − 2f, hence S{2f+1,··· ,n} acts ∈D ∈P faithfully on it. This implies zd1dJ =0foralld1 f ,J f .Thusx =0and the lemma is proved. Corollary 3.11. Let d ∈D ,ν = ν .Then ν f (f) ∩ ⊗ (f+1) ⊕ ˜−1 B ann (vc vk)d = B Kd1 Ef σd2 . ∈ k If d= d˜1,d2∈Dν σ∈S{2f+1,··· ,n} Hence B(f) ∩ ann (v ⊗ v )d = B(f+1). d∈Dν k∈If c k Proof. First, we claim that for any d,˜ d ∈D with d˜= d, 2 ν ˜−1 (vc ⊗ vk)d d Ef =0, and thus the right-hand side of the above equality is contained in the left-hand side. ˜−1 By Lemma 3.9, it suffices to consider the case where dd = w ∈ S(2f,n−2f).By ˜ ˜ ˜ Lemma 3.8, we can write d = d1dJ , d = d1dL,whered1, d1 ∈Df ,J,L ∈Pf .Since dJ ,dL are distinguished right coset representatives of S(2f,n−2f) in Sn, we deduce ˜ that J = L. Hence d1 = wd1 and hence w ∈ S2f . By Lemma 3.7, we can write ˜ ˜ w = zd3,wherez ∈ Ψ,d3 ∈Df .Sinced = d, it follows that d1 = d1. Therefore, by the decomposition given in Lemma 3.7, we know that d1 = wd˜1 = zd3d˜1 implies ˜−1 −1 −1 that d3 = 1. Therefore, by Lemma 3.9, vcdd Ef = vcd3 z Ef =0. This proves our first claim. ⊗ ∈ ∈D (f) Second, note that the annihilator of vc vk d (k If ,d νf )inB is −1 (f) precisely d ann(vc ⊗ vk) ∩ B . By Lemma 3.10, to complete the proof of the −1 −1 ∈D ∈ corollary, it suffices to show that each d d1 Ef σd2,where1= d1,d2 ν ,σ ˜−1 ! ˜ ˜ ˜ ∈D ! ∈ S{2f+1,··· ,n}, can be written in the form d1 Ef σd2,whered = d1, d2 ν , σ S{2f+1,··· ,n}. In fact, assume that d1d = zd3dJ ,wherez ∈ Ψ,d3 ∈Df ,J ∈Pf . Then by the decomposition given in Lemma 3.7, d1 = 1 implies that d3dJ = d. Note that Ψ is generated by s1,s2is2i+1s2i−1s2i,i =1, ··· ,f. Using the fact that s1e1 = e1 and s2is2i+1s2i−1s2ie2i−1e2i+1 = s2is2i+1e2ie2i−1e2i+1 = e2i+1e2ie2i−1e2i+1 = e2i+1e2ie2i+1e2i−1 = e2i−1e2i+1, it is easy to see that zEf = Ef for any z ∈ Ψ. It follows that −1 −1 −1 −1 −1 d d1 Ef σd2 =(d3dJ ) z Ef σd2 =(d3dJ ) Ef σd2, as required. This completes the proof of the corollary. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 203 ⊗n (1) Proof of Theorem 3.4. We have seen in Lemma 3.5 that ann − V ⊆ B , Bn( 2m) ⊗n (f+1) and Corollary 3.11 implies that ann − V ⊆ B provided that Bn( 2m) ⊗n ⊆ (f) ⊗n ⊆ annBn(−2m) V B . Thus by induction on f we have annBn(−2m) V B(f) for all natural numbers f.SinceB(f+1) =0forf>[n/2] it follows that ⊗n ≥ annBn(−2m) V =0.Inotherwords,ϕ is injective if m n. Suppose furthermore K is an infinite field. By (2.8) the natural homomorphism sy from the group algebra KGSp(V ) to the symplectic Schur algebra SK (m, n)is sy ∼ ∼ surjective. Note that since SK (m, n) is a quasi-hereditary algebra and V = L(ε1) = ∼ ⊗n sy (ε1) = (ε1), it follows that V is also a tilting module over SK (m, n). By the general theory from tilting modules (e.g. [DPS, Lemma 4.4 (c)]), ⊗n ⊗n ⊗ sy ⊗ EndKGSp(V ) V K K =EndS (m,n) V K K K ⊗n ⊗n =End sy V =End V , S (m,n) K KGSp(VK ) K K ⊗n ⊗n where V := V ⊗K K, and dim EndSsy(m,n) V = dim EndSsy (m,n) VC . K K K C Therefore dim End V ⊗n KGSp(V ) 2m ⊗n = dim EndC C (C ) GSp2m( ) = (dim S!λ)2 (by the fact that m ≥ n and [GW, (10.3.3)]) 0≤f≤[n/2] λn−2f =dimBn(−2m), !λ where S is the cell module for Bn(−2m) associated to λ. By comparing dimen- sions, we see that ϕ is in fact an isomorphism. This completes the proof of Theorem 3.4 and hence the proof of Theorem 1.4 in the case m ≥ n. 4. The case m License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 204 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU ! with respect to V and without this symbol for V . The notion of the signs !ij for ∈{ ··· ··· } i, j 1, ,m0,m0, , 1 extends the ij defined in the beginning for V . We have a natural embedding of Sp(V )intoSp(V!), that is, ! (4.1) Sp(V )= g ∈ Sp(V ) gv˜j =˜vj,foreachm +1≤ j ≤ (m +1) . ⊗n ! ⊗n ! ⊗n ⊗n The tensor space V is a direct summand of V ;letπK : V → V be the corresponding projection. That is, πK sends all simple tensors which contain a tensor factorv ˜i orv ˜i for m +1≤ i ≤ m0 to zero. The symplectic form defines a KSp(V )-isomorphism ι from V onto V ∗ := ∗ ⊗n HomK (V,K), taking v ∈ V to ι(v):=(v, −) ∈ V ;thusV and hence V are self-dual KSp(V )-modules. The analogous statement holds for V! and KSp(V!). ∗ We identify EndK (V )withV ⊗ V in the standard way. If we represent a K- endomorphism of V as a matrix (di,j )(i, j ∈{1, ··· ,m,m ··· , 1 }), relative to a ∗ basis (vi), then the corresponding vector of V ⊗ V is dij vi ⊗ ι(vj ) . i,j Note that ι(vj )(vs)=δs,j for any 1 ≤ s ≤ 1 . This construction extends easily to a tensor product by ⊗n ∼ ⊗n ⊗n ∗ ∼ ⊗n ∗ ⊗n EndK V = V ⊗ V = V ⊗ (V ) ! ∈ → ⊗n and works similarly for V .Ifg Sp(V ), ρ : KSp(V ) EndK V is the ⊗n representation afforded by the tensor space, then ρ(g)actsonEndK V by con- ⊗n jugation. Hence EndK V is naturally a KSp(V )-module and the isomorphisms above are KSp(V )-module maps. In particular ⊗n ∼ ⊗n ∗ ⊗n Sp(V ) EndKSp(V ) V = V ⊗ (V ) , where the latter denotes the invariants of V ⊗n ⊗ (V ∗)⊗n under the left diagonal ∼ ∗ action of KSp (V ). Using the fact that V = V as a KSp(V )-module, we obtain ⊗n ∼ ⊗2n EndK V = V , and hence ⊗n ∼ ⊗2n Sp(V ) EndKSp(V ) V = V . Note that the above isomorphism sends a homomorphism represented by the matrix (di,j )(i,j ∈ I(2m, n)), relative to a basis (vi), to the vector (4.2) di,j vi ⊗ vj , i,j∈I(2m,n) ··· ··· ··· where i =(i1, ,in),j =(j1, ,jn),j =(j1, ,jn). Therefore, we can express our problem in terms of invariants. A similar construction works for V! and Sp(V!). Since Sp(V ) ≤ Sp(V!) we may restrict V! ⊗2n to Sp(V ), and it is easy to see that Sp(V! ) the projection π : V! ⊗2n → V ⊗2n is KSp(V )-linear. In particular, π V! ⊗2n K K Sp(V ) ⊆ V ⊗2n . Lemma 4.3. Let θ : Bn(−2m0) → Bn(−2m) be the K-linear isomorphism which is defined on the common basis of these algebras, consisting of Brauer diagrams, as License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 205 the identity. Then the following diagram ! ! ϕ ! ⊗n ∼ ! ⊗2n Sp(V ) B (−2m ) −−−−→ End ! V −−−−→ V n ⏐ 0 KSp(⏐V ) ⏐ ⏐ ⏐ ⏐ θ πK πK ϕ ⊗n ∼ ⊗2n Sp(V ) Bn(−2m) −−−−→ EndKSp(V ) V −−−−→ V ! ⊗n is commutative, where πK maps an endomorphism of V to its restriction to ⊗n ! ⊗n V ⊆ V followed by the projection πK . Proof. We use the same symbols to denote the standard generators for the two Brauer algebras B (−2m ),B (−2m). By definition, n 0 n −1 ··· −1 ··· θ d1 e1e3 e2f−1σd2 = d1 e1e3 e2f−1σd2, f for any 0 ≤ f ≤ [n/2], σ ∈ S{2f+1,··· ,n}, d1,d2 ∈Dν ,whereν := ((2 ), (n − 2f)). Note that θ is a K-linear map, but does not respect multiplication, since θ(e1e1)= − − 2m0e1 = 2me1 = θ(e1)θ(e1).ThesameistrueforπK . { ··· ··· } ! ⊗n Let I = m +1, ,m0,m0, , (m +1) . We identify endomorphisms of V ⊗n (resp., of V ) with their matrices relative to the basis (˜vi) (resp., the basis (vi)). The map πK just sends a matrix (di,j ) to its submatrix obtained by deleting those rows and columns indexed by elements in I, while πK sends all simple tensors which contain a tensor factorv ˜i for i ∈ I to zero. Using (4.2), one sees easily that the ! right square diagram is commutative. It remains to show that πK ϕ = ϕθ. ≤ ≤ ∈ We identify πK with πK . We have to show that for any 0 f [n/2], σ f S{2f+1,··· ,n}, d1,d2 ∈Dν ,whereν := ((2 ), (n − 2f)), −1 −1 π ϕ! d e e ···e − σd = ϕθ d e e ···e − σd K 1 1 3 2f 1 2 1 1 3 2f 1 2 −1 ··· = ϕ d1 e1e3 e2f−1σd2 , or equivalently, ! −1 ! ··· ! −1 ··· (4.4) πK ϕ d1 ϕ e1e3 e2f−1 ϕ σd2 = ϕ d1 ϕ e1e3 e2f−1 ϕ σd2 . Note that for any w ∈ Sn where both ϕ!(w)andϕ(w) are given by right place ! permutation, it is trivial that πK ϕ(w)=ϕ(w). Now m0 ! ! ⊗···⊗ ⊗ ⊗ − ⊗ ⊗ ⊗ ϕ(ei)= jiji+1 v˜j1 v˜ji−1 (˜vk v˜k v˜k v˜k ) v˜ji+2 j∈I(2m,n) k=1 ···⊗v˜j ⊗ v˜j ⊗···⊗v˜j ⊗ v˜j ⊗···⊗v˜j , n 1 i i+1 n m ⊗···⊗ ⊗ ⊗ − ⊗ ⊗ ⊗ ϕ(ei)= jiji+1 vj1 vji−1 (vk vk vk vk ) vji+2 j∈I(2m,n) k=1 ···⊗v ⊗ v ⊗···⊗v ⊗ v ⊗···⊗v . jn j1 ji ji+1 jn ! ··· ··· It is also easy to see that πK ϕ(e1e3 e2f−1)=ϕ(e1e3 e2f−1). Therefore, to prove (4.4), it suffices to show that for any x, y ∈ Sn, ! ! ··· ! ! ! ··· ! πK ϕ x ϕ e1e3 e2f−1 ϕ y = πK ϕ x πK ϕ e1e3 e2f−1 πK ϕ y . But this follows from direct verification (although πK is in general not an algebra homomorphism). This completes the proof of the lemma. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 206 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU ≥ ! Henceforth we assume that m0 n. By Theorem 3.4, ϕ is an isomorphism; ! ⊗n ⊗n hence ϕ is surjective if and only if πK EndKSp(V! ) V =EndKSp(V ) V ,or ! ! ⊗2n Sp(V ) ⊗2n Sp(V ) equivalently, πK V = V . This means that every KSp(V )- ⊗n ! ! ! ⊗n endomorphism f of V canbeextendedtoaKSp(V )-endomorphism f of V ! ⊗2n such that πK f = f. It also means that every Sp(V )-invariant v of V can be ! ! ⊗2n extended to a Sp(V )-invariant v! of V such that πK (v!)=v. To accomplish this we replace the groups Sp(V )andSp(V!) by their Lie algebras −1 g = sp2m and !g = sp2m .LetA := Z[v, v ], where v is an indeterminate over 0 ! Z,andletQ(v) be its quotient field. Let UA respectively UA be Lusztig’s A- form (see [Lu3]) in the quantized enveloping algebra of g respectively !g. For any commutative integral domain R and any invertible q ∈ R we write UR := UA ⊗A R, where we consider R as an A-module by the specialization v → q.Furthermore, taking q =1∈ Z and taking the quotient by the ideal generated by the Ki − 1for i =1, ··· ,m, one gets the Kostant’s Z-form (see [Ko], [Lu2, (8.15)] and the proof of [Lu1, (6.7)(c), (6.7)(d)]) ∼ ∼ UZ = UA ⊗A Z /K − 1, ··· ,K − 1 = UZ/K − 1, ··· ,K − 1 1 m 1 m ∼ = UA/K1 − 1, ··· ,Km − 1 ⊗A Z in the ordinary enveloping algebra of the complex Lie algebra sp2m(C), and the hyperalgebra ∼ ∼ UK = UZ ⊗Z K = UA ⊗A Z /K1 − 1, ··· ,Km − 1⊗Z K ∼ = UK /K1 − 1, ··· ,Km − 1 ! of the simply connected simple algebraic group Sp2m(K). Similarly we define UR, ! ! UZ and UK . It is well known that (see [Ja]) there is an equivalence of categories between {rational Sp2m(K)-modules} and {locally finite UK -modules} such that the trivial Sp2m(K)-module corresponds to the trivial UK -module, where the trivial UK - module is the one dimensional module which affords the counit map of the Hopf algebra UK .TheSp2m(K)-action on tensor space gives rise to a locally finite UK -action on tensor space. Therefore ⊗n ⊗n ∼ ⊗2n UK ⊗2n Sp(V ) EndKSp(V ) V =EndUK V = V = V . This works in the same way for V!. Hence π is a U -linear map which maps the K K U! U invariants V! ⊗2n K into V ⊗2n K . ! ! ⊗2n UK ⊗2n UK Our goal is to show that πK V = V . For this purpose, we U U! have to investigate certain nice bases of V ⊗2n K respectively V! ⊗2n K .Let ! A ··· ··· VA (resp., VA)bethefree -module generated by v1, ,vm0 ,vm , ,v1 (resp., ! 0 ! by v1, ··· ,vm,vm , ··· ,v1 ). Recall that there is an action of UQ(v) on VQ(v) := License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 207 ! VA ⊗A Q(v) which is defined on generators as follows. ⎧ ⎪ ⎨v˜i, if j = i +1, v˜m0 , if j = m0, Eiv˜j := v˜ , if j = i , Em v˜j := ⎩⎪ (i+1) 0 0, otherwise, 0, otherwise; ⎧ ⎨⎪v˜i+1, if j = i, v˜ , if j = m , m0 0 Fiv˜j := v˜ , if j =(i +1), Fm v˜j := ⎩⎪ i 0 0, otherwise, 0, otherwise; ⎧ ⎨⎪vv˜j , if j = i or j =(i +1), −1 Kiv˜j := v v˜ , if j = i +1orj = i , ⎩⎪ j v˜ , otherwise; ⎧ j 2 ⎨⎪v v˜j , if j = m0, −2 Km v˜j := v v˜ , if j = m , 0 ⎩⎪ j 0 v˜j , otherwise, where 1 ≤ i License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 208 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU ! ∗ ! For each 1 ≤ i ≤ 1 , ι(˜vi)=(˜vi, −) ∈ VA := HomA VA, A .Thenι(˜v1)isa highest weight vector of weight ε1.Themap˜v1 → ι(˜v1) extends naturally to a ! ! ∼ ! ∗ ! UA-module isomorphism ι : VA = VA = UAι(˜v1). One checks easily that i−1 2m0+1−i ι (˜vi)=v ι(˜vi),ι(˜vi )=v ι(˜vi ), ∀ 1 ≤ i ≤ m0. Using the isomorphism ι,wegetthat ! ! UA UA ! ⊗n ∼ ! ⊗n ∼ ! ⊗n ! ⊗n ∗ ! ⊗ EndUA VA = End VA = VA VA ! UA ! ∼ ! ⊗n ! ∗ ⊗n ∼ ! ⊗2n UA = VA ⊗ (VA) = VA . ⊗n ∼ ⊗2n UA Similarly, EndUA VA = VA .Consequently,foranyfieldk and any spe- cialization v → q ∈ k×, U! U! ! ⊗2n k ∼ ! ⊗n ∼ ! ⊗n ∼ ! ⊗2n A V = End! V = End! VA ⊗A k = VA ⊗A k. k Uk k UA U U ⊗2n k ∼ ⊗n ∼ ⊗n ∼ ⊗2n A U U ⊗A ⊗A Similarly, Vk = End k Vk = End A VA k = VA k.Note ⊗2n that when specializing q to 1, each Ki acts as the identity on the tensor space V . It follows that ⊗2n UZ ∼ ⊗n ∼ ⊗n VZ = EndUZ VZ = EndUZ VZ ∼ ⊗n ⊗ Z ∼ ⊗2n UA ⊗ Z = EndUA VA A = VA A and ⊗2n UK ∼ ⊗n ∼ ⊗n ∼ ⊗n ⊗ V = EndU V = EndU V = EndUA VA A K K K K K K ∼ ⊗2n UA ∼ ⊗2n UA ∼ ⊗2n UZ = VA ⊗A K = VA ⊗A Z ⊗Z K = VZ ⊗Z K. Similar results hold for V!, U! and U!. In [Lu3, (27.1.2)], Lusztig introduced the notion of a based module and by [Lu3, ! ! ⊗2n (27.3)], the UQ(v)-module M := (VQ(v)) is a based module. That is, there is a canonical basis B! of M , in Lusztig’s notation ([Lu3, (27.3.2)]), each element in B! is ··· ··· ⊗···⊗ of the formv ˜i1 ˜v˜i2 ˜ ˜v˜i2n ,and˜vi1 ˜ ˜v˜i2n is equal tov ˜i1 v˜i2n plus a linear ⊗···⊗ ··· ··· combination of elementsv ˜j1 v˜j2n with (˜vj1 , , v˜j2n ) < (˜vi1 , , v˜i2n )and with coefficients in v−1Z[v−1], where “<” is a partial order defined in [Lu3, (27.3.1)]. ! ! ⊗2n ⊗2n In particular, B is an A-basis of VA . Similarly, we define M := (VQ(v)) as a module over UQ(v), and we have a canonical basis B of M. Each element of B is of the form vi vi ···vi . ! 1 2 2n ! Let X+ be the set of all the dominant weights of !g.Forλ ∈ X+,wedenoteby ! ∆Q(v)(λ) the irreducible UQ(v)-module of highest weight λ. We define M [λ]:= N.! N ⊆M ∼ N=∆Q(v)(λ) Then M = M [λ]. λ∈X+ ! ! ! For each λ ∈ X+,letM[>λ]:= M[µ] and define B[>λ]:=B ∩ M[>λ]. λ<µ∈X+ ! By [Lu3, (27.1.8)(b)], B[>λ]isaQ(v)-basis of M[>λ]. We define M[>λ]A := License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 209 A b∈B![>λ] b. By [Lu3, (27.1.2)(b), (27.1.8)], it is easy to see that M[>λ]A is ! stable under UA. Hence for any specialization v → q = 0 in some field K, ! ! M[>λ] := ! Kb is U -stable and the set b b ∈ B[>λ] forms a K b∈B[>λ] K K-basis of M[>λ]K .LetM[=0]:= λ=0 M[λ]. By [Lu3, (27.2.5)], M [=0]= M [≥ µ]. µ∈X+−{0} In particular, B![=0]:= B![λ]formsaQ(v)-basis of M [= 0]. We define λ=0 A U! M[=0]A := b∈B![=0] b.ThenM[=0]A is stable under A. Hence for any ! specialization v → q = 0 in some field K, M[=0] := ! Kb is U -stable K b∈B[=0] K ∈ ! and the set b b B[=0] forms a K -basis of M[=0]K . The isomorphism ι ! ⊗2n ∼ ! ⊗2n ∗ induces a natural isomorphism VA = VA , which we still denote by ι .Itis U! ∗ ! ⊗2n Q(v) ! ⊗2n clear that ι maps VQ(v) isomorphically onto VQ(v) /M[=0] .Inparticular, ! ! ! ! ⊗2n UA ! ⊗2n UA ι (a)vanishesonB[= 0] for every a ∈ VA . Therefore, ι maps VA into ∗ ! ⊗2n V /M[=0] A . By comparing dimensions, we conclude that for each field K A U! ∗ A ! ⊗2n K ! ⊗2n which is an -algebra, ι maps VK isomorphically onto VK /M[=0]K . ! ∗ ! ⊗2n UA ! ⊗2n As a consequence, ι also maps VA isomorphically onto VA /M[=0] A . Similarly, one can define X+ (the set of all the dominant weights of g), and for each λ ∈ X+, one can define M[λ], M[>λ],B[>λ],M[=0]and B[= 0]. One has that ∗ ⊗2n UA ⊗2n M = M[λ], and V is canonically isomorphic to V /M [=0] A . λ∈X+ A A Recall that (see [Lu3, (27.2.1)]), B = B[λ], B! = B![λ]. λ∈X + λ∈X+ ! ! ⊗2n By [Lu3, (27.2.5)], the image of B[0] (resp., B[0]) in VA /M[=0] A (resp., in ⊗2n ! ⊗2n ⊗2n VA /M [=0] A) forms an A-basis of VA /M[=0] A (resp., of VA /M [=0] A). Let J0 := (i1, ··· ,i2n) ∈ I(2m, 2n) vi ···vi ∈ B[0] , 1 2n ! ··· ∈ ··· ∈ ! J0 := (i1, ,i2n) I(2m0, 2n) v˜i1 ˜ ˜v˜i2n B[0] . Corollary 4.5. With the above notation the set ⊗···⊗ ··· ∈ vi1 vi2n + M[> 0]A (i1, ,i2n) J0 ⊗2n forms an A-basis of VA /M [> 0]A. ⊗2n Proof. This is clear, by the fact that the image of B[0] in VA /M [> 0]A is an A- ··· ⊗···⊗ basis and each vi1 vi2n is equal to vi1 vi2n plus a linear combination of ⊗···⊗ ··· ··· elements vj1 vj2n with (vj1 , ,vj2n ) < (vi1 , ,vi2n ) and with coefficients in v−1Z[v−1]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 210 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU Similarly, the set ⊗···⊗ ··· ∈ ! (4.6) v˜i1 v˜i2n + M[> 0]A (i1, ,i2n) J0 ! ⊗2n forms an A-basis of VA /M[> 0]A. ! Theorem 4.7. With the above notation, J0 ⊆ J0. ! Proof. For each 1 ≤ i ≤ m0,lete!i, fi (resp., ei, fi) be the Kashiwara operators of U! U U! ! Q(v) (resp., of Q(v)). The Q(v)(sp2m0 )-crystal structure on VQ(v) is given below: − − −→1 −→···2 m−→0 1 −→m0 m−→···0 1 −→2 −→1 1 2 m0 m0 2 1 , where i ˜ −1 −1 j −→ k ⇔ fiv¯j ≡ v˜k (mod v M) ⇔ v˜j ≡ e˜iv˜k (mod v M). Similarly, the UQ(v)-crystal structure on VQ(v) is as below: − − 1 −→1 2 −→···2 m−→1 m −→m m m−→···1 −→2 2 −→1 1 . Comparing with the two crystal graphs, it is easy to see that for each 1 ≤ i ≤ m and each j ∈{1, ··· ,m}∪{m, ··· , 1}, ≥ !k ∈ −1 ≥ k ∈ −1 max k 0 ei v˜j v M =max k 0 ei vj v M , ≥ !k ∈ −1 ≥ k ∈ −1 max k 0 fi v˜j v M =max k 0 fi vj v M . Moreover, for each m +1≤ i ≤ m0 and each j ∈{1, ··· ,m}∪{m , ··· , 1 }, −1 ! −1 e!iv˜j ∈ v M, fiv˜j ∈ v M. Let B (resp., B!) be the canonical basis of V ⊗n (resp., of V! ⊗n) constructed ! from the canonical basis of V (resp., of V ); see [Lu3, (27.3.1)]. For each λ ∈ X+ ! lo hi lo hi (resp., λ ∈ X+), let B [λ] ,B [λ] (resp., B [λ] , B [λ] ) be as defined in [Lu3, (27.2.3)]. Now [Lu3, (17.2.4)] gave the rules for the action of the Kashiwara operators ! e!i, fi,ej,fj on tensor products. As a consequence, our previous discussion shows that for any 1 ≤ i1, ··· ,in ≤ 2m, ··· ∈ hi ⇐⇒ ··· ∈ hi vi1 vin B [λ] v˜i1 ˜ ˜v˜in B [λ] , ··· ∈ lo ⇐⇒ ··· ∈ lo vi1 vin B [λ] v˜i1 ˜ ˜v˜in B [λ] . Now applying [Lu3, (27.3.8)], which said that ! ! lo ! hi B[0] = b ˜ b b ∈ B [−w0(λ )] ,b ∈ B [λ ] , λ∈X + lo hi B[0] = b b b ∈ B [−w0(λ )] ,b ∈ B [λ ] , λ∈X+ our theorem follows immediately. Proof of Theorem 1.4. We regard Z as an A-algebra by specializing v to 1 ∈ Z,and regard K as a Z-algebra as usual. Then it is easy to see that ι ⊗A 1K coincides ∗ with the canonical Sp2m-module isomorphism V → V ,v → ι(v):=(v, −) for any ! ! v ∈ V .LetVZ := VA ⊗A Z, M[=0] Z := M[=0] A ⊗A Z. We have similar notation for License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 211 ! ! ⊗2n UK ⊗2n UK VZ, M[=0] Z. We claim that the natural projection map V → V is surjective. In fact, we have the following commutative diagram: ∗ ⊗ ∗ ! ∼ ! ⊗2n ∼ ! 2n ! ⊗2n UK V VZ −−−−→ −−−−→ ⊗Z V K ⏐ M[=0]⏐ K M[=0]Z⏐ ⏐ ⏐ ⏐ ∗ ∗⊗ πK jK jZ Z1 ∗ ⊗ ∗ ∼ ⊗2n ∼ 2n ⊗2n UK V VZ V −−−−→ −−−−→ ⊗Z K M[=0] K M[=0] Z where the rightmost vertical homomorphism is induced from the canonical homo- ⊗2n ! ⊗2n morphism jZ : VZ /M [=0] Z → VZ /M[=0] Z.NotethatjZ is well defined as M[=0] Z ⊆ M[=0] Z (which follows from the fact that for each λ ∈ X+ with λ =0, MC[λ] should be contained in MC[=0]). By (4.6) and Theorem 4.7, the image of ⊗···⊗ ··· ∈ vi1 vi2n + M[> 0]K (i1, ,i2n) J0 under jK := jZ ⊗Z 1K is always linearly independent, which shows that jK is ∗ ∗ ⊗ injective. Hence jK := jZ Z 1K is surjective. It follows that ! UK ! ⊗2n UK ⊗2n πK V = V , as required. Now using Lemma 4.3 and Theorem 3.4, we complete the proof of Theorem 1.4 when K is algebraically closed. Now suppose that K is an arbitrary infinite field. Let K denote the algebraic closure of K. Note that the image of ϕ is generated (as an algebra) by ϕ(e1), ··· ,ϕ(en−1),ϕ(s1), ··· ,ϕ(sn−1) , and the canonical homomorphism End V ⊗n ⊗ K =End V ⊗n ⊗ K KSp(VK ) K K UK K K ⊗n ⊗n → EndU V =End V K K KSp(VK ) K ⊗ ⊗ ∼ ⊗ is an isomorphism, where UK = UZ Z K, UK = UZ Z K = UK K K. It follows that the dimension of im(ϕ) is constant under field extensions K ⊆ K. The proof is completed. Remark 4.8. The argument above in the proof of Theorem 1.4 actually shows that ! ! ⊗2n UZ ⊗2n UZ πZ VZ = VZ , ! ⊗n ⊗n or equivalently, πZ End ! V =End V . UZ Z UZ Z Acknowledgment The authors are grateful for the referee’s helpful comments and suggested im- provements to the earlier version of this paper. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 212 RICHARD DIPPER, STEPHEN DOTY, AND JUN HU References [B] R. Brauer, On algebras which are connected with semisimple continuous groups, Ann. of Math. 38 (1937), 857–872. MR1503378 [B1] W. P. Brown, An algebra related to the orthogonal group, Michigan Math. J. 3 (1955– 1956), 1–22. MR0072122 (17:232a) n [B2] W. P. Brown, The semisimplicity of ωf , Ann. of Math. 63 (1956), 324–335. MR0075931 (17:821g) [BW] J. Birman and H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. (1) 313 (1989), 249–273. MR992598 (90g:57004) [CC] C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976), 330–354. MR0422314 (54:10305) [CL] R. W. Carter and G. Lusztig, On the modular representations of general linear and sym- metric groups, Math. Z. 136 (1974), 193–242. MR0354887 (50:7364) [CP] V. Chari and A. Pressley, “A guide to quantum groups,” Cambridge University Press, Cambridge, 1994. MR1300632 (95j:17010) [DD] R. Dipper and S. Donkin, Quantum GLn, Proc. London Math. Soc. 63 (1991), 165–211. MR1105721 (92g:16055) [Do1] S. Donkin, On Schur algebras and related algebras I, J. Alg. 104 (1986), 310–328. MR866778 (89b:20084a) [Do2] S. Donkin, Good filtrations of rational modules for reductive groups, Arcata Conf. on Representations of Finite Groups. Proceedings of Symp. in Pure Math., 47 (1987), 69–80. MR933351 (89f:20048) [DPS] J. Du, B. Parshall and L. Scott, Quantum Weyl reciprocity and tilting modules, Commun. Math. Phys. 195 (1998), 321–352. MR1637785 (99k:17026) [Dt] S. Doty, Polynomial representations, algebraic monoids, and Schur algebras of classic type, J. Pure Appl. Algebra 123 (1998), 165–199. MR1492900 (98j:20057) [E] J. Enyang, Cellular bases for the Brauer and Birman-Murakami-Wenzl algebras, J. Alg. 281 (2004), 413–449. MR2098377 (2005f:20008) [GL] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996),1–34. MR1376244 (97h:20016) [Gr] J. A. Green, “Polynomial representations of GLn,” Lect. Notes in Math. Vol. 830, Springer- Verlag, 1980. MR606556 (83j:20003) [Gri] D. Ju. Grigor’ev, An analogue of the Bruhat decomposition for the closure of the cone of a Chevalley group of the classical series, Sov. Math. Doklady 23 (1981), 393–397. [GW] R. Goodman and N. R. Wallach, “Representations and invariants of classical groups,” Cambridge University Press, 1998. MR1606831 (99b:20073) [Ja] J. C. Jantzen, “Representations of Algebraic Groups,” Academic Press, Inc, 1987. MR899071 (89c:20001) [KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184. MR560412 (81j:20066) [Ko] B. Kostant, Group over Z, Proceedings of Symp. in Pure Math., 9 (1966), 90–98. MR0207713 (34:7528) [Lu1] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257–296. MR1013053 (91e:17009) [Lu2] G. Lusztig, Quantum groups at roots of 1, Geometriae Dedicata 35 (1990), 89–114. MR1066560 (91j:17018) [Lu3] G. Lusztig, “Introduction to Quantum Groups,” Progress in Math., 110,Birkh¨auser, Boston, 1990. MR1227098 (94m:17016) [M] J. Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. (4) 26 (1987), 745–758. MR927059 (89c:57007) [Mu] E. Murphy, The representations of Hecke algebras of type An, J. Alg. 173 (1995), 97–121. MR1327362 (96b:20013) [Oe] S. Oehms, Centralizer coalgebras, FRT-construction, and symplectic monoids, J. Algebra (1) 244 (2001), 19–44. MR1856529 (2002h:16062) [Sc] I. Schur, Uber¨ die rationalen Darstellungen der allgemeinen linearen Gruppe, (1927). Reprinted in I. Schur, Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BRAUER ALGEBRAS AND SCHUR-WEYL DUALITY 213 [T] R. Tange, The symplectic ideal and a double centraliser theorem, preprint, arXiv:0705.0377, (2007). [W] H. Weyl, “The classical groups, their invariants and representations,” Princeton University Press, 1946. MR1488158 (98k:01049) [Xi] C. C. Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math. (2) 154 (2000), 280–298. MR1784677 (2001g:20008) Mathematisches Institut B, Universitat¨ Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany E-mail address: [email protected] Department of Mathematics and Statistics, Loyola University Chicago, 6525 North Sheridan Road, Chicago, Illinois 60626 E-mail address: [email protected] Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use