R matrix for generalized quantum group of type A
aa r X i v : . [ m a t h . QA ] J a n R MATRIX FOR GENERALIZED QUANTUM GROUP OF TYPE A JAE-HOON KWON AND JEONGWOO YU
Abstract.
The generalized quantum group U ( ǫ ) of type A is an affine analogue of quan-tum group associated to a general linear Lie superalgebra gl M | N . We prove that thereexists a unique R matrix on tensor product of fundamental type representations of U ( ǫ )for arbitrary parameter sequence ǫ corresponding to a non-conjugate Borel subalgebraof gl M | N . We give an explicit description of its spectral decomposition, and then as anapplication, construct a family of finite-dimensional irreducible U ( ǫ )-modules which havesubspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type A (1) M − or A (1) N − . Contents
1. Introduction 12. Generalized quantum group U ( ǫ ) of type A
33. Schur-Weyl duality and polynomial representations of ˚ U ( ǫ ) 84. R matrix for finite-dimensional U ( ǫ )-modules 145. Kirillov-Reshetikhin modules 22References 261. Introduction
A generalized quantum group U ( ǫ ) associated to ǫ = ( ǫ , . . . , ǫ n ) with ǫ i ∈ { , } is aHopf algebra introduced in [17], which appears in the study of solutions to the tetrahedronequation or the three-dimensional Yang-Baxter equation.The generalized quantum group U ( ǫ ) of type A is equal to the usual quantum affinealgebra of type A (1) n − , when ǫ is homogeneous, that is, ǫ i = ǫ j for all i = j . But it becomes amore interesting object when ǫ is non-homogeneous, which is closely related to the quantizedenveloping algebra associated to an affine Lie superalgebra [22], or which can be viewed asan affine analogue of the quantized enveloping algebra of the general linear Lie superalgebra gl M | N [21], where M and N are the numbers of 0 and 1 in ǫ , respectively. We remark thatthe subalgebra ˚ U ( ǫ ) of U ( ǫ ) associated to the Lie superalgebra gl M | N was also introduced Mathematics Subject Classification.
Key words and phrases. quantum group, crystal base, Lie superalgebra.J.-H.K. was supported by the National Research Foundation of Korea(NRF) grant funded by the Koreagovernment(MSIT) (No. 2019R1A2C108483311). in [7] independently, as symmetries appearing in the study of wave functions of quantummechanical systems [23].When the parameter ǫ is standard, that is, ǫ M | N = (0 M , N ), it is shown in [17] thatthere exists a unique R matrix on the tensor product of finite-dimensional U ( ǫ M | N )-modules W s,ǫ ( x ), which correspond to fundamental representations of type A (1) N − with spectral pa-rameter x when N ≥
3. Indeed, the R matrix is obtained by reducing the solution of thetetrahedron equation, and the uniqueness follows from the irreducibility of tensor product W l,ǫ ( x ) ⊗ W m,ǫ ( y ) for generic x and y . An explicit spectral decomposition of the associated R matrix is obtained by analyzing the maximal vectors with respect to ˚ U ( ǫ M | N ).By applying fusion construction using the R matrix in [17], a family of irreducible U ( ǫ M | N )-modules is constructed in [16], which are parametrized by rectangular partitionsinside an ( M | N )-hook. Moreover the existence of their crystal base is proved together witha combinatorial description of the associated crystal graphs. It can be viewed as a naturalsuper-analogue of Kirillov-Reshetikhin modules (simply KR modules) of type A (1) ℓ , which isa most important family of finite-dimensional irreducible modules of quantum affine algebras(cf. [5, 14]).The results in [17] and [16] suggests that there is a close connection between finite-dimensional representations of U ( ǫ M | N ) and U q ( A (1) n − ). The purpose of this paper is toextend the results in [17] and [16] to arbitrary parameter sequence ǫ , and find a moreconcrete connection between the finite-dimensional representations of U ( ǫ ) and U q ( A (1) ℓ ).From a viewpoint of representations of gl M | N , the sequence ǫ represents the type of Borelsubalgebras of gl M | N , which are not conjugate to each other. It is not obvious whether therepresentation theory of U ( ǫ ) is the same under a different choice of permutations of ǫ M | N .For example, if we change the Borel in the generalized quantum group, then the definingrelations and the crystal structure associated to U ( ǫ )-modules become much different fromthe ones with respect to ǫ M | N as ǫ gets far from ǫ M | N (cf. [2, 15]).We first show that there exists a unique R matrix on the tensor product of finite-dimensional U ( ǫ )-modules W s,ǫ ( x ) of fundamental type (Theorem 4.10). Since the existenceof R matrix for arbitrary ǫ was shown in [17], it suffices to prove the irreducibility of ten-sor product W l,ǫ ( x ) ⊗ W m,ǫ ( y ) for generic x and y . We use a method completely differentfrom [17]. Indeed, motivated by the work [6], we introduce a functor called truncation, andshow that it sends any U ( ǫ )-module with polynomial weights to a U ( ǫ ′ )-module, preservingthe comultiplications in tensor product, where ǫ ′ is a subsequence of ǫ . This in particularenables us to define an oriented graph structure on W l,ǫ ( x ) ⊗ W m,ǫ ( y ) when x = y = 1 withadditional arrows other than the ones associated to U ( ǫ ). With this structure, we prove theconnectedness of the crystal (Theorem 4.8), and hence the irreducibility for generic x and y . Next, we prove that the truncation functor is compatible with the R matrix. This imme-diately implies that the spectral decomposition of the R matrix for U ( ǫ ) is the same as thatof type A (1) ℓ (Theorem 5.2) and hence does not depend on the choice of ǫ . As an application,we construct a family of irreducible U ( ǫ )-modules W ( r ) s,ǫ which yields the usual KR modules under truncation (Theorem 5.3). We conjecture that W ( r ) s,ǫ has a crystal base as in the caseof ǫ = ǫ M | N . We expect that the compatibility of truncation with the R matrix will alsoplay a crucial role in understanding arbitrary finite-dimensional U ( ǫ )-modules in connectionwith those of type A (1) ℓ .There are other recent works on the finite-dimensional representations of quantum affine superalgebra associated to gl M | N [24, 25, 26]. It would be interesting to compare with theseresults.The paper is organized as follows. In Section 2, we review basic materials for a generalizedquantum group and its crystal base. In Section 3, we present the classical Schur-Weyl dualityfor ˚ U ( ǫ ) and then realize the irreducible polynomial representation of ˚ U ( ǫ ). In Section 4, weprove the main theorem on the existence of the R matrix. In Section 5, we construct KRtype modules of U ( ǫ ) using the R matrix. Acknowledgement
The authors would like to thank Euiyong Park and Masato Okadofor helpful discussions and thank Shin-Myung Lee for careful reading of the manuscript andcomments. 2.
Generalized quantum group U ( ǫ ) of type A Generalized quantum group.
We fix a positive integer n ≥
4. Let ǫ = ( ǫ , · · · , ǫ n )be a sequence with ǫ i ∈ { , } for 1 ≤ i ≤ n . We denote by I the linearly ordered set { < < · · · < n } with Z -grading given by I = { i | ǫ i = 0 } and I = { i | ǫ i = 1 } . Weassume that M is the number of i with ǫ i = 0 and N is the number of i with ǫ i = 1 in ǫ .We denote by ǫ M | N the sequence when ǫ = · · · = ǫ M = 0 and ǫ M +1 = · · · = ǫ n = 1.Let P = L i ∈ I Z δ i be the free abelian group generated by δ i with a symmetric bilinearform ( · | · ) given by ( δ i | δ j ) = ( − ǫ i δ ij for i, j ∈ I . Let { δ ∨ i | i ∈ I } ⊂ P ∨ := Hom Z ( P, Z ) bethe dual basis such that h δ i , δ ∨ j i = δ ij for i, j ∈ I .Let I = { , , . . . , n − } and α i = δ i − δ i +1 , α ∨ i = δ ∨ i − ( − ǫ i + ǫ i +1 δ ∨ i +1 ( i ∈ I ) . Throughout the paper, we understand the subscript i ∈ I modulo n . When ǫ = ǫ M | N , theDynkin diagram associated to the Cartan matrix ( h α j , α ∨ i i ) ≤ i,j ≤ n is (cid:13) · · · (cid:13) (cid:13) · · · NN (cid:13) α α M α α n − α M − α M +1 where N denotes an isotropic simple root. JAE-HOON KWON AND JEONGWOO YU
Let q be an indeterminate. We put I even = { i ∈ I | ( α i | α i ) = ± } and I odd = { i ∈ I | ( α i | α i ) = 0 } , and set q i = ( − ǫ i q ( − ǫi = q if ǫ i = 0 , − q − if ǫ i = 1 , ( i ∈ I ) . Definition 2.1.
We define U ( ǫ ) to be the associative Q ( q )-algebra with 1 generated by q h , e i , f i for h ∈ P ∨ and i ∈ I satisfying q = 1 , q h + h ′ = q h q h ′ ( h, h ′ ∈ P ∨ ) , (2.1) ω j e i ω − j = q h α i ,δ ∨ j i j e i , ω j f i ω − j = q −h α i ,δ ∨ j i j f i , (2.2) e i f j − f j e i = δ ij ω i ω − i +1 − ω − i ω i +1 q − q − , (2.3) e i = f i = 0 ( i ∈ I odd ) , (2.4)where ω j = q ( − ǫi δ ∨ j ( j ∈ I ), and the Serre-type relations e i e j − e j e i = f i f j − f j f i = 0 , ( | i − j | > ,e i e j − ( − ǫ i [2] e i e j e i + e j e i = 0 ,f i f j − ( − ǫ i [2] f i f j f i + f j f i = 0 , ( i ∈ I even and | i − j | = 1) , (2.5)and(2.6) e i e i − e i e i +1 − e i e i +1 e i e i − + e i +1 e i e i − e i − e i − e i e i +1 e i + ( − ǫ i [2] e i e i − e i +1 e i = 0 ,f i f i − f i f i +1 − f i f i +1 f i f i − + f i +1 f i f i − f i − f i − f i f i +1 f i + ( − ǫ i [2] f i f i − f i +1 f i = 0 , ( i ∈ I odd ) . We call U ( ǫ ) the generalized quantum group of affine type A associated to ǫ (see [17]).Put k i = ω i ω − i +1 for i ∈ I . Then we have for i, j ∈ Ik i e j k − i = D ij e j , k i f j k − i = D − ij f j , e i f j − f j e i = δ ij k i − k − i q − q − , where D ij = q h α j ,δ ∨ i i i q −h α j ,δ ∨ i +1 i i +1 . There is a Hopf algebra structure on U ( ǫ ), where thecomultiplication ∆, the antipode S , and the couint ε are given by∆( q h ) = q h ⊗ q h , ∆( e i ) = e i ⊗ k − i ⊗ e i , ∆( f i ) = f i ⊗ k i + 1 ⊗ f i , (2.7) S ( q h ) = q − h , S ( e i ) = − e i k − i , S ( f i ) = − k i f i ,ε ( q h ) = 1 , ε ( e i ) = ε ( f i ) = 0 , for h ∈ P ∨ and i ∈ I . Let η be the anti-automorphism on U ( ǫ ) defined by η ( q h ) = q h , η ( e i ) = q i f i k − i , η ( f i ) = q − i k i e i , for h ∈ P ∨ and i ∈ I . It satisfies η = id and∆ ◦ η = ( η ⊗ η ) ◦ ∆ . We have an isomorphism between U ( ǫ ) and U ( e ǫ ) where e ǫ is obtained from ǫ by permutationof ǫ i ’s, which is not an isomorphism of Hopf algebras [20, Theorem 2.7] (cf. [19, 37.1]). Theorem 2.2.
For ≤ i ≤ n − , let e ǫ = ( e ǫ , . . . , e ǫ n ) be the sequence given by exchanging ǫ i and ǫ i +1 in ǫ . Then there exists an isomorphism of algebras τ i : U ( ǫ ) −→ U ( e ǫ ) given by τ i ( k i ) = k − i , τ i ( e i ) = − f i k i , τ i ( f i ) = − k − i e i ,τ i ( k j ) = k i k j , τ i ( e j ) = [ e i , e j ] D ij , τ i ( f j ) = [ f j , f i ] D − ij ( | i − j | = 1) ,τ i ( k j ) = k j , τ i ( e j ) = e j , τ i ( f j ) = f j ( | i − j | > , where the inverse map is given by τ − i ( k i ) = k − i , τ − i ( e i ) = − k − i f i , τ − i ( f i ) = − e i k i ,τ − i ( k j ) = k i k j , τ − i ( e j ) = [ e j , e i ] D ij , τ − i ( f j ) = [ f i , f j ] D − ij ( | i − j | = 1) ,τ − i ( k j ) = k j , τ − i ( e j ) = e j , τ − i ( f j ) = f j ( | i − j | > . (cid:3) Crystal base of U ( ǫ ) -modules. For a U ( ǫ )-module V and µ = P i µ i δ i ∈ P , let V µ = { u ∈ V | ω i u = q µ i i u ( i ∈ I ) } be the µ -weight space of V . For a non-zero vector u ∈ V µ , we denote by wt( u ) = µ theweight of u . Let P ≥ = P i ∈ I Z ≥ δ i and let O ≥ be the category of U ( ǫ )-modules withobjects V such that(2.8) V = M µ ∈ P ≥ V µ with dim V µ < ∞ . which is closed under taking submodules, quotients and tensor products. Remark 2.3.
There is another comultiplication on U ( ǫ ) given by∆ + ( q h ) = q h ⊗ q h , ∆ + ( e i ) = 1 ⊗ e i + e i ⊗ k i , ∆ + ( f i ) = k − i ⊗ f i + f i ⊗ , (2.9)(while ∆ op+ is used in [16]). Let ⊗ and ⊗ + denote the tensor product with respect to ∆and ∆ + , respectively. For U ( ǫ )-modules M and N , we have a U ( ǫ )-linear isomorphism ψ : M ⊗ N −→ M ⊗ + N given by(2.10) ψ ( u ⊗ v ) = Y i ∈ I q µ i ν i i ! u ⊗ v, for u ∈ M µ and v ∈ N ν with µ = P i µ i δ i and ν = P i ν i δ i . JAE-HOON KWON AND JEONGWOO YU
Let us recall the notion of crystal base for V ∈ O ≥ [16] (cf. [2]). The Kashiwara operators˜ e i and ˜ f i on V for i ∈ I are defined as follows. Suppose that u ∈ V µ is given. Case 1 . Suppose that i ∈ I odd and ( ǫ i , ǫ i +1 ) = (0 , e i u = η ( f i ) u = q − i k i e i u, ˜ f i u = f i u. Case 2.
Suppose that i ∈ I odd and ( ǫ i , ǫ i +1 ) = (1 , e i u = e i u, ˜ f i u = η ( e i ) u = q i f i k − i u. Case 3.
Suppose that i ∈ I even and ( ǫ i , ǫ i +1 ) = (0 , ζ : U q ( sl ) → U ( ǫ ) i be the Q ( q )-algebra isomorphism given by ζ ( e ) = e i , ζ ( f ) = f i and ζ ( k ) = k i , where U q ( sl ) = h e, f, k ± i is the usual quantum group for sl with relation kek − = q e , kf k − = q − f , ef − f e = k − k − q − q − . The induced comultiplication ∆ ζ := ( ζ − ⊗ ζ − ) ◦ ∆ ◦ ζ on U q ( sl ) is∆ ζ ( k ± ) = k ± ⊗ k ± , ∆ ζ ( e ) = k − ⊗ e + e ⊗ , ∆ ζ ( f ) = 1 ⊗ f + f ⊗ k. So we define ˜ e i and ˜ f i on V to be the usual Kashiwara operators on the lower crystal baseof U q ( sl )-module induced from ζ . In other words, if u = P k ≥ f ( k ) i u k , where f ( k ) i = f ki / [ k ]!and e i u k = 0 for k ≥
0, then we define˜ e i u = X k ≥ f ( k − i u k , ˜ f i u = X k ≥ f ( k +1) i u k . Case 4.
Suppose that i ∈ I even and ( ǫ i , ǫ i +1 ) = (1 , ξ : U q ( sl ) → U ( ǫ ) i be the Q ( q )-algebra homomorphism given by ξ ( e ) = − e i , ξ ( f ) = f i and ξ ( k ) = k − i . Then theinduced comultiplication ∆ ξ on U q ( sl ) is∆ ξ ( k ± ) = k ± ⊗ k ± , ∆ ξ ( e ) = k ⊗ e + e ⊗ , ∆ ξ ( f ) = 1 ⊗ f + f ⊗ k − . So we define ˜ e i and ˜ f i on V to be the Kashiwara operators on the upper crystal base of U q ( sl )-module induced from ξ . In other words, if u = P k ≥ f ( k ) i u k , where e i u k = 0 for k ≥ l k = h wt( u k ) , α ∨ i i , then we define˜ e i u = X k ≥ q − l k +2 k − f ( k − i u k , ˜ f i u = X k ≥ q l k − k − f ( k +1) i u k . Let A be the subring of Q ( q ) consisting of f ( q ) /g ( q ) with f ( q ) , g ( q ) ∈ Q [ q ] and g (0) = 0. Definition 2.4.
Let V ∈ O ≥ be given. A pair ( L, B ) is a crystal base of V it if satisfiesthe following conditions:(1) L is an A -lattice of V and L = L µ ∈ P ≥ L µ , where L µ = L ∩ V µ ,(2) B is a signed basis of L/qL , that is B = B ∪ − B where B is a Q -basis of L/qL ,(3) B = F µ ∈ P ≥ B µ where B µ ⊂ ( L/qL ) µ ,(4) ˜ e i L ⊂ L, ˜ f i L ⊂ L and ˜ e i B ⊂ B ∪ { } , ˜ f i B ⊂ B ∪ { } for i ∈ I , (5) ˜ f i b = b ′ if and only if ˜ e i b ′ = ± b for i ∈ I and b, b ′ ∈ B .Let us call B/ {± } a crystal of V , which is an I -colored oriented graph. We have a tensorproduct rule for crystals (see [2] and [16, Proposition 3.4]). Proposition 2.5.
Let V , V ∈ O ≥ be given. Suppose that ( L i , B i ) is a crystal base of V i for i = 1 , . Then ( L ⊗ L , B ⊗ B ) is a crystal base of V ⊗ V , where B ⊗ B ⊂ ( L /qL ) ⊗ ( L /qL ) = L ⊗ L /qL ⊗ L . Moreover, e e i and e f i act on B ⊗ B as follows: (1) if i ∈ I odd and ( ǫ i , ǫ i +1 ) = (0 , , then e e i ( b ⊗ b ) = b ⊗ e e i b , if h wt( b ) , α ∨ i i > , e e i b ⊗ b , if h wt( b ) , α ∨ i i = 0 , e f i ( b ⊗ b ) = b ⊗ e f i b , if h wt( b ) , α ∨ i i > , e f i b ⊗ b , if h wt( b ) , α ∨ i i = 0 , (2.11)(2) if i ∈ I odd and ( ǫ i , ǫ i +1 ) = (1 , , then e e i ( b ⊗ b ) = b ⊗ e e i b , if h wt( b ) , α ∨ i i = 0 , e e i b ⊗ b , if h wt( b ) , α ∨ i i > , e f i ( b ⊗ b ) = b ⊗ e f i b , if h wt( b ) , α ∨ i i = 0 , e f i b ⊗ b , if h wt( b ) , α ∨ i i > , (2.12)(3) if i ∈ I even and ( ǫ i , ǫ i +1 ) = (0 , , then e e i ( b ⊗ b ) = b ⊗ e e i b , if ϕ i ( b ) ≥ ε i ( b ) , e e i b ⊗ b , if ϕ i ( b ) < ε i ( b ) , e f i ( b ⊗ b ) = b ⊗ e f i b , if ϕ i ( b ) > ε i ( b ) , e f i b ⊗ b , if ϕ i ( b ) ≤ ε i ( b ) , (2.13)(4) if i ∈ I even and ( ǫ i , ǫ i +1 ) = (1 , , then e e i ( b ⊗ b ) = e e i b ⊗ b , if ϕ i ( b ) ≥ ε i ( b ) ,b ⊗ σ i e e i b , if ϕ i ( b ) < ε i ( b ) , e f i ( b ⊗ b ) = e f i b ⊗ b , if ϕ i ( b ) > ε i ( b ) ,b ⊗ σ i e f i b , if ϕ i ( b ) ≤ ε i ( b ) , (2.14) where σ i = ( − (wt( b ) ,α i ) . Proof.
The proof is almost the same as in [16, Proposition 3.4], where the order of tensorproduct is reversed due to a different convention of comultiplication. (cid:3)
Remark 2.6.
Let V = L i ∈ I Q ( q ) v i denote the U ( ǫ )-module, where(2.15) ω i v j = q δ ij i v j , e k v j = δ k j − v j − , f k v j = δ kj v j +1 , JAE-HOON KWON AND JEONGWOO YU for i, j ∈ I and k ∈ I . It is clear that the pair L = L i ∈ I A v i and B = { ± v i (mod q L ) | i ∈ I } is a crystal base of V . The crystal structure on B ⊗ ℓ / {± } for ℓ ≥ Schur-Weyl duality and polynomial representations of ˚ U ( ǫ )3.1. Schur-Weyl duality.
Put ˚ I = I \ { } . Let ˚ U ( ǫ ) be the Q ( q )-subalgebra of U ( ǫ )generated by q h and e i , f i for h ∈ P ∨ and i ∈ ˚ I .Let us consider V = L i ∈ I Q ( q ) v i in (2.15) as a ˚ U ( ǫ )-module. Fix ℓ ≥
2. Let Φ ℓ : ˚ U ( ǫ ) −→ End Q ( q ) ( V ⊗ ℓ ) denote the action of ˚ U ( ǫ ) on V ⊗ ℓ via (2.7). Note that V ⊗ ℓ is semisimple (see[16, Corollary 4.1]).Assume that ǫ = 0. We have a ˚ U ( ǫ )-linear map R : V ⊗ −→ V ⊗ given by R ( v i ⊗ v j ) = q − q − i v i ⊗ v j , if i = j,q − v j ⊗ v i , if i > j, ( q − − v i ⊗ v j + q − v j ⊗ v i , if i < j, (3.1)satisfying the Yang-Baxter equation; R R R = R R R , where R ij denotes the map acting as R on the i -th and the j -th component and the identityelsewhere on V ⊗ (cf. [11]).Let H ℓ ( q − ) be the Iwahori-Hecke algebra of type A over Q ( q ) generated by h i for i ∈{ , . . . , ℓ − } subject to the relations;( h i − q − )( h i + 1) = 0 ,h i h j = h j h i , ( | i − j | > ,h i h j h i = h j h i h j ( | i − j | = 1) , for i, j ∈ { , . . . , ℓ − } . Let W be the symmetric group on { , . . . , ℓ } and s i = ( i i + 1) bethe transposition for 1 ≤ i ≤ ℓ −
1. For w ∈ W , ℓ ( w ) denote the length of w and let h ( w )be the element in H ℓ ( q − ) associated to w such that h ( s i ) = h i for 1 ≤ i ≤ ℓ − H ℓ ( q − ) on V ⊗ ℓ , say, Ψ ℓ : H ℓ ( q − ) −→ End Q ( q ) ( V ⊗ ℓ ), where Ψ ℓ ( h i ) acts as R on the i -th and ( i + 1)-th compo-nent and the identity elsewhere. Then we have an analogue of Schur-Weyl duality for ˚ U ( ǫ )(cf. [11]) as follows. The proof is similar to the case when ǫ i = 0 for all i . Theorem 3.1.
We have
End H ℓ ( q − ) ( V ⊗ ℓ ) = Φ ℓ (˚ U ( ǫ )) , End ˚ U ( ǫ ) ( V ⊗ ℓ ) = Ψ ℓ ( H ℓ ( q − )) . (cid:3) Polynomial representations of ˚ U ( ǫ ) . Recall that M is the number of i ’s with ǫ i = 0and N is the number of i ’s with ǫ i = 1 in ǫ .Let P be the set of all partitions. A partition λ = ( λ i ) i ≥ ∈ P is called an ( M | N )-hook partition if λ M +1 ≤ N (cf. [4]). We denote the set of all ( M | N )-hook partitions by P M | N . For a Young diagram λ , a tableau T obtained by filling λ with letters in I is calledsemistandard if (1) the letters in each row (resp. column) are weakly increasing from leftto right (resp. from top to bottom), (2) the letters in I (resp. I ) are strictly increasingin each column (resp. row). Let SST ǫ ( λ ) be the set of all semistandard tableaux of shape λ . Then SST ǫ ( λ ) is non-empty if and only if λ ∈ P M | N . For T ∈ SST ǫ ( λ ), let w ( T ) bethe word given by reading the entries in T column by column from left to right, and frombottom to top in each column.For T ∈ SST ǫ (1 r ) with r ≥
1, let d ( T ) = P u 2, and let W denote the symmetric group on { , . . . , ℓ } . Suppose that λ ∈ P isgiven with P i ≥ λ i = ℓ . Let T λ + be the standard tableau obtained by filling λ with { , . . . , ℓ } row by row from top to bottom and from left to right in each row, and let T λ − be the tableauobtained by filling λ with { , . . . , ℓ } column by column from left to right and from top tobottom in each column.Let w λ ∈ W be such that w λ ( T λ − ) = T λ + , where w λ ( T λ − ) is the tableau obtained by acting w λ on the letters in T λ − . Let W λ + (resp. W λ − ) be the Young subgroup of W stabilizing therows (resp. columns) of T λ + (resp. T λ − ). Then the q -deformed Young symmetrizer is givenby(3.2) Y λ ( q ) = h ( w − λ ) e λ + h ( w λ ) e λ − , ([10]) where e λ + = X w ∈ W λ + h ( w ) , e λ − = X w ∈ W λ − ( − q ) − ℓ ( w ) h ( w ) . For 1 ≤ u < v ≤ ℓ , let W uv = h s i | u ≤ i ≤ v − i . Suppose that a is a letter in T λ − suchthat a + 1 is located in the same column. We put C a = 1 + h a . Then we have Y λ ( q ) C a = 0 , (3.3)Next, suppose that a is a letter in T λ − , where there is another letter d to the right. Let b bethe letter at the bottom of column where a is placed, and c = b + 1 the letter at the top ofthe column where d is placed. Let G λa be the set of minimal length right coset representativesof W ab × W cd in W ad . We define the Garnir element at a to be(3.4) G λa = X w ∈G λa ( − q ) ℓ ( w ) h ( w ) . The collection of boxes in the Young diagram λ corresponding to the letters from a to d in T λ − is called a Garnir belt at a . Then we have the following relations [3, (15)]; Y λ ( q ) G λa = 0 . (3.5)Let T be a tableau of shape λ with letters in I , and let T ( i ) be the letter in T at theposition corresponding to i in T λ − for 1 ≤ i ≤ ℓ . Let v T = Y λ ( q ) (cid:0) v T (1) ⊗ . . . ⊗ v T ( ℓ ) (cid:1) . For σ ∈ W , let T σ be the tableau given by replacing T ( i ) with T ( σ ( i )) for 1 ≤ i ≤ ℓ .Let a be a letter in T λ − with d to the right in the same row and with b, c as above. Let w bethe longest element in W ab × W cd , and let ˜ G λa = w G λa w . Let u , . . . , u s and u s +1 , . . . , u r + s be the letters in T corresponding to c, . . . , d and a, . . . , b in T λ − , respectively. Then we mayidentify σ ∈ ˜ G λa with a permutation on { , . . . , r + s } satisfying σ (1) < · · · < σ ( s ) and σ ( s + 1) < · · · < σ ( s + r ) so that T σ is the tableau obtained from T by replacing u i ’swith u σ ( i ) ’s for 1 ≤ i ≤ r + s . With this identification, we let ˜ ℓ ( σ ) be the length of σ as apermutation on { , . . . , r + s } , and put X σ = { i | ≤ i ≤ s, s + 1 ≤ σ − ( i ) ≤ s + r } ,Y σ = { j | s + 1 ≤ j ≤ s + r, ≤ σ − ( j ) ≤ r } . Lemma 3.2. Suppose that T is column-semistandard such that either T ( a ) = T ( d ) ∈ I or T ( a ) > T ( d ) . Then under the above hypothesis, we have v T = − X σ ∈ ˜ G λa ,σ =1 ( − q ) ˜ ℓ ( σ )+ m ( σ,T ) v T σ , where m ( σ, T ) = − (cid:12)(cid:12) { ( i, j ) | ≤ i < j ≤ s, i / ∈ X σ , j ∈ X σ , u i = u j } (cid:12)(cid:12) − (cid:12)(cid:12) { ( k, l ) | s + 1 ≤ k < l ≤ s + r, k ∈ Y σ , l / ∈ Y σ , u k = u l } (cid:12)(cid:12) + (cid:12)(cid:12) { ( x, y ) | ≤ x ≤ s, s + 1 ≤ y ≤ s + r, x ∈ X σ or y ∈ Y σ , u x = u y } (cid:12)(cid:12) . Proof. We have v T = Y λ ( q ) v , where v = (cid:0) v T (1) ⊗ . . . ⊗ v T ( ℓ ) (cid:1) . Following the abovenotations, we have v = v ′ ⊗ v u s ⊗ · · · ⊗ v u r + s ⊗ v u ⊗ · · · v u s ⊗ v ′′ . Note that v T w = Y λ ( q ) (cid:0) v ′ ⊗ v u r + s ⊗ · · · ⊗ v u s ⊗ v u s ⊗ · · · v u ⊗ v ′′ (cid:1) , where u r + s ≥ . . . u s = T ( a ) ≥ u s = T ( d ) ≥ · · · ≥ u .For w ∈ G λa , we have by (3.1) and (3.3) h ( w ) (cid:0) v ′ ⊗ v u r + s ⊗ · · · ⊗ v u s ⊗ v u s ⊗ · · · v u ⊗ v ′′ (cid:1) = q − ℓ ( w ) ( − q ) m ( σ,T w ) Ä v ′ ⊗ v u σ ( r + s ) ⊗ · · · ⊗ v u σ (1+ s ) ⊗ v u σ ( s ) ⊗ · · · ⊗ v u σ (1) ⊗ v ′′ ä , (3.6)where σ is the permutation on { , . . . , r + s } corresponding to w ww and m ( σ, T w ) = (cid:12)(cid:12) { ( i, j ) | i < j, σ − ( i ) < σ − ( j ) , u i = u j } (cid:12)(cid:12) . Hence it follows from (3.5) and that (3.6)0 = Y λ ( q ) G λa ( q ) (cid:0) v ′ ⊗ v u r + s ⊗ · · · ⊗ v u s ⊗ v u s ⊗ · · · v u ⊗ v ′′ (cid:1) = Y λ ( q ) X w ∈G λa ( − q ) ℓ ( w ) h ( w ) (cid:0) v ′ ⊗ v u r + s ⊗ · · · ⊗ v u s ⊗ v u s ⊗ · · · v u ⊗ v ′′ (cid:1) = Y λ ( q ) X w ∈G λa ( − q ) ℓ ( w )+ m ( σ,T w ) Ä v ′ ⊗ v u σ ( r + s ) ⊗ · · · ⊗ v u σ (1+ s ) ⊗ v u σ ( s ) ⊗ · · · v u σ (1) ⊗ v ′′ ä = X w ∈G λa ( − q ) ℓ ( w )+ m ( σ,T w ) v T w w = X σ ∈ ˜ G λa ( − q ) ˜ ℓ ( σ )+ m ( σ,T w ) v T σw . We have(3.7) X σ ∈ ˜ G λa ( − q ) ˜ ℓ ( σ )+ m ( σ,T w ) v ( T σ ) w = 0 . For σ ∈ ˜ G λa , let U σ be the subtableau of T σ corresponding to the Garnir belt at a , where U = U id . We define d a ( T σ ) in the same way as in d ( T ) only by using the letters in U σ . Let l p > · · · > l ≥ r q > · · · > r be the distinct letters appearing in U , where l i and r j arelocated in the left and right columns of U , respectively.Let m i (resp. n j ) be the number of occurrences of l i ’s (resp. r j ’s) in U , which remainin the same column after applying σ . Let m ′ i (resp. n ′ i ) be the number of l i ’s (resp. r j ’s)which are placed on the right (resp. left) column of U σ after applying σ to U . Note that P i m ′ i = P j n ′ j Case 1 . Suppose that l = r q . We have d a ( T ) = X ≤ i Case 2 . Suppose that Suppose that l = r q . In this case, d a ( T ) is the same as in Case 1 ,and d a ( T ) − d a ( T σ ) = − X ≤ i ≤ p m i m ′ i − X ≤ j ≤ q n j n ′ j + m p n ′ + m ′ p n . Note that m ( σ, T w ) = (cid:12)(cid:12) { ( i, j ) | ≤ i < j ≤ s, i ∈ X σ , j X σ , u i = u j } (cid:12)(cid:12) + (cid:12)(cid:12) { ( k, l ) | s + 1 ≤ k < l ≤ s + r, k Y σ , l ∈ Y σ , u k = u l } (cid:12)(cid:12) + (cid:12)(cid:12) { ( x, y ) | ≤ x ≤ s, s + 1 ≤ y ≤ s + r, x ∈ X σ , y ∈ Y σ , u x = u y } (cid:12)(cid:12) , where the last summand is equal to m ′ p n ′ . By similar arguments as in (3.10), we have d a ( T ) − d a ( T σ ) = m ( σ, T ) − m ( σ, T w ) . This also proves the identity in the lemma as in (3.11). (cid:3) For λ ∈ P M | N with P i λ i = ℓ , let(3.12) V ǫ ( λ ) = X T ∈ SST ǫ ( λ ) Q ( q ) v T . Let H λ be the tableau in SST ǫ ( λ ), which is defined inductively as follows:(1) Fill the first row (resp. column) of λ with 1 if ǫ = 0 (resp. ǫ = 1).(2) Suppose that we have filled a subdiagram of λ from 1 to i . Then fill the first row(resp. column) of the remaining diagram with i + 1 if ǫ i +1 = 0 (resp. ǫ i +1 = 1). Example 3.3. Suppose that n = 5, ǫ = (0 , , , , 0) and λ = (6 , , , , H λ = 1 1 1 1 1 12 3 4 4 42 3 5 52 32 Proposition 3.4. We have the following. (1) V ǫ ( λ ) is a ˚ U ( ǫ ) -submodule of V ⊗ ℓ . (2) V ǫ ( λ ) is an irreducible ˚ U ( ǫ ) -module with basis { v T | T ∈ SST ǫ ( λ ) } . (3) v H λ is a highest weight vector in V ǫ ( λ ) . Proof. (1) It is clear that V ǫ ( λ ) is invariant under q h for h ∈ P ∨ . It suffices to check f i V ǫ ( λ ) ⊂ V ǫ ( λ ) for i ∈ ˚ I since the proof for e i is the same. The proof is similar to thecase when ǫ = (0 , . . . , 0) (cf. [9]). For column-semistandard tableaux U and V of shape λ , we define U < V if there exists 1 ≤ k ≤ ℓ such that U ( k ) < V ( k ) and U ( k ′ ) = V ( k ′ ) for k < k ′ ≤ ℓ .Suppose that T ∈ SST ǫ ( λ ) is given. By (2.7), f i v T is a Q ( q )-linear combination of v T ′ ’s, where we may assume that T is column-semistandard by (3.3). If such T ′ is notsemistandard, then we may apply Lemma 3.2 to T ′ so that v T ′ is a linear combinationof T ′′ ’s which is column-semistandard and T ′ < T ′′ . Repeating this process finitely manytimes, we conclude that f i T is a linear combination of v S ’s for some S ∈ SST ǫ ( λ ). Therefore,have f i V ǫ ( λ ) ⊂ V ǫ ( λ ).(2) Since V ǫ ( λ ) = Y λ ( q ) V ⊗ ℓ and Y λ ( q ) is a primitive idempotent up to scalar multiplica-tion [10], it follows from Theorem 3.1 that V ǫ ( λ ) is an irreducible ˚ U ( ǫ )-module. Recall thatthe dimension of the irreducible H ℓ ( q − )-module S λ generated by Y λ ( q ) is the number ofstandard tableaux of shape λ . We may have an analogue of the Robinson-Schensted typecorrespondence, which is a bijection from the set of words of length ℓ with letters in I to theset of pair of standard tableau and semistandard tableau of shape λ (cf. [4]). Comparingthe dimensions of V ⊗ ℓ and its decomposition into H ( q − ) ⊗ ˚ U ( ǫ )-module S λ ⊗ V ǫ ( λ ), weconclude that dim Q ( q ) V ǫ ( λ ) is equal to | SST ǫ ( λ ) | , and hence { v T | T ∈ SST ǫ ( λ ) } is a linearbasis of V ǫ ( λ ).(3) The character of V ǫ ( λ ) is equal to that of polynomial representations of the generallinear Lie superalgebra gl M | N corresponding to λ ∈ P M | N , and wt( v H λ ) is maximal [8,Theorem 2.55]. This implies that e i v H λ = 0 for all i ∈ ˚ I and hence v H λ is a highest weightvector. (cid:3) Remark 3.5. The character of V ǫ ( λ ) for λ ∈ P M | N is called a hook Schur polynomial[4], which depends only on ǫ up to permutations. The tensor product of two polynomialrepresentations is completely reducible and the multiplicity of each irreducible componentis given by usual Littlewood-Richardson coefficient.3.3. Crystal base of V ǫ ( λ ) . Let λ ∈ P M | N be given. We may define an ˚ I -colored orientedgraph structure by identifying T with w ( T ) rev , the reverse word of w ( T ).Let L ǫ ( λ ) = M T ∈ SST ǫ ( λ ) A v ∗ T ,B ǫ ( λ ) = { ± v ∗ T (mod qL ǫ ( λ )) | T ∈ SST ǫ ( λ ) } , (3.13)where v ∗ T = q − d ( T ) v T for T ∈ SST ǫ ( λ ). Lemma 3.6. When λ = (1 r ) or ( r ) for r ≥ , ( L ǫ ( λ ) , B ǫ ( λ )) is a crystal base of V ǫ ( λ ) , andthe crystal B ǫ ( λ ) / {± } is isomorphic to SST ǫ ( λ ) . Proof. The proof is similar to that of [16, Proposition 3.3]. (cid:3) Proposition 3.7. Suppose that ǫ = ǫ M | N . For λ ∈ P M | N , ( L ǫ ( λ ) , B ǫ ( λ )) is a crystal baseof V ǫ ( λ ) . Proof. The proof is similar to that of [18, Theorem 4.4]. Let ( L ( λ ) , B ( λ )) be given by L ( λ ) = X r ≥ , i ,...,i r ∈ ˚ I A e x i · · · e x i r v λ , B ( λ ) = { ± e x i · · · e x i r v λ mod qL ( λ ) | r ≥ , i , . . . , i r ∈ ˚ I } \ { } , where v λ is a highest weight vector in V ǫ ( λ ) and x = e, f for each i k . Following the samearguments in [2], it is shown in [16] that ( L ( λ ) , B ( λ )) is a crystal base of V ǫ ( λ ). The crystal B ( λ ) / {± } is equal to SST ǫ ( λ ) which is connected.Let µ = ( µ , . . . , µ r ) = λ ′ be the conjugate partition of λ , and V µǫ = V ǫ ((1 µ )) ⊗ . . . ⊗ V ǫ ((1 µ r )) . Let I µǫ be the subspace of V µǫ spanned by the vectors induced from the relation (3.5), whichincludes the relations in Lemma 3.2. Since I µǫ is a ˚ U ( ǫ )-submodule, the quotient V µǫ /I µǫ isisomorphic to V ǫ ( λ ) by Proposition 3.4. So we have a well-defined ˚ U ( ǫ )-linear map π µ : V µǫ −→ V ǫ ( λ )given by π µ ( v T ⊗ . . . ⊗ v T r ) = v T where T is the column semistandard tableau whose i -thcolumn (from the left) is T i for 1 ≤ i ≤ r . Since the decomposition of V µ is equal to theusual Pieri rule of Schur functions, it has exactly one component isomorphic to V ǫ ( λ ). Hence π µ is equal to the projection onto V ǫ ( λ ) up to scalar multiplication.Let L µǫ = L ǫ ((1 µ )) ⊗ . . . ⊗ L ǫ ((1 µ r )) be the crystal lattice of V µǫ . By [16, Theorem 4.14], π µ ( L µǫ ) is a crystal lattice of V ǫ ( λ ) whose wt( H λ )-weight space is equal to A v ∗ H λ . Since thecrystal of V ǫ ( λ ) is connected, we conclude that { v ∗ T | T ∈ SST ǫ ( λ ) } is an A -basis of π µ ( L µǫ )which is equal to L ǫ ( λ ). (cid:3) Remark 3.8. For arbitrary ǫ , the ˚ I -colored oriented graph SST ǫ ( λ ) is not in general con-nected (see [15] for more details). Furthermore, it is not known yet whether V ǫ ( λ ) has acrystal base for any λ ∈ P M | N . We expect that ( L ǫ ( λ ) , B ǫ ( λ )) in (3.13) is a crystal base of V ǫ ( λ ). 4. R matrix for finite-dimensional U ( ǫ ) -modules Finite-dimensional U ( ǫ ) -modules of fundamental type. Let Z + be the set ofnon-negative integers. Let Z n + ( ǫ ) = { m = ( m , . . . , m n ) | m i ∈ Z + if ǫ i = 0, m i ∈ { , } if ǫ i = 1, ( i ∈ I ) } . For m ∈ Z n + ( ǫ ), let | m | = m + · · · + m n . For i ∈ I , put e i = (0 , · · · , , · · · , 0) where 1appears only in the i -th component.For s ∈ Z + , let W s,ǫ = M m ∈ Z n + ( ǫ ) , | m | = s Q ( q ) | m i be the Q ( q )-vector space spanned by | m i for m ∈ Z n + ( ǫ ) with | m | = s . For a parameter x ∈ Q ( q ), we denote by W s,ǫ ( x ) a U ( ǫ )-module V , where V = W s,ǫ as a Q ( q )-space and the actions of e i , f i , ω j are given by e i | m i = x δ i, q m i +1 − m i − [ m i +1 ] | m + e i − e i +1 i , if m + e i − e i +1 ∈ Z n + ( ǫ ) , , otherwise ,f i | m i = x − δ i, q m i − m i +1 − [ m i ] | m − e i + e i +1 i , if m − e i + e i +1 ∈ Z n + ( ǫ ) , , otherwise ,ω j | m i = q m j j | m i , for i ∈ I , j ∈ I , and m = ( m , . . . , m n ) ∈ Z n + ( ǫ ). Here we understand e = e n . Remark 4.1. We may identify W s,ǫ ( x ) with V ǫ (( s )) (3.12) as a ˚ U ( ǫ )-module, where | m i corresponds to v T , where T is the tableau of shape ( s ) with m i the number of occurrencesof i in T ( i ∈ I ). Also the map(4.1) φ ( | m i ) = q − P i For s ∈ Z + , the pair ( L s,ǫ , B s,ǫ ) is a crystal base of W s,ǫ , where thecrystal B s,ǫ / {± } is connected. Proof. It follows from the same arguments as in Lemma 3.6 that ( L s,ǫ , B s,ǫ ) is a crystal baseof W s,ǫ . The crystal SST ǫ (( s )) is connected with highest element H ( s ) . Since the crystal B s,ǫ / {± } of W s,ǫ is equal to SST ǫ (( s )) as an ˚ I -colored graph, B s,ǫ / {± } is connected asan I -colored oriented graph. (cid:3) Subalgebra U ( ǫ ′ ) . Suppose that n ≥ ǫ = ( ǫ , . . . , ǫ n ) be given. Let ǫ ′ =( ǫ ′ , . . . , ǫ ′ n − ) be the sequence obtained from ǫ by removing ǫ i for some i ∈ I . We furtherassume that ǫ ′ is homogeneous when n = 4, that is, ǫ ′ = (000) or (111).Put I ′ = { , , · · · , n − } . Let us denote by ω ′ l , e ′ j , and f ′ j the generators of U ( ǫ ′ ) for1 ≤ l ≤ n − j ∈ I ′ , where k ′ j = ω ′ j ( ω ′ j +1 ) − . Let us define K j , E j , F j for j ∈ I ′ asfollows: Case 1 . Assume that 2 ≤ i ≤ n − 1. For j ∈ I ′ , put K j = k j , if j ≤ i − ,k i − k i , if j = i − ,k j +1 , if j ≥ i,E j = e j , if j ≤ i − , [ e i − , e i ] D i − i , if j = i − ,e j +1 , if j ≥ i, F j = f j , if j ≤ i − , [ f i , f i − ] D − i − i , if j = i − ,f j +1 , if j ≥ i. (4.3) Case 2 . Assume that i = n . For j ∈ I ′ , put K j = k j , if j = 0 ,k n − k , if j = 0 ,E j = e j , if j = 0 , [ e n − , e ] D n − , if j = 0 , F j = f j , if j = 0 , [ f , f n − ] D − n − , if j = 0 . (4.4) Case 3 . Assume that i = 1. For j ∈ I ′ , put K j = k k , if j = 0 ,k j +1 , if j = 0 ,E j = [ e , e ] D , if j = 0 ,e j +1 , if j = 0 , F j = [ f , f ] D − 10 1 , if j = 0 ,f j +1 , if j = 0 . (4.5) Theorem 4.3. There exists a homomorphism of Q ( q ) -algebras φ : U ( ǫ ′ ) −→ U ( ǫ ) such that φ ( k ′ j ) = K j , φ ( e ′ j ) = E j , φ ( f ′ j ) = F j ( j ∈ I ′ ) . Proof. Let us prove Case 1 since the the proof of the other cases are similar. Let e ǫ =( e ǫ , . . . e ǫ n ) be the sequence obtained from ǫ by exchanging ǫ i and ǫ i +1 , and let τ i : U ( ǫ ) −→U ( e ǫ ) be the isomorphism in Theorem 2.2.Put Ω j = ω j for 1 ≤ j ≤ i − ω j +1 for i ≤ j ≤ n − 1, and let φ ( ω ′ j ) = Ω j , φ ( e ′ j ) = E j , and φ ( f ′ j ) = F j for j = 1 , · · · , n − 1. Let us check that Ω j , E j , F j satisfy the relations inDefinition 2.1. Note that D i − i = q − i .First, the relations (2.1) and (2.2) are trivial. Let us check that (2.3) holds. Let E j and F l be given for j, l ∈ I ′ . If j = l or j = l = i − 1, then it is clear. When j = l = i − 1, wehave τ − i ( e i − ) = [ e i − , e i ] D i − i = E i − , τ − ( f i − ) = F i − , and τ − ( k i − ) = K i − . Hence(2.3) holds. We can check the relation (2.4) by the same argument.Next, consider the relations (2.5). The first one is immediate. So it is enough to show thesecond one. We may only consider four non-trivial cases when the pair of relevant indicesin I ′ are ( i − , i − , ( i − , i − , ( i − , i ) , ( i, i − 1) with the first index in the pair in I ′ even . In case of ( i − , i − E i − E i − − ( − ǫ i − [2] E i − E i − E i − + E i − E i − = e i − e i − e i − q − i e i − e i e i − − ( q i − + q − i − ) e i − e i − e i e i − +( q i − + q − i − ) q − i e i − e i e i − e i − + e i − e i e i − − q − i e i e i − e i − . which is zero, since e i − e i − + e i − e i − = ( q i − + q − i − ) e i − e i − e i − and hence e i − e i − e i − ( q i − + q − i − ) e i − e i − e i e i − + e i − e i e i − = 0 , − q − i e i − e i e i − + q − i ( q i − + q − i − ) e i − e i e i − e i − − q − i e i e i − e i − = 0 . The proof for ( i, i − 1) is the same. In case of ( i − , i − 2) and ( i − , i ), the proof reducesto the case of ( i − , i − 1) or ( i, i − 1) by applying τ i to E l ’s for l = i − , i − , i .Finally let us check the relation (2.6). We may only consider the cases when the relevanttriple of indices in I ′ are ( i − , i − , i − , ( i − , i − , i ) , ( i − , i, i + 1) with the index inthe middle in I ′ odd . In case of ( i − , i, i + 1) and i ∈ I ′ odd , we have E i E i − E i E i +1 − E i E i +1 E i E i − + E i +1 E i E i − E i − E i − E i E i +1 E i + ( − ǫ i [2] E i E i − E i +1 E i = e i +1 ( e i − e i − q − i e i e i − ) e i +1 e i +2 − e i +1 e i +2 e i +1 ( e i − e i − q − i e i e i − )+ e i +2 e i +1 ( e i − e i − q − i e i e i − ) e i +1 − ( e i − e i − q − i e i e i − ) e i +1 e i +2 e i +1 + ( − ǫ i +1 [2] e i +1 ( e i − e i − q − i e i e i − ) e i +2 e i +1 , which is zero by (2.6) for U ( ǫ ) with respect to i + 1 ∈ I odd . The proof for ( i − , i − , i − i − , i − , i ) reduces to the previous cases by applying τ i to E l for l = i − , i − , i . We leave the proof for F j ’s to the reader. (cid:3) Truncation to U ( ǫ ′ ) -modules. Let ǫ ′ be as in Section 4.2. Suppose that M ′ is thenumber of j ’s with ǫ ′ j = 0 and N ′ is the number of j ’s with ǫ ′ j = 1 in ǫ ′ .For a submodule V of V ⊗ ℓ ( ℓ ≥ tr ǫǫ ′ ( V ) = M µ ∈ wt( V )( µ | δ i )=0 V µ , where wt( V ) is the set of weights of V . For any submodules V, W of a tensor power of V , itis clear that tr ǫǫ ′ ( V ⊗ W ) = tr ǫǫ ′ ( V ) ⊗ tr ǫǫ ′ ( W ) , as a vector space. Lemma 4.4. Let V ′ = tr ǫǫ ′ ( V ) . Then (1) V ′ is isomorphic to the natural representation of ˚ U ( ǫ ′ ) given in (2.15) , (2) tr ǫǫ ′ ( V ⊗ ℓ ) is isomorphic to V ′⊗ ℓ as a ˚ U ( ǫ ′ ) -module. Proof. (1) Let us assume that 2 ≤ i ≤ n − j ∈ ˚( I ′ ) and k ∈ I \ { i } given. It is clear from (4.3) that E j v k = v j , if k = j + 1 , , if k = j + 1 , ( j ≤ i − , E j v k = v j +1 , if k = j + 2 , , if k = j + 2 , ( j ≥ i ) . When j = i − 1, we have E i − = e i − e i − q − i e i e i − , and E i − v k = v i − , if k = i + 1 , , if k = i + 1 . We have similar formulas for F j for j ∈ ˚( I ′ ). Hence V ′ is invariant under the action of ˚ U ( ǫ ′ ).In fact, V ′ is isomorphic to the natural representation of ˚ U ( ǫ ′ ) (2.15).(2) We see that the actions of E j , F j , K j ( j ∈ ˚( I ′ )) on V ′ ⊗ V ′ are equal to those of(4.7) K − j ⊗ E j + E j ⊗ , ⊗ F j + F j ⊗ K j , K j ⊗ K j , respectively. This implies that V ′ ⊗ V ′ and hence ( V ′ ) ⊗ ℓ are invariant under the action of˚ U ( ǫ ′ ). For example, in case of E i − = e i − e i − q − i e i e i − , we have∆( E i − ) = ∆( e i − )∆( e i ) − q − i ∆( e i )∆( e i − )= k − i − k − i ⊗ e i − e i + k − i − e i ⊗ e i − + k − i e i − ⊗ e i + e i − e i ⊗ − q − i ( k − i k − i − ⊗ e i e i − + k − i e i − ⊗ e i + k − i − e i ⊗ e i − + e i e i − ⊗ . Then the action of ∆( E i − ) on V ′ ⊗ V ′ is equal to k − i k − i − ⊗ e i − e i + e i − e i ⊗ 1, and hence K − i − ⊗ E i − + E i − ⊗ (cid:3) Proposition 4.5. Let λ ∈ P M | N be given. (1) tr ǫǫ ′ ( V ǫ ( λ )) is a ˚ U ( ǫ ′ ) -submodule of V ǫ ( λ ) via φ . (2) tr ǫǫ ′ ( V ǫ ( λ )) is non-zero if and only if λ ∈ P M ′ | N ′ . In this case, we have tr ǫǫ ′ ( V ǫ ( λ )) ∼ = V ǫ ′ ( λ ) , as a ˚ U ( ǫ ′ ) -module. Proof. (1) It follows from Lemma 4.4 and(4.8) tr ǫǫ ′ ( V ǫ ( λ )) = tr ǫǫ ′ ( Y λ ( q ) V ⊗ ℓ ) = Y λ ( q ) tr ǫǫ ′ ( V ⊗ ℓ ) = Y λ ( q ) tr ǫǫ ′ ( V ) ⊗ ℓ = Y λ ( q )( V ′ ) ⊗ ℓ . (2) Note that SST ǫ ′ ( λ ) ⊂ SST ǫ ( λ ). By Proposition 3.4 and (4.8), we see that tr ǫǫ ′ ( V ǫ ( λ ))is a Q ( q )-span of { v T ′ | T ′ ∈ SST ǫ ′ ( λ ) } , which in fact forms a basis. This implies that tr ǫǫ ′ ( V ǫ ( λ )) is non-zero if and only if λ ∈ P M − | N when ǫ i = 0, and λ ∈ P M | N − when ǫ i = 1. Hence, tr ǫǫ ′ ( V ǫ ( λ )) is isomorphic to V ǫ ′ ( λ ) when it is non-zero by (4.7) and Proposition3.4. (cid:3) Corollary 4.6. Let V, W be submodules of a tensor power of V . Then (1) tr ǫǫ ′ ( V ) , tr ǫǫ ′ ( W ) , and tr ǫǫ ′ ( V ⊗ W ) are ˚ U ( ǫ ′ ) -modules via φ , (2) tr ǫǫ ′ ( V ⊗ W ) ∼ = tr ǫǫ ′ ( V ) ⊗ tr ǫǫ ′ ( W ) as ˚ U ( ǫ ′ ) -modules. Proof. Since V ⊗ ℓ is completely reducible, it follows from Proposition 4.5 and (4.7). (cid:3) We may define tr ǫǫ ′ and have similar results for U ( ǫ )-modules in O ≥ . Proposition 4.7. (1) For s ∈ Z + and x ∈ Q ( q ) , tr ǫǫ ′ ( W s,ǫ ( x )) is a U ( ǫ ′ ) -submodule of W s,ǫ ( x ) via φ , and tr ǫǫ ′ ( W s,ǫ ( x )) ∼ = W s,ǫ ′ ( x ) . Moreover, ( tr ǫǫ ′ ( L s,ǫ ) , tr ǫǫ ′ ( B s,ǫ )) is a crystal base of tr ǫǫ ′ ( W s,ǫ ) isomorphic to ( L s,ǫ ′ , B s,ǫ ′ ) . (2) For l, m ∈ Z + and x, y ∈ Q ( q ) , tr ǫǫ ′ ( W l,ǫ ( x ) ⊗ W m,ǫ ( y )) is a U ( ǫ ′ ) -module via φ , and tr ǫǫ ′ ( W l,ǫ ( x ) ⊗ W m,ǫ ( y )) ∼ = tr ǫǫ ′ ( W l,ǫ ( x )) ⊗ tr ǫǫ ′ ( W m,ǫ ( y )) , as U ( ǫ ′ ) -modules. Proof. The proof is the same as in Proposition 4.5. (cid:3) Irreducibility of W l,ǫ ( x ) ⊗W m,ǫ ( y ) . Let us show that W l,ǫ ( x ) ⊗W m,ǫ ( y ) is irreduciblefor l, m ∈ Z + and generic x, y ∈ Q ( q ). When ǫ = ǫ M | N , the irreducibility is shown in [17].In this paper, we give a different proof of it, which is also available for arbitrary ǫ . Theorem 4.8. For l, m ∈ Z + , W l,ǫ ⊗ W m,ǫ is irreducible. Proof. Let us assume without loss of generality that M, N ≥ ǫ = 0.Let ( L s,ǫ , B s,ǫ ) be the crystal base of W s,ǫ in (4.2) for s = l, m . By Proposition 2.5,( L l,ǫ ⊗ L m,ǫ , B l,ǫ ⊗ B m,ǫ ) is a crystal base of W l,ǫ ⊗ W m,ǫ . If M = 1, then it is provedin [16] that B l,ǫ ⊗ B m,ǫ / {± } is connected. Since dim Q ( q ) ( W l,ǫ ⊗ W m,ǫ ) ( l + m ) δ = 1 and B l,ǫ ⊗ B m,ǫ / {± } is connected, it follows from [2, Lemma 2.7] that W l,ǫ ⊗ W m,ǫ is irreducible.We assume that M ≥ 2. Set ǫ ′ = ǫ M | , which is the subsequence of ǫ obtained byremoving all ǫ i = 1’s. Note that the length of ǫ ′ may be less than 4 so that U ( ǫ ′ ) is notwell-defined, but tr ǫǫ ′ can be defined in the same way as in (4.6). We put W s,ǫ ′ := tr ǫǫ ′ ( W s,ǫ ) , L s,ǫ ′ := tr ǫǫ ′ ( L s,ǫ ) ⊂ L s,ǫ , B s,ǫ ′ := tr ǫǫ ′ ( B s,ǫ ) ⊂ B s,ǫ . Let 1 ≤ j < · · · < j M ≤ n be such that ǫ j k = 0 for 1 ≤ k ≤ M . By Theorem 4.3, wehave a U q ( sl )-action on W l,ǫ ′ ⊗ W m,ǫ ′ corresponding to the pair ( ǫ j k , ǫ j k +1 ) or ( ǫ j M , ǫ j ).For 0 ≤ k ≤ M − 1, let us denote by e e k ′ and e f k ′ the Kashiwara operators corresponding to( ǫ j k , ǫ j k +1 ) when k = 0 and to ( ǫ j M , ǫ j ) when k = 0.If we put I ′ = { k ′ | k = 0 , . . . , M − } , then L l,ǫ ′ ⊗ L m,ǫ ′ is invariant under e e k ′ and e f k ′ for k ′ ∈ I ′ , and hence B l,ǫ ′ ⊗ B m,ǫ ′ / {± } is an I ′ -colored oriented graph. Since L l,ǫ ′ ⊗ L m,ǫ ′ ⊂L l,ǫ ⊗ L m,ǫ and B l,ǫ ′ ⊗ B m,ǫ ′ ⊂ B l,ǫ ⊗ B m,ǫ , we may regard B l,ǫ ⊗ B m,ǫ / {± } as an ( I ⊔ I ′ )-colored oriented graph.Let b = | m i ⊗ | m i ∈ B l,ǫ ⊗ B m,ǫ be given. We will show that b is connected to | l e i⊗| m e i , which implies that B l,ǫ ⊗B m,ǫ / {± } is connected as an ( I ⊔ I ′ )-colored orientedgraph. Let us write m i = ( m i , . . . , m in ) for i = 1 , i , . . . , i r ∈ I such that ( ǫ i k , ǫ i k +1 ) = (0 , 0) for1 ≤ k ≤ r and(4.9) b ′ := e x i . . . e x i r b ≡ | m ′ i ⊗ | m ′ i (mod q L l,ǫ ′ ⊗ L m,ǫ ′ ) , for some | m ′ i ∈ W l,ǫ ′ and | m ′ i ∈ W m,ǫ ′ , where e x i s = e e i s or e f i s for each 1 ≤ s ≤ r .Suppose that there exists k with ǫ k = 1 such that m k = 1 or m k = 1. Let i and j be themaximal and minimal indices respectively such that i < k < j and ǫ i = ǫ j = 0. If there is nosuch ( i, j ), then we have ǫ = ǫ M | N and identify this case with the one of ǫ = (0 M − , N , i , . . . , i r in { i, i + 1 , . . . , j − } , we may assume for simplicity that m ab = 0 for a = 1 , b 6∈ { i, . . . , j } .Let us use induction on L = | m | + | m | . Suppose that L = 1. If m k = 1, then e f j − e f j − . . . e f k b satisfies (4.9). If m k = 1, then e e i e e i +1 . . . e e k − b satisfies (4.9).Suppose that L > 1. We may assume that e f i +1 b = e f i +2 b = · · · = e f j − b = 0. Then bytensor product rule in Proposition 2.5 we have m = m i e i + X x ≤ u ≤ y e u + X z ≤ v ≤ j − e v + m j e j , m = m i e i + X y +1 ≤ v ≤ j − e v + m j e j , (4.10)for some i < x < y < z < j . Here we assume that P z ≤ v ≤ j − e v in m is empty if there isno such z . Now we take the following steps to construct b ′ in (4.9).Step 1. If there exists z such that y < z < j and m z = · · · = m j − = 1, then byapplying e f z e f z − . . . e f j − to b , m in (4.10) is replaced by(4.11) m i e i + X x ≤ u ≤ y e u + X z +1 ≤ v ≤ j − e v + ( m j + 1) e j . Repeating this step, (4.11) is replaced by m i e i + X x ≤ u ≤ y e u + ( m j + j − z ) e j . Hence we may assume that m in (4.10) is of the form m i e i + P x ≤ u ≤ y +1 e u + m j e j .Step 2. If m j = 0, then we have e f j − b = | m i ⊗ | m − e j − + e j i . Hence we may apply the induction hypothesis to conclude (4.9).Step 3. If m ij = 0, then by applying e e i e e i +1 . . . e e j − e e j − to b , m and m are replaced by m i e i + X x ≤ u ≤ y +1 e u + ( m j − e j , ( m i + 1) e i + X y +2 ≤ v ≤ j − e v + m j e j , respectively. Repeating this step d times such that m j − d ≥ y + d + 1 ≤ j , m and m are replaced by m i e i + X x ≤ u ≤ y + d e u + ( m j − d ) e j , ( m i + d ) e i + X y + d +1 ≤ v ≤ j − e v + m j e j , respectively. We may keep this process until m j − d = 0, which belongs to the case in Step 2 ,or P y + d +1 ≤ v ≤ j − e v is empty. In the latter case, m is replaced by m i e i + P x ≤ u ≤ j − e u +( m j − d ) e j so that we may apply e f j − and use induction hypothesis to have b ′ . This provesthe claim.By construction of b ′ and its weight, we have b ′ − | m ′ i ⊗ | m ′ i ∈ ( L l,ǫ ′ ⊗ L m,ǫ ′ ) ∩ ( q L l,ǫ ⊗ L m,ǫ ) = q L l,ǫ ′ ⊗ L m,ǫ ′ , and hence b ′ ∈ ( L l,ǫ ′ ⊗ L m,ǫ ′ /q L l,ǫ ′ ⊗ L m,ǫ ′ ) ⊂ ( L l,ǫ ⊗ L m,ǫ /q L l,ǫ ⊗ L m,ǫ ). If M = 2, then itis easy to show that b ′ = | m ′ i ⊗ | m ′ i ∈ B l,ǫ ′ ⊗ B m,ǫ ′ is connected to | l e i ⊗ | m e i under e e k ′ and e f k ′ for k = 0 , 1. If M ≥ 3, then we can also show that b ′ = | m ′ i ⊗ | m ′ i ∈ B l,ǫ ′ ⊗ B m,ǫ ′ is connected to | l e i ⊗ | m e i by using the fact that B l,ǫ ′ ⊗ B m,ǫ ′ / {± } is a connected crystalof type A (1) M − (cf. [1]).Finally, since dim Q ( q ) ( W l,ǫ ⊗ W m,ǫ ) ( l + m ) δ = 1 and B l,ǫ ⊗ B m,ǫ / {± } is connected, itfollows from [2, Lemma 2.7] that W l,ǫ ⊗ W m,ǫ is irreducible. This completes the proof. (cid:3) Corollary 4.9. For l, m ∈ Z + and generic x, y ∈ Q ( q ) , W l,ǫ ( x ) ⊗ W m,ǫ ( y ) is irreducible. Proof. It follows from [12, Lemma 3.4.2]. (cid:3) Existence of R matrix. For l, m ∈ Z + and x, y ∈ Q ( q ), consider a non-zero Q ( q )-linear map R on W l,ǫ ( x ) ⊗ W m,ǫ ( y ) such that∆ op ( g ) ◦ R = R ◦ ∆( g ) , (4.12)for g ∈ U ( ǫ ), where ∆ op is the opposite coproduct of ∆ in (2.7), that is, ∆ op ( g ) = P ◦ ∆( g ) ◦ P and P ( a ⊗ b ) = b ⊗ a . We denote it by R ( z ), where z = x/y , since R depends only on z .We say that R ( z ) satisfies the Yang-Baxter equation if we have(4.13) R ( u ) R ( uv ) R ( v ) = R ( v ) R ( uv ) R ( u ) , on W s ,ǫ ( x ) ⊗ W s ,ǫ ( x ) ⊗ W s ,ǫ ( x ) with u = x /x and v = x /x for ( s ) , ( s ) , ( s ) ∈ P M | N . Here R ij ( z ) denotes the map which acts as R ( z ) on the i -th and the j -th componentand the identity elsewhere. We call R ( z ) the (quantum) R matrix . Theorem 4.10. Let l, m ∈ Z + given with ( l ) , ( m ) ∈ P M | N . Suppose that ǫ = 0 . Thereexists a unique non-zero linear map R ( z ) ∈ End Q ( q ) ( W l,ǫ ( x ) ⊗ W m,ǫ ( y )) satisfying (4.12) and (4.13) , and R ( z )( | l e i ⊗ | m e i ) = | l e i ⊗ | m e i for generic x, y ∈ Q ( q ) . Proof. The existence of such a map for arbitrary ǫ is proved in [17, Theorem 5.1] withrespect to ∆ + in (2.9), say R + . Let(4.14) χ = ψ ◦ ( φ ⊗ φ ) , where ψ and φ are given in (2.10) and (4.1), respectively. Then R := χ − ◦ R + ◦ χ satisfies the conditions (4.12) and (4.13), and R ( z )( | l e i⊗| m e i ) = | l e i⊗| m e i with respectto ∆. The uniqueness follows from the irreducibility in Corollary 4.9 and normalization by R ( z )( | l e i ⊗ | m e i ) = | l e i ⊗ | m e i . (cid:3) Remark 4.11. If M = 0, then the existence of R matrix is already known. Hence we mayassume that M ≥ 1. If ǫ = 1, then we may choose the smallest i ∈ I so that there existsa unique R matrix satisfying R ( z )( | l e i i ⊗ | m e i i ) = | l e i i ⊗ | m e i i .5. Kirillov-Reshetikhin modules Spectral decomposition. Suppose that ǫ = ( ǫ , . . . , ǫ n ) is given with n ≥ 4. Recallthat M is the number of i ’s with ǫ i = 0 and N is the number of i ’s with ǫ i = 1 in ǫ .Let l, m ∈ Z + be given. Let R ǫ ( z ) be the R matrix on W l,ǫ ( x ) ⊗ W m,ǫ ( y ) in Theorem4.10. We have as a ˚ U ( ǫ )-module, W l,ǫ ( x ) ⊗ W m,ǫ ( y ) ∼ = M t ∈ H ( l,m ) V ǫ (( l + m − t, t )) , where H ( l, m ) = { t | ≤ t ≤ min { l, m } , ( l + m − t, t ) ∈ P M | N } .Let us take a sequence ǫ ′′ = ( ǫ ′′ , . . . , ǫ ′′ n ′′ ) of 0 , n ′′ ≫ n satisfying the following:(1) ǫ is a subsequence of ǫ ′′ ,(2) we have as a ˚ U ( ǫ ′′ )-module W l,ǫ ′′ ( x ) ⊗ W m,ǫ ′′ ( y ) ∼ = M ≤ t ≤ min { l,m } V ǫ ′′ (( l + m − t, t )) , (3) if ǫ ′ = ǫ M ′′ | with M ′′ = |{ i | ǫ ′′ i = 0 }| ≫ 0, then we have as a ˚ U ( ǫ ′ )-module W l,ǫ ′ ( x ) ⊗ W m,ǫ ′ ( y ) ∼ = M ≤ t ≤ min { l,m } V ǫ ′ (( l + m − t, t )) , Let R ǫ ′′ ( z ) and R ǫ ′ ( z ) denote the R matrices on W l,ǫ ′′ ( x ) ⊗ W m,ǫ ′′ ( y ) and W l,ǫ ′ ( x ) ⊗W m,ǫ ′ ( y ), respectively. Lemma 5.1. For ǫ = ǫ or ǫ ′ , we have the following commutative diagram: W l,ǫ ′′ ( x ) ⊗ W m,ǫ ′′ ( y ) P R ǫ ′′ ( z ) / / tr ǫ ′′ ǫ (cid:15) (cid:15) W m,ǫ ′′ ( y ) ⊗ W l,ǫ ′′ ( x ) tr ǫ ′′ ǫ (cid:15) (cid:15) W l, ǫ ( x ) ⊗ W m, ǫ ( y ) P R ǫ ( z ) / / W m, ǫ ( y ) ⊗ W l, ǫ ( x ) Proof. For ǫ = ǫ or ǫ ′ , the restriction of P R ǫ ′′ ( z ) on tr ǫ ′′ ǫ ( W l,ǫ ( x ) ⊗ W m,ǫ ( y )), which gives awell-defined U ( ǫ )-linear endomorphism. By Proposition 4.7 and Theorem 4.10, the restricted R matrix is the quantum R matrix on W l, ǫ ( x ) ⊗ W m, ǫ ( y ), which proves the commutativityof the diagram. (cid:3) For 0 ≤ t ≤ min { l, m } , let v ′ ( l, m, t ) be the highest weight vectors of V ǫ ′ (( l + m − t, t )) in W l,ǫ ′ ( x ) ⊗ W m,ǫ ′ ( y ) such that v ′ ( l, m, t ) ∈ L l,ǫ ′ ⊗ L m,ǫ ′ ,v ′ ( l, m, t ) ≡ | l e i ⊗ | ( m − t ) e + t e i (mod q L l,ǫ ′ ⊗ L m,ǫ ′ ) . We also define v ′ ( m, l, t ) in the same manner. For 0 ≤ t ′ ≤ min { l, m } , we may regard V ǫ (( l + m − t, t )) ⊂ V ǫ ′′ (( l + m − t, t )) , V ǫ ′ (( l + m − t, t )) ⊂ V ǫ ′′ (( l + m − t, t )) as a Q ( q )-space, and let P l,mt : W l,ǫ ′′ ( x ) ⊗ W m,ǫ ′′ ( y ) −→ W m,ǫ ′′ ( y ) ⊗ W l,ǫ ′′ ( x ) be a ˚ U ( ǫ ′′ )-linear map given by P l,mt ( v ′ ( l, m, t ′ )) = δ tt ′ v ′ ( m, l, t ′ ). Then we have the following spectraldecomposition of P R ǫ ′′ ( z ) P R ǫ ′′ ( z ) = X ≤ t ≤ min { l,m } ρ t ( z ) P l,mt , for some ρ t ( z ) ∈ Q ( q ). By Proposition 4.5 and Lemma 5.1, we also have the followingspectral decomposition of P R ǫ ( z ) P R ǫ ′ ( z ) = X ≤ t ≤ min { l,m } ρ t ( z ) P l,mt ,P R ǫ ( z ) = X t ∈ H ( l,m ) ρ t ( z ) P l,mt , (5.1)where we understand P l,mt as defined on W l, ǫ ( x ) ⊗ W m, ǫ ( y ). Then we have the followingexplicit description of P R ǫ ( z ), which is proved in case of ǫ = ǫ M | N [17]. Theorem 5.2. We have (5.2) P R ǫ ( z ) = min { l,m } X t =max { l + m − n, } Ñ min { l,m } Y i = t +1 z − q l + m − i +2 − q l + m − i +2 z é P l,mt ( M = 0) , (5.3) P R ǫ ( z ) = min { l,m,n − } X t =0 t Y i =1 − q l + m − i +2 zz − q l + m − i +2 ! P l,mt ( M = 1) , (5.4) P R ǫ ( z ) = min { l,m } X t =0 t Y i =1 − q l + m − i +2 zz − q l + m − i +2 ! P l,mt (2 ≤ M ≤ n ) , where we assume that ρ min { l,m } ( z ) = 1 in (5.2) and ρ ( z ) = 1 in (5.3) and (5.4) . Proof. We may consider the case of 1 ≤ M ≤ n only since the case when M = 0 is known(see [16, (5.6)] or [17, (6.10)]). It is well-known that P R ǫ ′ ( z ) for ǫ ′ = ǫ M ′′ | has the followingspectral decomposition P R ǫ ′ ( z ) = X ≤ t ≤ min { l,m } ρ ′ t ( z ) P l,mt , where ρ ′ ( z ) = 1 , ρ ′ t ( z ) = t Y i =1 − q l + m − i +2 zz − q l + m − i +2 (1 ≤ t ≤ min { l, m } ) , (cf. [16, (5.8)] or [17, (6.16)]). We remark that χ ( v ′ ( l, m, t )) and χ ( v ′ ( m, l.t )) for 0 ≤ t ≤ min { l, m } are the same scalar multiplications of the highest weight vectors in [17, (6.14)],where χ is as in (4.14). Hence it follows from (5.1) that ρ t ( z ) = ρ ′ t ( z ) ( t ∈ H ( l, m )) , which completes the proof. (cid:3) Kirillov-Reshetikhin modules. As an application of Theorem 5.2, let us construct afamily of irreducible U ( ǫ )-modules in O ≥ which corresponds to usual Kirillov-Reshetikhinmodules under truncation. Let us assume that 1 ≤ M ≤ n − M ∈ { , n } are well-known [13].Fix s ≥ V x = W s,ǫ ( x ) for x ∈ Q ( q ). We take a normalizationˇ R ( z ) = s Y i =1 z − q s − i +2 − q s − i +2 z ! P R ( z ) , where R ( z ) is the R matrix on V x ⊗ V y . Since ( s ) P M | N if and only if M = 1 and s > n − 1, we haveˇ R ( z ) = P n − t =0 Å Q si = t +1 z − q s − i +2 − q s − i +2 z ã P s,st , if ( s ) P M | N , P s,ss + P s − t =0 Å Q si = t +1 z − q s − i +2 − q s − i +2 z ã P s,st , if ( s ) ∈ P M | N . For r ≥ 2, let W denote the group of permutations on r letters generated by s i = ( i i + 1)for 1 ≤ i ≤ r − 1. By Theorem 4.10, we have U ( ǫ )-linear mapsˇ R w ( x , . . . , x r ) : V x ⊗ · · · ⊗ V x r −→ V x w (1) ⊗ · · · ⊗ V x w ( r ) for w ∈ W and generic x , . . . , x r satisfying the following:ˇ R ( x , . . . , x r ) = id V x ⊗···⊗ V xr , ˇ R s i ( x , . . . , x r ) = Ä ⊗ ji +1 id V xj ä , ˇ R ww ′ ( x , . . . , x r ) = ˇ R w ′ ( x w (1) , . . . , x w ( r ) ) ˇ R w ( x , . . . , x r ) , for w, w ′ ∈ W with ℓ ( ww ′ ) = ℓ ( w ) + ℓ ( w ′ ). Let w denote the longest element in W . ByTheorem 5.2, ˇ R w ( x , . . . , x r ) does not have a pole at q k for k ∈ Z + as a function in x , . . . , x r . Hence we have a U ( ǫ )-linear mapˇ R r := ˇ R w ( q r − , q r − , · · · , q − r ) : V q r − ⊗ · · · ⊗ V q − r −→ V q − r ⊗ · · · ⊗ V q r − . Then we define a U ( ǫ )-module(5.5) W ( r ) s,ǫ := Im ˇ R r . It is proved in [16] that W ( r ) s,ǫ is irreducible when ǫ = ǫ M | N , where the proof uses thecrystal base of polynomial representation of U M | N ( ǫ ). Now we give another proof of theirreducibility of W ( r ) s,ǫ , which is available for arbitrary ǫ . Theorem 5.3. Let r, s ≥ be given. Then W ( r ) s,ǫ is non-zero if and only if ( s r ) ∈ P M | N .In this case, W ( r ) s,ǫ is irreducible, and it is isomorphic to V ǫ (( s r )) as a ˚ U ( ǫ ) -module. Proof. Let us take a sequence ǫ ′′ = ( ǫ ′′ , . . . , ǫ ′′ n ′′ ) of 0 , ǫ is a subsequence of ǫ ′′ , (2) we have as a ˚ U ( ǫ ′′ )-module(5.6) V ǫ ′′ (( s )) ⊗ r ∼ = M λ ∈ P V ǫ ′′ ( λ ) ⊕ K λ ( sr ) , where K λ ( s r ) is the Kostka number associated to λ and ( s r ) (cf. Remark 3.5),(3) if ǫ ′ = ǫ M ′′ | with M ′′ = |{ i | ǫ ′′ i = 0 }| , then we have as a ˚ U ( ǫ ′ )-module(5.7) V ǫ ′ (( s )) ⊗ r ∼ = M λ ∈ P V ǫ ′ ( λ ) ⊕ K λ ( sr ) . Let us define a U ( ǫ ′′ )-module W ( r ) s,ǫ ′′ by the same way as in (5.5), where ˇ R ′′ r and V ′′ x denotethe corresponding ones. We define W ( r ) s,ǫ ′ , ˇ R ′ r and V ′ x similarly.By Lemma 5.1, we have the following commutative diagram: V ′′ q r − ⊗ · · · ⊗ V ′′ q − r ˇ R ′′ r / / tr ǫ ′′ ǫ ′ (cid:15) (cid:15) V ′′ q − r ⊗ · · · ⊗ V ′′ q r − tr ǫ ′′ ǫ ′ (cid:15) (cid:15) V ′ q r − ⊗ · · · ⊗ V ′ q − r ˇ R ′ r / / V ′ q − r ⊗ · · · ⊗ V ′ q r − By (5.6), (5.7) and Proposition 4.5, the decomposition of W ( r ) s,ǫ ′′ into polynomial ˚ U ( ǫ ′′ )-modules is the same as that of W ( r ) s,ǫ ′ into polynomial ˚ U ( ǫ ′ )-modules. It is well-known that W ( r ) s,ǫ ′ is irreducible and isomorphic to V ǫ ′ (( s r )) as a ˚ U ( ǫ ′ )-module since U ( ǫ ′′ ) ∼ = U q ( A (1) M ′′ − ).Therefore, W ( r ) s,ǫ ′′ is irreducible and isomorphic to V ǫ ′′ (( s r )) as a ˚ U ( ǫ ′′ )-module.Again by Lemma 5.1, we have the following commutative diagram: V ′′ q r − ⊗ · · · ⊗ V ′′ q − r ˇ R ′′ r / / tr ǫ ′′ ǫ ′ (cid:15) (cid:15) V ′′ q − r ⊗ · · · ⊗ V ′′ q r − tr ǫ ′′ ǫ ′ (cid:15) (cid:15) V q r − ⊗ · · · ⊗ V q − r ˇ R r / / V q − r ⊗ · · · ⊗ V q r − Since tr ǫ ′′ ǫ ′ ( V ǫ ′′ (( s r ))) is non-zero if and only if ( s r ) ∈ P M | N , which is equal to V ǫ (( s r )) inthis case, it follows that W ( r ) s,ǫ is non-zero if and only if ( s r ) ∈ P M | N . This implies in thiscase that W ( r ) s,ǫ is irreducible, and it is isomorphic to V ǫ (( s r )) as a ˚ U ( ǫ )-module. (cid:3) The following can be proved by similar arguments. Corollary 5.4. Suppose that ( s r ) ∈ P M | N is given. (1) If r ≤ M and M ≥ , then tr ǫǫ ′ Ä W ( r ) s,ǫ ä is the Kirillov-Reshetikhin module of type A (1) M − corresponding to the partition ( s r ) , where ǫ ′ = ǫ M | . (2) If s ≤ N and N ≥ , then tr ǫǫ ′ Ä W ( r ) s,ǫ ä is the Kirillov-Reshetikhin module of type A (1) N − corresponding to the partition ( r s ) , where ǫ ′ = ǫ | N . Remark 5.5. As in case of ǫ = ǫ M | N [16], we also expect that W ( r ) s,ǫ has a crystal base forarbitrary ǫ (cf. Remark 3.8). One may use a similar argument as in the proof of Theorem 5.3 to prove the irreducibilityof a tensor product of W l,ǫ ( x )’s and its image under R matrix in some special cases. Let l , . . . , l r ∈ Z + and x , . . . , x r ∈ Q ( q ) be given and let ǫ ′ = ǫ M | . Proposition 5.6. If M is sufficiently large and W l ,ǫ ′ ( x ) ⊗ · · · ⊗ W l r ,ǫ ′ ( x r ) is irreducible,then W l ,ǫ ( x ) ⊗ · · · ⊗ W l r ,ǫ ( x r ) is also irreducible. Proof. Suppose that W l ,ǫ ( x ) ⊗ · · · ⊗ W l r ,ǫ ( x r ) is not irreducible and let W be a propernon-trivial submodule. Since M is sufficiently large, the multiplicity of V ǫ ( λ ) for λ ∈ P in W l ,ǫ ( x ) ⊗ · · · ⊗ W l r ,ǫ ( x r ) is equal to that of V ǫ ′ ( λ ) for λ ∈ P in W l ,ǫ ′ ( x ) ⊗· · · ⊗ W l r ,ǫ ′ ( x r ) (cf. Remark 3.5). This also holds for W and tr ǫǫ ′ ( W ), which implies that tr ǫǫ ′ ( W ) is a proper non-zero subspace of tr ǫǫ ′ ( W l ,ǫ ( x ) ⊗ · · · ⊗ W l r ,ǫ ( x r )). Since tr ǫǫ ′ ( W ) = W ∩ tr ǫǫ ′ ( W l ,ǫ ( x ) ⊗ · · · ⊗ W l r ,ǫ ( x r )), it follows that tr ǫǫ ′ ( W ) is a proper non-zero U ( ǫ ′ )-submodule, which is a contradiction. (cid:3) Remark 5.7. Proposition 5.6 together with the irreducibility of W l,ǫ ′ ⊗ W m,ǫ ′ also impliesTheorem 4.8 when M ≥ 3. But we do not know whether it holds for M = 2. We also wouldlike to point out that the proof of Theorem 4.8 has its own interest since it describes a newconnected crystal graph structure on B l,ǫ ⊗ B m,ǫ / {± } . Proposition 5.8. Suppose that x i /x i +1 q − Z + for ≤ i ≤ r − . If M is sufficientlylarge and the image of ˇ R ′ w ( x , . . . , x r ) : W l ,ǫ ′ ( x ) ⊗ · · · ⊗ W l r ,ǫ ′ ( x r ) −→ W l r ,ǫ ′ ( x r ) ⊗ · · · ⊗ W l ,ǫ ′ ( x ) is irreducible, then the image of ˇ R w ( x , . . . , x r ) : W l ,ǫ ( x ) ⊗ · · · ⊗ W l r ,ǫ ( x r ) −→ W l r ,ǫ ( x r ) ⊗ · · · ⊗ W l ,ǫ ( x ) is also irreducible, where ˇ R ′ w ( x , . . . , x r ) is the restriction of ˇ R w ( x , . . . , x r ) on W l ,ǫ ′ ( x ) ⊗· · · ⊗ W l r ,ǫ ′ ( x r ) . Proof. It follows from Lemma 5.1 and the same argument as in Proposition 5.6. (cid:3) References [1] T. Akasaka, M. Kashiwara, Finite-dimensional representations of quantum affine algebras , Publ. Res.Inst. Math. Sci. (1997) 839–867.[2] G. Benkart, S.-J. Kang, M. 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