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CRYSTAL DUALITY AND LITTLEWOOD-RICHARDSON RULE OF EXTREMAL WEIGHT CRYSTALS 9 0 JAE-HOONKWON 0 2 Abstract. We consider a category of gl -crystals, whose object is a disjoint ∞ p unionofextremalweightcrystalswithboundednon-negativelevelandfinitemul- e S tiplicity for each connected component. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti- 7 isomorphic to an Ore extension of the character ring of integrable lowest gl - ∞ ] modules with respect to derivations shifting the fundamental weight characters. A ALittlewood-Richardson ruleofextremalweightcrystalswithnon-negativelevel Q is described explicitly in terms of classical Littlewood-Richardson coefficients. A . h doublecrystalstructureonbinarymatricesofvariousshapesandassociatedcrys- t a tal dualities are used as our main tools. m [ 1 Contents v 6 2 1. Introduction 1 1 2. Crystals 5 1 . 3. Double crystal structure on binary matrices 7 9 0 4. Realization of extremal weight crystals 16 9 5. Tensor product of extremal weight crystals 22 0 : 6. Differential operators on lowest weight characters 31 v i References 42 X r a 1. Introduction Let U (g) be the quantized enveloping algebra associated with a symmetrizable q Kac-Moody algebra g. In [12], Kashiwara introduced a class of integrable modules over U (g) called extremal weight modules, which is a natural generalization of q integrable highest or lowest modules. There exist a global crystal base and a crystal base of an extremal weight module, and the crystal base of an extremal weight crystal, simply an extremal weight crystal, appears as a subcrystal of that of the modified quantized enveloping algebra U (g) [12]. Suppose that g is an affine Kac- q Moody algebra of finite rank. Then an extremal weight crystal of positive (resp. e This work was supported by KRFGrant 2008-314-C00004. 1 2 JAE-HOONKWON negative)levelisisomorphictothecrystalbaseofanintegrablehighest(resp. lowest) weight module. In [1, 14], the structure of level zero extremal weight modules has been studied in detail, and it was conjectured by Kashiwara [14] that an extremal weightcrystaloflevelzeroisisomorphictotheproductofthecrystalbaseofatensor product of fundamental representations and a set of Laurent Schur polynomials. In [3], Beck and Nakajima proved this conjecture (see also [2, 22] for the case when g is symmetric), and furthermore based on the study of extremal weight modules they proved the Kashiwara’s conjecture on Peter-Weyl decomposition and the Lusztig’s conjecture on two-sided cell structures of U (g) in a purely algebraic way, though q there is a geometric background related with quiver varieties. e A natural question arises whether there is a nice description of extremal weight crystals and their tensor products when g is an infinite rank affine Lie algebra. In [19] the author studied extremal weight crystals of type A . It is shown that +∞ an extremal weight crystal is isomorphic to the tensor product of a lowest weight crystal and a highest weight crystal, and the Grothendieck ring generated by the isomorphism classes of extremal weight crystals is isomorphic to the Weyl algebra of infinite rank. A Littlewood-Richardson rule of extremal weight crystals is then described explicitly using the operators induced from the multiplication by Schur functions together with their adjoints. The purpose of this paper is to study extremal weight crystals of type A (or ∞ extremal weight gl -crystals), wheregl denotes the infiniterank affineLie algebra ∞ ∞ of type A . It should be noted that (1) an extremal weight gl -crystal is always ∞ ∞ connected, (2) there are extremal weight crystals of non-zero level, which are isomorphic to neither a highest nor lowest weight crystal, and (3) the extremal weight crystals of non-negative (or non-positive) levels are closed under tensor product. These are important features of extremal weight gl -crystals, which do not neces- ∞ sarily hold in other affine types of finite rank. Also as in case of A , we need +∞ certain non-commuting operators to describe the tensor product of extremal gl - ∞ crystals because of the non-existence of characters and the non-commutativity of tensor products. Let us explain our results in detail. For an integral weight Λ of level k ≥ 0, we denote by B(Λ) the extremal weight crystal with extremal weight Λ. Let B be the crystal of the natural representation of U (gl ). The connected components of B⊗n q ∞ (n ≥ 1) are parameterized by partitions λ of n, say B . Note that the crystal B is λ λ not isomorphic to the crystal of a highest weight or lowest weight module. Then we showthatB(Λ)isconnectedandthereexistuniquepartitionsµ,ν andthedominant integral weight Λ′ of level k such that B(Λ) ≃ B ⊗B∨⊗B(Λ′), µ ν LITTLEWOOD-RICHARDSON RULE OF EXTREMAL WEIGHT CRYSTALS 3 where B∨ is the dual crystal of B (Theorem 4.6). Note that B(Λ′) is the crystal ν ν of the highest weight module with highest weight Λ′. An extremal weight crystal of non-negative level is also characterized from the above result by taking its dual. Next, we consider a category C of gl -crystals, whose object is a disjoint union ∞ of extremal weight crystals with bounded non-negative level and finite multiplicity for each connected component. We show that C is a monoidal category undertensor productof crystals (Theorem 5.1). In particular, thetensor productof two extremal weight crystals of non-negative levels is isomorphic to a disjoint union of extremal weight crystals, and the multiplicity of each connected component is finite. We remark that the tensor product in C is not necessarily commutative. LetK betheGrothendieckgroupassociatedwithC. ThenK isanon-commutative Z-algebra with 1, where the multiplication in C is induced from tensor product of crystals. Let z = {z |k ∈ Z} be a set of formal commuting variables, and let R k be the ring of formal power series in z of bounded degree over Z. Note that the ring generated by the characters of the integrable highest weight representations of gl is isomorphic to R. Now, let D be an Ore extension of R associated with a ∞ commuting family of derivations γ± = (−1)n−1 z ∂ (n ∈ Z). We may n k∈Z k∓n∂zk view D as a non-commutative polynomial ring over R in γ± = {γ±|n ∈ Z}. Then P n we show that there exists a Z-algebra isomorphism K −∼→ Dopp (Theorem 5.7), where isomorphism classes of an integrable highest weight crystal and a level zero extremal weight crystal are mapped to polynomials in z and γ±, respectively. Here Dopp denotes the opposite algebra of D. Based on the above results, we obtain a Littlewood-Richardson rule for extremal weight crystals of non-negative levels (Theorem 5.14), which is given explicitly in terms of classical Littlewood-Richardson coefficients. In fact, the tensor product of level zero extremal weight crystals corresponds to the product of double symmetric functions, whose decomposition can be given by the classical Littlewood-Richardson rule due to a crystal (gl ,gl )-duality on the nth exterior algebra generated by the ∞ n natural representation ofgl (Proposition 3.9), and the decomposition of thetensor ∞ productof integrable highest weight gl -crystals is explained by usinga crystal ver- ∞ sion of the (gl ,gl )-duality on the level n fermionic Fock space due to Frenkel [6] ∞ n (Proposition 3.10) following [27]. Hence the only non-trivial part is the decomposition of the tensor product B(Λ)⊗B ⊗B∨, where Λ is a dominant integral weight µ ν and µ,ν are partitions, and it is obtained by analyzing non-trivial commutation relations for monomials in z and γ±, equivalently Pieri rules for extremal weight crystals (Proposition 4.12). 4 JAE-HOONKWON Finally, we study the action of D on R as differential operators. Let C∨ be the category of gl -crystals consisting of dual crystals B∨ for B ∈ C and let Cl.w. be its ∞ subcategorywhereaconnectedcomponentofitsobjectisisomorphictoanintegrable lowest weight crystal. Wedenote byK ∨ andK l.w. thecorrespondingGrothendieck groups. TheisomorphismsK ∨ ≃ D,K l.w. ≃ R inducealeftK ∨-modulestructure on K l.w.. It corresponds to the composite of two functors as follows; C∨×Cl.w. −⊗→ C∨ −p→r Cl.w., where pr is the natural projection. Using the Littlewood-Richardson rule, we obtain an explicit combinatorial description of the action of K ∨ on K l.w.. We observe that the action of level zero extremal weight crystals are transitive on the set of integrable lowest weight crystals with a fixed level. As an application, we obtain a newinterpretation ofaone-to-one correspondencebetween levelnintegrablehighest (or lowest) weight gl -modules and finite dimensional gl -modules which comes ∞ n from the (gl ,gl )-duality on level n Fock space [6] (Theorem 6.8). As another ∞ n application, we construct an action of the Hall-Littlewood vertex operators [10] on Z[t] ⊗ K l.w. (Theorem 6.15), which naturally yields an A -analogue of Hall- ∞ Littlewood function. For a combinatorial realization, most gl -crystals in this paper are embedded in ∞ a set of binary matrices of various shapes, equivalently (infinite) abacus model, and a double crystal structure on binary matrices [4, 20], which produces various dualities, plays a crucial role to prove our main results, while the rational semistandard tableaux for gl [24, 25] was used to understand extremal weight A -crystals [19]. n +∞ We also expect a similar result for the other infinite rank affine Lie algebras, that is, roughly speaking, the Grothendieck ring generated by extremal weight crystals can be realized as a ring of differential operators on the integrable irreducible characters. The paper is organized as follows. In Section 2, we review briefly the notion of crystals. In Section 3, we introduce a double crystal (or bicrystal) structure on binary matrices, which is our main method. In Section 4, wegive a characterization of extremal weight crystals. In Section 5, we introduce the monoidal category C, char- acterize its Grothendieck ring, and give a Littlewood-Richardson rule for extremal weight crystals. In Section 6, we study the action of K ∨ on K l.w. induced from that of D on R and discuss its applications. LITTLEWOOD-RICHARDSON RULE OF EXTREMAL WEIGHT CRYSTALS 5 2. Crystals 2.1. The Lie algebra gl . Let gl denote the Lie algebra of complex matrices ∞ ∞ (a ) with finitely many non-zero entries and let U (gl ) be its quantized en- ij i,j∈Z q ∞ veloping algebra. Let E be the elementary matrix with 1 at the i-th row and the j-th column and ij zero elsewhere. Let h= CE betheCartan subalgebraof gl and h·,·i denote i∈Z ii ∞ the natural pairing on h∗×h. We denote by Π∨ = {h = E −E |i ∈ Z} the L i ii i+1,i+1 set of simple coroots, and Π = {α = ǫ −ǫ |i ∈ Z} the set of simple roots, where i i i+1 ǫ ∈ h∗ is determined by hǫ ,E i = δ . i i jj ij For i∈ Z, let r be the simple reflection given by r (λ) = λ−hλ,h iα for λ ∈h∗. i i i i Let W be the Weyl group of gl , that is, the subgroup of GL(h∗) generated by r ∞ i for i∈ Z. LetP = Zǫ ⊕ZΛ = ZΛ theweightlatticeofgl ,whereΛ isdefined i∈Z i 0 i∈Z i ∞ 0 by hΛ ,E i = −hΛ ,E i = 1 for j ≥ 1, and Λ is given by Λ − 0 ǫ 0 −jL+1,−j+1 0 Ljj 2 i 0 k=i k (resp. Λ + i ǫ ) fori < 0(resp. i> 0). We call Λ the i-th fundamentalweight. 0 k=1 k i P A partial ordering on P is defined as usual. P For k ∈ Z, let P = kΛ + Zǫ be the set of integral weights of level k. Let k 0 i∈Z i P+ = {Λ ∈ P |hΛ,h i ≥ 0, i ∈ Z} = Z Λ be the set of dominant integral i L i∈Z ≥0 i weights. We also put P+ = P+ ∩P for k ∈ Z. k k P Note that for Λ = c Λ ∈ P, the level of Λ is c since ǫ = Λ −Λ i∈Z i i i∈Z i i i i−1 for i∈ Z. If we put Λ = c Λ , then Λ= Λ −Λ with Λ ∈ P+. P± i;ci≷0 i i + P− ± 2.2. Review on crystalsP. Let us briefly recall the notion of crystals (see [13] for a general review and references therein). A gl -crystal is a set B together with the maps wt : B → P, ε ,ϕ : B → ∞ i i Z∪{−∞} and e ,f : B → B∪{0} for i∈ Z, which satisfy the following conditions; i i (1) for b ∈B, we have e e ϕ (b) = hwt(b),h i+ε (b), i i i (2) if e b ∈ B for b ∈ B, then i ε (e b)= ε (b)−1, ϕ (e b) = ϕ (b)+1, wt(e b)= wt(b)+α , i i i i i i i i e (3) if f b ∈B for b ∈B, then i e e e ε (f b)= ε (b)+1, ϕ (f b)= ϕ (b)−1, wt(f b)= wt(b)−α , ei i i i i i i i (4) f b = b′ if and only if b = e b′ for all i ∈ Z, b,b′ ∈ B, i e ei e (5) If ϕ (b) = −∞, then e b = f b = 0, i i i e e where 0 is a formal symbol and −∞ is the smallest element in Z∪{−∞} such that e e −∞+n = −∞ for all n ∈ Z. 6 JAE-HOONKWON For Λ ∈ P+, we denote by B(±Λ) the crystal base of the irreducible U (gl )- q ∞ module with highest (resp. lowest) weight vector u of weight ±Λ, which are ±Λ gl -crystals. For Λ ∈ P, let T = {t } be a gl -crystal with wt(t ) = Λ, e t = ∞ Λ Λ ∞ Λ i Λ f t = 0, and ε (t )= ϕ (t ) = −∞ for i∈ Z. We denote by B and B∨ the crystal i Λ i Λ i Λ base of the natural representation of U (gl ) and its dual respectively, wheere the q ∞ e associated colored oriented graphs are as follows; −3 −2 −1 0 1 2 3 B : ··· −→ −2 −→ −1 −→ 0 −→ 1 −→ 2 −→ 3−→ ··· , B∨ : ··· −3→ 3∨ −2→ 2∨ −1→ 1∨ −0→ 0∨ −−→1 −1∨ −−→2 −2∨ −−→3 ··· . Let I be an interval in Z. We denote by gl the subalgebra of gl spanned I ∞ by E for i,j ∈ I. One can define gl -crystals in the same way as in gl . For ij I ∞ p,q ∈ Z, we put [p,q] = {p,p + 1,...,q} (p < q), and [p,∞) = {p,p + 1,...}, (−∞,q]= {...,q−1,q}. For n≥ 1, we denote [1,n] by [n] for simplicity. In the rest of this section, let us review some necessary terminologies on crystals. Here a crystal means a gl -crystal, where I is an interval in Z (see [13] for more I details). Let B be a crystal. Then B is equipped with a colored oriented graph structure, where b →i b′ if and only if b′ = f b (i ∈ Z). B is called connected if it is connected i as a graph. We say that B is regular if ε (eb) = max{k|ekb 6= 0} and ϕ (b) = max{k|fkb 6= i i i i 0} for all i. For example, B(±Λ) is regular for Λ ∈ P+. If B is regular, then there exists an action of the Weyl group on B as foellows; for a simple reflection reand i b ∈ B fhwt(b),hiib, if hwt(b),h i≥ 0, S b = i i ri e−hwt(b),hiib, if hwt(b),h i≤ 0, ei i where we denote by S the operator corresponding to an Weyl group element w. w  Also we defineemaxb = eεi(b)b aendfmaxb = fϕi(b)b for i andan element b in aregular i i i i crystal. The dual cryestal B∨ eof B is defiened to beethe set {b∨|b ∈ B} with wt(b∨) = −wt(b), ε (b∨)= ϕ (b), ϕ (b∨)= ε (b), i i i i ∨ e (b∨)= f b , f (b∨) = (e b)∨, i i i i (cid:16) (cid:17) for b ∈ B and all i. We aessume thate0∨ = 0e. e Let B and B be crystals. A morphism ψ : B → B is a map from B ∪{0} to 1 2 1 2 1 B ∪{0} such that 2 LITTLEWOOD-RICHARDSON RULE OF EXTREMAL WEIGHT CRYSTALS 7 (1) ψ(0) = 0, (2) wt(ψ(b)) = wt(b), ε (ψ(b)) = ε (b), andϕ (ψ(b)) = ϕ (b) whenever ψ(b) 6= 0, i i i i (3) ψ(e b)= e ψ(b) for b ∈B and i such that ψ(b) 6= 0 and ψ(e b)6= 0, i i 1 i (4) ψ(f b)= f ψ(b) for b ∈ B and i such that ψ(b) 6= 0 and ψ(f b) 6= 0. i i 1 i e e e We call ψ an embedding and B a subcrystal of B when ψ is injective, and call ψ 1 2 e e e strict if ψ : B ∪{0} → B ∪{0} commutes with e and f for all i, where we assume 1 2 i i that e 0= f 0 = 0. i i A tensor product of B and B is defined to bee the seet B ⊗B = {b ⊗b |b ∈ 1 2 1 2 1 2 i B (ie= 1,2)e} with i wt(b ⊗b )= wt(b )+wt(b ), 1 2 1 2 ε (b ⊗b )= max(ε (b ),ε (b )−hwt(b ),h i), i 1 2 i 1 i 2 1 i ϕ (b ⊗b ) = max(ϕ (b )+hwt(b ),h i,ϕ (b )), i 1 2 i 1 2 i i 2 e b ⊗b , if ϕ (b )≥ ε (b ), i 1 2 i 1 i 2 e (b ⊗b )= i 1 2  b ⊗e b , if ϕ (b )< ε (b ),  1 i 2 i 1 i 2 e e  f b ⊗eb , if ϕ (b )> ε (b ), i 1 2 i 1 i 2 f (b ⊗b )= i 1 2  b ⊗f b , if ϕ (b )≤ ε (b ), e1 i 2 i 1 i 2 e for all i, where we assume that 0⊗b = b ⊗0 = 0. Then B ⊗B is a crystal.  2e 1 1 2 Note that (B ⊗B )∨ ≃ B∨⊗B∨. 1 2 2 1 For b ∈ B (i = 1,2), we say that b is (gl -)equivalent to b , and write b ≡ b i i 1 I 2 1 2 if there exists a crystal isomorphism C(b ) → C(b ) sending b to b , where C(b ) 1 2 1 2 i denote the connected component of B including b (i = 1,2). i i 3. Double crystal structure on binary matrices 3.1. Crystal operators on binary matrices. For intervals I,J in Z, let M be I,J the set of I ×J matrices A = (a ) with a ∈ {0,1}. We denote by A and Aj the ij ij i i-th row and the j-th column of A for i∈ I and j ∈ J, respectively. Suppose that A ∈ M and k ∈ J with k+1 ∈ J are given. For each i ∈ I, we I,J define A −E +E , if a = 1 and a = 0, i k k+1 ik ik+1 (3.1) f A = k i 0, otherwise,  e where Ek is the row vector whose components are indexed by J with 1 in the k-th component and 0 elsewhere. In a similar way, we define e A = A +E −E if k i i k k+1 a = 0 and a = 1, and 0 otherwise. ik ik+1 e 8 JAE-HOONKWON Consider the sequence ǫ (A) = (ǫ (A )) , where ǫ (A ) = + if f A 6= 0, − if k k i i∈I k i k i e A 6= 0, and · otherwise. We say that A is k-admissible if there exist M,N ∈ I k i e such that e (1) ǫ (A ) 6= + for all i < M, k i (2) ǫ (A ) 6= − for all i > N. k i Note that if I is finite, then A is k-admissible for all k ∈J. Supposethat A is k-admissible. We replace a pair (ǫ (A ),ǫ (A )) = (+,−) such k s k s′ that M ≤ s < s′ ≤ N and ǫ (A ) = · for s < t < s′ by (·, ·) in ǫ (A), and repeat k t k this process as far as possible until we get a sequence with no + placed to the left of −. We call the right-most − (resp. left-most +) in this reduced form of ǫ (A) the k k-good − sign (resp. k-good + sign). We define e A (resp. f A) to be the matrix k k in M given by applying e (resp. f ) to the row of A having k-good − (resp. +) I,J k k sign, and 0 if there is no k-good sign. Note that ief A is k-admeissible, then enA and k fnA are well-defined for allen ≥ 1. e k Next, we define another operators E and F on A ∈M for l ∈I with le+1∈ I. l l I,J e Define a bijection e e ρ: M −→ M I,J −J,I byρ(A) = (a )∈ M , where−J = {−j|j ∈ J}. We say thatAis l-admissible i,−j −J,I if ρ(A) is l-admissible in the above sense. Then for an l-admissible A, we define (3.2) E (A) = ρ−1(e ρ(A)), F (A) = ρ−1 f ρ(A) . l l l l (cid:16) (cid:17) The following leemma followsefrom [16, Peroposition 4.e2] (see also [4, 20], where essentially the same facts are stated in a slightly different way). Lemma 3.1. Let A ∈ M be given. If A is both k-admissible and l-admissible for I,J some l ∈ I and k ∈ J, then x X A= X x A, k l l k where x= e,f and X = E,F. e e e e For A ∈ M , we say that A is J-admissible (resp. I-admissible) if A is k- I,J admissible (resp. l-admissible) for all k ∈ J with k + 1 ∈ J (resp. l ∈ I with l +1 ∈ I). Note that if J (resp. I) is a finite set, then A is always I-admissible (resp. J-admissible). If A is I-admissible, then we obtain a connected I-colored oriented graph obtained by applying E , F ’s to A, which we denote by C (A). k k I Lemma 3.1 implies the following immediately. e e Lemma 3.2. Let A ∈ M be given. Suppose that A is I-admissible and k- I,J admissible for some k ∈ J. If x A 6= 0 (x = e,f), then there exists an isomorphism k C (A) −→ C (x A) of I-colored oriented graph sending A to x A. I I k k e e e LITTLEWOOD-RICHARDSON RULE OF EXTREMAL WEIGHT CRYSTALS 9 We have similar statements when A is J-admissible and l-admissible for some l ∈I. For A = (a )∈ M , put ij I,J (3.3) A∨ = (1−a ) . ij i∈I,j∈J Suppose that A and A∨ are k-admissible for some k ∈ J. Then we have ∨ (3.4) f A = e A∨ ∨, e A= f A∨ . k k k k (cid:16) (cid:17) (cid:0) (cid:1) The same statement holeds for Eel and Fl, wehen A aned A∨ are l-admissible for some l ∈I. We call A∨ the dual of A. e e 3.2. Crystals of semistandard tableaux. Let P denote the set of partitions. We identify a partition λ = (λ ) with a Young diagram as usual (see [21]). The i i≥1 number of non-zero parts in λ is denoted by ℓ(λ), and called the length of λ. We denote by λ′ the conjugate partition of λ. For a skew Young diagram λ/µ, |λ/µ| denotes the number of dots or boxes in the diagram. Let A be a linearly ordered set. A tableau T obtained by filling λ/µ with entries in A is called a semistandard tableau of shape λ/µ if the entries in each row are weakly increasing from left to right, and the entries in each column are strictly increasing from top to bottom. We denote bySSTA(λ/µ) theset of all semistandard tableaux of shapeλ/µ with entries in A. For T ∈ SSTA(λ/µ), let w(T)col (resp. w(T)row) denote the word of T with respect to column (resp. row) reading, where we read the entries column by column (resp. row by row) from right to left (resp. top to bottom), and in each column (resp. row) from top to bottom (resp. right to left). LetAdenoteoneofthecrystalsBandB∨,whicharelinearlyorderedwithrespect to the partial ordering on P. Note that the set of all finite words with letters in A is a gl -crystal, where each word of length r ≥ 1 is identified with an element ∞ in A⊗r = A⊗···⊗A (r times). Given a skew Young diagram λ/µ, the injective image of SSTA(λ/µ) in the set of all finite words under the map T 7→ w(T)col (or w(T) ) together with {0} is invariant under e ,f . Hence it is a gl -crystal [15]. row i i ∞ In particular, for λ ∈ P, we have e e SSTB(λ)∨ ≃ SSTB∨(λ∨), where λ∨ is the skew Young diagram obtained from λ by 180◦-rotation. For µ ∈ P, we put (3.5) Bµ = SSTB(µ), and we identify B∨µ with SSTB∨(µ∨). Note that Bµ has neither a highest nor lowest weight vector. 10 JAE-HOONKWON Proposition 3.3. B ⊗B∨ and B∨⊗B are connected for µ,ν ∈ P. µ ν ν µ Proof. First, we claim that B is connected. Suppose that S,T ∈ B are given. µ µ Choose p ∈ Z such that all entries in S and T are greater than p. Then S is an element in SST (µ), which is a connected gl -crystal with a unique highest [p,∞) [p,∞) [p,∞) weight element, say u . This implies that S and T are contained in the same µ connected component, and hence B is connected. µ Let S ⊗T ∈ B ⊗B∨ be given. Choose p ∈ Z such that S ∈ SST (µ). Then µ ν [p,∞) [p,∞) we have e ···e S = u for some i ,...,i ∈ [p,∞). By tensor product rule of i1 ir µ 1 r crystals, we also have e e em1···emr(S ⊗T)= u[p,∞)⊗T′, i1 ir µ for some m ,...,m ≥ 1 and T′ ∈ B∨. 1 r e e ν Ifpissufficientlysmall,thenwemayassumethatalltheentriesinT (andhencein T′)aresmallerthan(p+ℓ(µ))∨. ChooseqsuchthatT′ ∈ (SST (ν))∨. Notethat (−∞,q] (SST (ν))∨ is a gl -crystal with a unique highest weight element v(−∞,q]. (−∞,q] (−∞,q] ν Hence e ···e T′ = v(−∞,q] for some j ,...,j ∈ (−∞,q]. Since {j ,...,j } does j1 js ν 1 s 1 s [p,∞) not intersect with the entries in u , we have µ e e e ···e u[p,∞)⊗T′ = u[p,∞)⊗v(−∞,q]. j1 js µ µ ν (cid:16) (cid:17) Now, let U⊗V ∈ B begiven. Then if p is sufficiently small and q is sufficiently µ,ν e e large, then it follows from the same argument that U ⊗ V is also connected to u[p,∞)⊗v(−∞,q]. This implies that B ⊗B∨ is connected. µ ν µ ν The proof of the connectedness of B∨⊗B is almost the same. So we leave it to ν µ the readers. (cid:3) Remark 3.4. In the proof of Proposition 3.3, we showed that for S⊗T ∈B ⊗B∨ µ ν there exist k ,...,k ∈ Z such that e ···e (S ⊗T) = u[p,∞) ⊗ v(−∞,q] for some 1 t k1 kt µ ν p < q. By applying suitable e ’s, we may assume that p ≪ 0 ≪ q. k e e For a ≥1, define an injective map e (3.6) σ :B −→ M a (1a) {1},Z by σ (S) = (a ) with wt(S) = a ǫ . It is easy to see that σ commutes with a 1i i∈Z i 1i i a e , f (k ∈ Z), where we assume that σ (0) = 0 (see Section 3.1 for the definitions k k P a of e and f on M ). Similarly, for b ≥ 1 we define an injective map k k {1},Z e e (3.7) τ : B∨ −→ M e e b (1b) {1},Z by τ (T) = (a ) with wt(T) = (a − 1)ǫ . Then τ commutes with e , f b 1i i∈Z i 1i i b k k (k ∈ Z). For convenience, we assume that σ is a map sending trivial crystal to zero P 0 e e