SCHUR-WEYL DUALITY FOR U (sl ) v,t n YANMIN YANG, HAITAO MA, ZHU-JUN ZHENG 7 1 Abstract. In [8], the authors get a new presentation of two-parameter quantum algebra 0 U (g). Their presentation can cover all Kac-Moody cases. In this paper, we construct v,t 2 a suitable Hopf pairing such that U (sl ) can be realized as Drinfeld double of certain v,t n n Hopf subalgebras with respect to the Hopf pairing. Using Hopf pairing, we construct a a R-matrixforU (sl )whichwillbeusedtogivetheSchur-WeyldualbetweenU (sl )and v,t n v,t n J HeckealgebraH (v,t). Furthermore,usingtheFusionprocedureweconstructtheprimitive k 3 orthogonal idempotents of H (v,t). As a corollary, we give the explicit construction of k 2 irreducible U (sl )-representations of V⊗k. v,t n ] A Q . 1. Introduction h t a Classical Schur-Weyl duality related irreducible finite-dimensional representations of the m general linear and symmetric groups [18]. The quantum version for the quantum enveloping [ algebra U (sl ) and the Hecke algebra H (S ) has been one of the pioneering examples q n q m 1 [13] in the fervent development of quantum groups. Two-parameter general linear quantum v groupswereintroducedbyTakeuchi in1990[17]. Therelatedreferences are[1,4,6,7,12,17]. 2 6 In 2001, Benkart and Witherspoon obtained the structure of two-parameter quantum groups 2 corresponding to the general linear Lie algebra gl and the special linear Lie algebra sl with 6 n n 0 a different motivation [3]. They showed that the quantum groups can be realized as Drinfeld 1. doubles of certain Hopf subalgebras with respect to Hopf pairings. Using Hopf pairing, 0 Benkarat and Witherspoon constructed R-matrix which is used to establish an analogue of 7 Schur-Weyl duality [2]. 1 : In [8], using geometric construction, the authors got a new presentation of generators and v i relations for a two-parameter quantum algebra Uv,t determined by a certain matrix which X may served as a generalized Cartan matrix. The two parameters v and t they used are r a different from the one (r,s). Furthermore, their presentation covered all Kac-Moody cases, unlike the one in literature which mainly studies finite type and some affine types. A two- parameter quantum algebra U is a two-cocycle deformation, depending only on the second v,t parameter t, of its one-parameter analogue. And the algebra U in [3], is a two-cocycle r,s deformation depending both parameters (r = s−1). Wefocusonthetwo-parameter quantumgroupU (sl )forthepurposeofgivingtheShur- v,t n Weyl dual between U (sl ) and H (v,t). We will show that U (sl ) also has a Drinfeld v,t n k v,t n double realization by two certain Hopf subalgebras, but the R-matrix constructed by similar way in [2] can not afford a representation of H (v,t), we need to take a suitable modification. k This paper is organized as follows. In section 2, we give a Hopf pairing between two certain Hopf subalgebras of U (sl ). Then we prove that U (sl ) can be realized as the v,t n v,t n Date: January 24, 2017. Key words and phrases. Schur-Weyl duality, two-parameter quantum algebra, Hecke algebra. 1 2 YANMINYANG,HAITAOMA, ZHU-JUNZHENG Drinfeld double of certain Hopf subalgebras with respect to the Hopf pairing. In section 3, we construct the tensor power representation V⊗k of U (sl ) and R-matrix R˜. In section 4, v,t n we prove that the U (sl )-module V⊗k affords a representation of Hecke algebra H (v,t). v,t n k This leads to a Schur-Weyl duality between U (sl ) and Hecke algebra H (v,t). In section v,t n k 5, we give a family of primitive orthogonal idempotents of H (v,t). In section 6, irreducible k representations of U (sl ) are constructed by using the fusion procedure. v,t n 2. Two-Parameter Quantum Group U (sl ) And Its Drinfeld Double v,t n In this section, we review the definition of two-parameter quantum algebra U (sl ) intro- v,t n duced by Fan and Li in [8] and its Hopf algebra structure. In particular, U (sl ) also can v,t n be realized as a Drinfeld double of its certain subalgebras. 2.1. Two-Parameter Quantum Group U (sl ). In this paper, we fix the Cartan datum v,t n (Ω ,·) of type A , where n−1 n−1 1 0 0 ... 0 0 −1 1 0 ... 0 0   0 −1 1 ... 0 0 Ω = (Ω ) =  . . . . . .  , n−1 ij  .. .. .. .. .. ..     0 0 0 ... 1 0     0 0 0 ... −1 1   (n−1)×(n−1) and for any 1 ≤ i,j ≤ n−1, denote hi,ji = Ω , then i·j can be defined as hi,ji+hj,ii. ij Definition 1 ([8]). The two-parameter quantum algebra U (sl ) associated to Ω is an v,t n n−1 associative Q(v,t)-algebra with 1 generated by symbols E ,F , K±1,K′±1, ∀1 ≤ i < n and i i i i subject to the following relations. (R1) K±1K±1 = K±1K±1, K′±1K′±1 = K′±1K′±1, i j j i i j j i K±1K′±1 = K′±1K±1, K±1K∓1 = 1 = K′±1K′∓1, i j j i i i i i (R2) K E K−1 = vi·jthi,ji−hj,iiE , K′iE K′−1 = v−i·jthi,ji−hj,iiE , i j i j i j i j K F K−1 = v−i·jthj,ii−hi,jiF , K′F K′−1 = vi·jthj,ii−hi,jiF , i j i j i j i j K −K′ (R3) E F −F E = δ i i, i j j i ij v −v−1 (R4) [E ,E ] = [F ,F ] = 0 if | i−j |> 1, i j i j (R5) E2E −t(v +v−1)E E E +t2E E2 = 0, i i+1 i i+1 i i+1 i E E2 −t(v +v−1)E E E +t2E2 E = 0, i i+1 i+1 i i+1 i+1 i (R6) F2F −t−1(v +v−1)F F F +t−2F F2 = 0, i i+1 i i+1 i i+1 i F F2 −t−1(v+v−1)F F F +t−2F2 F = 0. i i+1 i+1 i i+1 i+1 i There is a Hopf algebra structure on the algebra U (sl ) with the comultiplication ∆ by v,t n ∆(K±1) = K±1 ⊗K±1, ∆(K′±1) = K′±1 ⊗K′, i i i i i i ∆(E ) = E ⊗1+K ⊗E , ∆(F ) = 1⊗F +F ⊗K′, i i i i i i i i SCHUR-WEYL DUALITY FOR Uv,t(sln) 3 the counit ε by ε(K±1) = ε(K′±1) = 1, ε(E ) = ε(F ) = 0, S(K±1) = K∓1, i i i i i i and the antipode S by S(K′±1) = K′∓1, S(E ) = −K−1E , S(F ) = −F K′−1. i i i i i i i i 2.2. Drinfeld Double Realization Of U (sl ). v,t n Definition 2. (See [14], 3.2.1) A Hopf paring of two Hopf algebras H and H′ is a bilinear form (−,−) : H′ ×H −→ K (a field) such that (1). (1,h) = ǫH(h), (h′,1) = ǫH′(h′); (2). (h′,hk) = (∆H′(h′),h⊗k) = (h′(1),h)(h′(2),k); (3). (h′k′,h) = (h′ ⊗k′,∆ (h)) =P(h′,h )(k′,h ); H (1) (2) for all h,k ∈ H, h′,k′ ∈ H′, where ǫHPand ǫH′ are the counits of H and H′ respectively, and ∆H and ∆H′ are their comultiplications. For h ∈ H, ∆(h) = h(1) ⊗h(2). P A direct consequence is that (1) (SH′(h′),h) = (h′,SH(h)) for all h ∈ H and h′ ∈ H′, where SH′ and SH are the antipodes of H′ and H respectively. Let B (resp. B′) be the Hopf subalgebra of U (sl ) generated by E , K±1 (resp. F ,K′±1) v,t n i i i i for 1 ≤ i < n. B′coop is the Hopf algebra having the opposite comultiplication to the Hopf algebra B′ and SB′coop = SB−′1, ∆B′coop = ∆op. Proposition 1. There exists a unique Hopf pairing (−,−) : B′coop×B −→ Q(v,t) such that δ ij (2) (F ,E ) = , i j v−1 −v (3) (K′,K ) = vj·ithj,ii−hi,ji, i j for any 1 ≤ i,j < n, and all other pairs of generators are 0. Moreover, we have (4) (S(a),S(b)) = (a,b) for any a ∈ B′coop, b ∈ B. Proof. Any Hopf pairing of bialgebras is determined by the values on the generators, so the uniqueness is clear. The process of proof reduces to the existence. The pairings defined by (2) and (3) in the proposition can be extended to a bilinear form on B′coop ×B by requiring that the conditions (1), (2) and (3) in definition 2 hold. We only ′ need to verify that the relations (2) and (3) in B and B are preserved. It is straightforward to check that the bilinear form preserves all the relations among the K±1 in B and the K′±1 in B′. Next, for any 1 ≤ i,j < n, we check i i (X,K E ) = (X,vj·ithj,ii−hi,jiE K ), j i i j where X is any word in the F and K′±1, 1 ≤ i < n. If X = K′F , the left hand side i i k l 4 YANMINYANG,HAITAOMA, ZHU-JUNZHENG (X,K E ) = (∆op(K′)∆op(F ),K ⊗E ) j i k l j i = (K′F ⊗K′ +K′K′ ⊗K′F ,K ⊗E ) k l k k l k l j i = (K′F ,K )(K′,E )+(K′K′,K )(K′F ,E ) k l j k i k l j k l i 1 = vj·ithj,ii−hi,ji(K′,K )(K′,K ) (l = i) v−1 −v k j k i the right hand side (X,vj·ithj,ii−hi,jiE K ) = vj·ithj,ii−hi,ji(∆op(K′F ),E ⊗K ) i j k l i j = vj·ithj,ii−hi,ji(K′F ⊗K′ +K′K′ ⊗K′F ,E ⊗K ) k l k k l k l i j 1 = vj·ithj,ii−hi,ji(K′,K )(K′,K ) (l = i) v−1 −v k j k i Hence, (X,K E ) = (X,vj·ithj,ii−hi,jiE K ). j i i j In particular, it can be similarly checked that the bilinear form preserves all the other relations in B and B′. (cid:3) Definition 3. (See [14], 3.2) If there is a Hopf pairing between Hopf algebras H and H′, then we may form the Drinfeld double D(H,H′coop), where H′coop is the Hopf algebra having the opposite coproduct to H. D(H,H′coop) is a Hopf algebra whose underlying vector space is H ⊗H′ with the tensor product coalgebra structure. The algebra structure is given by as follows: (a⊗f)(a′ ⊗f′) = (SH′coop(f(1)),a′(1))(f(3),a′(3))aa′(2) ⊗f(2)f′ X for a,a′ ∈ H and f,f′ ∈ H′. And the antipode S is given by S(a⊗f) = (1⊗SH′coop(f))(SH(a)⊗1). Clearly, the algebras H and H′coop are identified with H⊗1 and 1⊗H′coop respectively in D(H,H′coop). Proposition 2. D(B,B′coop) is isomorphic to U (sl ). v,t n Proof. Define the embedding maps ι : B −→ D(B,B′coop) E 7→ E := ι(E ) = E ⊗1 i i i i ±1 K±1 7→ K := ι(K±1) = K±1 ⊗1, i ci i i and c ι′ : B′coop −→ D(B,B′coop) F 7→ F := ι(F ) = 1⊗F i i i i K′±1 7→ K ′±1 := ι(K′±1) = 1⊗K′±1. i bi i i c SCHUR-WEYL DUALITY FOR Uv,t(sln) 5 ThenBandB′coop canbeviewedassubalgebrasinD(B,B′coop). AmapϕbetweenD(B,B′coop) and U (sl ) is defined as follows: v,t n ϕ : D(B,B′coop) −→ U (sl ) v,t n E 7−→ ϕ(E ) = E , i i i F 7−→ ϕ(F ) = F , ci ci i ±1 ±1 K 7−→ ϕ(K ) = K±1, bi bi i K ′±1 7−→ ϕ(K ′±1) = K′±1. ci ci i Note that, ϕ preserves the coalgebra structures. Next we will check that ϕ preserves the c c ′ ′ relations (R1-R6) in U (sl ). Consider ϕ(K K −K K ). By definition, v,t n j i i j ′ K K = (K ⊗1c)(1c⊗K′c) =cK ⊗K′. j i j i j i ′ To calculate K K , we havce c i j c c∆2(Kj) = Kj ⊗Kj ⊗Kj, (∆op)2(Ki′) = Ki′ ⊗Ki′ ⊗Ki′, so that ′ K K = (1⊗K′)(K ⊗1) i j i j c c = (SB′coop(Ki′),Kj)(Ki′,Kj)Kj ⊗Ki′ = K ⊗K′. j i ′ ′ That is, ϕ(K K −K K ) = 0 = K K′ −K′K . Other relations in B (resp. in B′) also can j i i j j i i j be verified by the same way. We verify the mixed relation R3. cc c c Similarly, E F = (E ⊗1)(1⊗F ) = E ⊗F . In order to calculate F E , we use i j i j i j j i cc ∆2(Ei) = Ei ⊗1⊗1+Ki ⊗Ei ⊗1+Ki ⊗Ki ⊗cEci, (∆op)2(F ) = 1⊗1⊗F +1⊗F ⊗K′ +F ⊗K′ ⊗K′, j j j j j j j so that F E = (1⊗F )(E ⊗1) j i j i cc = (SB′coop(Fj),Ei)(Kj′,1)1⊗Kj′ +(1,Ki)(Kj′,1)Ei ⊗Fj +(1,Ki)(Fj,Ei)Ki ⊗1 ′ = (−F (K′)−1,E )(K′,1)K +(1,K )(K′,1)E F +(1,K )(F ,E )K j j i j j i j i j i j i i δij ′ c δij cc c = − K +E F + K . j i j i v−1 −v v−1 −v That is, [E ,F ] = δicj (Kc′ −cK ). Applyicng ϕ gives the desired relation (R3) in U (sl ). i j v−1−v j i v,t n (cid:3) c c c c Let U0 denote the subalgebra of U (sl ) generated by K±1, K′±1, 1 ≤ i < n. Let U+ v,t n i i (resp. U−) denote the subalgebra of B (resp. B′) generated by E (resp. F ), 1 ≤ i < n. i i Then we have D(B,B′) ∼= U+ ⊗U0 ⊗U−. Corollary 1. The algebra U (sl ) has a triangular decomposition v,t n U (sl ) ∼= U− ⊗U0 ⊗U+. v,t n 6 YANMINYANG,HAITAOMA, ZHU-JUNZHENG 3. Finite Dimensional Representations Of U (sl ) v,t n 3.1. The Natural Representation Of U (sl ). Set Λ = Zǫ ⊕Zǫ ⊕···⊕Zǫ . For any v,t n 1 2 n n λ = λ ǫ ∈ Λ, one defines the algebra homomorphism λ : U0 → Q(v,t) by j j j=1 P b n n n n P λji·j P λj(hi,ji−hj,ii)−1 − P λji·j P λj(hi,ji−hj,ii)−1 (5) λ(K ) = vj=1 tj=1 , λ(K′) = v j=1 tj=1 , i i b 0, 1 ≤ i l , then 1 2 k i i+1 (R )2(v ⊗···⊗v ) = R (v ⊗···⊗v ⊗(v ⊗v +(v−1 −v)tv ⊗v )⊗···⊗v ) i l1 lk i l1 li−1 li+1 li li li+1 lk = t2(v ⊗···⊗v )+(v−1 −v)tR (v ⊗···⊗v ) l1 lk i l1 lk = (t2 +(v−1 −v)tR )(v ⊗···⊗v ). i l1 lk If l = l , we have i i+1 (t2 +(v−1 −v)tR )(v ⊗···⊗v ) = (t2 +(v−1 −v)v−1t2)(v ⊗···⊗v ) i l1 lk l1 lk = v−2t2(v ⊗···⊗v ) l1 lk = (R )2(v ⊗···⊗v ). i l1 lk SCHUR-WEYL DUALITY FOR Uv,t(sln) 9 For the last case l < l , we have i i+1 (R )2(v ⊗···⊗v ) = t2R (v ⊗···⊗v ⊗v ⊗v ⊗···⊗v ) i l1 lk i l1 li−1 li+1 li lk = t2(v ⊗···⊗v )+(v−1 −v)t3(v ⊗···⊗v ⊗v ⊗v ⊗···⊗v ) l1 lk l1 li−1 li+1 li lk = (t2 +(v−1 −v)tR )(v ⊗···⊗v ). i l1 lk (cid:3) This leads to the Schur-Weyl duality between U (sl ) and H (v,t). v,t n k Theorem 1. Assume v2 is not a root of unity. Then (1) δ (H (v,t)) = End (V⊗k); n k Uv,t(sln) n (2) for n ≥ k, we have End (V⊗k) ∼= H (v,t). Uv,t(sln) n k Proof. For conclusion (1), the proof is similarly to [3]. We consider the conclusion (2). Assume f ∈ End (V⊗k), v = v ⊗v ⊗···⊗v ∈ V⊗k, then f(v) must be the linear Uv,t(sln) n 1 2 k combinationsofv ⊗v ⊗···⊗v forsomeσ ∈ S . Wewillshowthatthereisanelement σ(1) σ(2) σ(k) k Tσ ∈ H (v,t) such that Rσ := δ (Tσ) in End (V⊗k), Rσ(v) = v ⊗v ⊗···⊗v . k n Uv,t(sln) n σ(1) σ(2) σ(k) For any element σ in S , σ can be written as a product of transpositions, denoted by k σ = τ ···τ , where τ = (i i +1). For distinct index j ,··· ,j , we set i1 im il l l i k ((v −v−1)tId+R )(v ⊗···⊗v ), if j > j ; Rτil(vj1 ⊗···⊗vjk) = (cid:26) t−2Ril(vj1 ⊗···⊗ivljk),j1 jk if jiill < jiill++11. Then defining Rσ = Rτim ◦ ···◦ Rτi1 in EndUv,t(sln)(Vn⊗k), it can be checked that Rσ(v) = v ⊗v ⊗···⊗v . Therefore, the map δ : H (v,t) → End (V⊗k) is a surjective, σ(1) σ(2) σ(k) n k Uv,t(sln) n and End (V⊗k) = span {Rσ|σ ∈ S }. Uv,t(sln) n Q(v,t) k Consequently, dim End (V⊗k) = k! = dim H (v,t). It follows that (2) holds. Q(v,t) Uv,t(sln) n Q(v,t) k (cid:3) Corollary 2. Assume v2 is not a root of unity. The space V⊗m as an U (sl )⊗H (v,t)- v,t n k module has the decomposition V⊗k ∼= V ⊗Vλ, λ M λ where the partition λ of k runs over the set of partitions such that l(λ) ≤ n, V is the λ U (sl )-module associated to λ, Vλ is the H (v,t)-module corresponding to λ. v,t n k 5. The Primitive Orthogonal Idempotents Of H (v,t) k For any i = 1,2,k −1, let s = (i,i+1) be the transposition in the symmetric group S . i k Choose a reduced decomposition w = s ···s for w ∈ S , denote T = T ···T . Then i1 il k w i1 il T does not depend on the reduced decomposition, and the set {T | w ∈ S } is a basis of w w k H (v,t) over Q(v,t). k The Jucys-Murphy elements y ,··· ,y of H (v,t) are defined inductively by 1 k k (9) y = 1, y = t−2T y T for i = 1,··· ,k−1. 1 i+1 i i i 10 YANMINYANG,HAITAOMA, ZHU-JUNZHENG These elements satisfy y T = T y for l 6= i, i−1. i l l i Furthermore, the elements y can be written as follows: i y = 1+(v−1 −v)t−1(T +T +···+T ), i (1 i) (2 i) (i−1 i) where T belong to H (v,t) associated to the transposition (m n) ∈ S . In particular, (m n) k k y ,··· ,y generate a commutative subalgebra of H (v,t). 1 k k Foranyi = 1,··· ,k, weletw denotetheuniquelongestelementofthesymmetricgroupS i i which is regarded as the natural subgroup of S . The corresponding elements T ∈ H (v,t) k wi k are then given by T = 1 and w1 T = T (T T )···(T T ···T )(T T ···T ) (10) wi 1 2 1 i−2 i−3 1 i−1 i−2 1 = (T ···T T )(T ···T T )···(T T )T , i = 2,··· ,k. 1 i−2 i−1 1 i−3 i−2 1 2 1 It is easily check that T T = T T , 1 ≤ j < i ≤ k, wi j i−j wi (11) T2 = t2(k−1)y y ···y , i = 1,··· ,k. wi 1 2 i Following [16], for any i = 1,··· ,k, we define the elements: (v−1 −v)x (12) T (x,y) = t−1T + , i i y −x where x and y are complex variables. We will regard the T (x,y) as rational functions in x i and y with values in H (v,t). These functions satisfy the braid relations: k (13) T (x,y)T (x,z)T (y,z) = T (y,z)T (x,z)T (x,y), i i+1 i i+1 i i+1 and (x−v−2y)(x−v2y) (14) T (x,y)T (y,x) = . i i (x−y)2 Following [5], we will identify a partition λ = (λ ,...,λ ) of k with its Young diagram 1 l which is a left-justified array of rows of cells such that the first row contains λ cells, the 1 second row contains λ cells, etc. A cell τ outside λ is called addable to λ if the union of the 2 cell τ and λ is a Young diagram. A tableau T of shape λ is obtained by filling in the cells of the diagram bijectively with the numbers 1,··· ,k. A tableau T is called standard if its entries increase along the rows and down the columns. If a cell occupied by i occurs in row m and column n, its (v,v−1)- content σ will be defined as v−2(n−m). i In accordance to [5], a set of primitive orthogonal idempotents {Eλ} of H (v,t), pa- T k rameterized by partitions λ of k and standard tableaux T of shape λ can be constructed inductively as follows. If k = 1, set Eλ = 1. For k ≥ 2, one defines inductively that T (y −ρ )···(y −ρ ) (15) Eλ = Eµ k 1 k l , T U (σ −ρ )···(σ −ρ ) 1 l where U is the tableau of shape µ obtained form T by removing the cell α occupied by k, and ρ ,··· ,ρ are the (v,v−1)-contents of all the addable cells of µ except for α, while σ is 1 l the (v,v−1)-content of α.

Schur-Weyl duality for $U_{v,t}(sl_{n})$

Similar Schur-Weyl duality for $U_{v,t}(sl_{n})$