Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space
aa r X i v : . [ m a t h . R T ] D ec Product formulas for the cyclotomic v -Schur algebraand for the canonical bases of the Fock space Toshiaki Shoji and Kentaro Wada ∗ To Gus Lehrer on the occasion of his 60th birthday
Graduate School of MathematicsNagoya UniversityChikusa-ku, Nagoya 464-8602, Japan
Abstract.
Let F q [ s ] be the q -deformed Fock space of level l with multi-charge s = ( s , . . . , s l ). Uglov defined canonical bases G ± ( λ, s ) of F q [ s ] for l -partitions λ ,and the polynomials ∆ ± λµ ( q ) ∈ Z [ q ± ] are defined as the coefficients of the transi-tion matrix between the canonical bases and the standard basis of F q [ s ]. In thispaper, we prove a product formula for ∆ ± λµ ( q ) for certain l -partitions λ, µ , whichexpresses ∆ ± λµ ( q ) as a product of such polynomials related to various smaller Fockspaces F q [ s [ i ] ]. Yvonne conjectures that ∆ + λµ ( q ) are related to the q -decompositionnumbers d λµ ( q ) of the cyclotomic v -Schur algebra S ( Λ ), where the parameters v, Q , . . . , Q l are roots of unity in C determined from s . In our earlier work, wehave proved a product formula for d λµ ( q ) of S ( Λ ), and the product formula for ∆ ± λµ ( q ) is regarded as a counter-part of that formula for the case of the Fock space. Introduction
For a given integer l >
0, let Π l be the set of l -partitions λ = ( λ (1) , . . . , λ ( l ) ).Here λ ( i ) = ( λ ( i )1 ≥ λ ( i )2 ≥ · · · ) is a partition for i = 1 , . . . , l . We denote by | λ | = P li =1 | λ ( i ) | , and call it the size of λ , where | λ ( i ) | = P j λ ( i ) j . For given integers l, n ≥ q , we consider the q -deformed Fock space F q [ s ] of level l with multi-charge s = ( s , . . . , s l ) ∈ Z l ≥ , which is a vector space over Q ( q ) withthe standard basis {| λ, s i | λ ∈ Π l } , equipped with an action of the affine quantumgroup U q ( b sl n ). In [U], Uglov constructed the canonical bases G ± ( λ, s ) for F q [ s ] withrespect to U q ( b sl n ), and for each λ, µ ∈ Π l , he defined a polynomial ∆ ± λ,µ ( q ) ∈ Q [ q ± ]by the formula G ± ( λ, s ) = X µ ∈ Π l ∆ ± λµ ( q ) | µ, s i , where ∆ ± λµ = 0 unless | λ | = | µ | . ∗ Both authors would like to thank B. Leclerc for valuable discussions.
The cyclotomic v -Schur algebra S ( Λ ) with parameters v, Q , . . . , Q l over a filed R , associated to H N,l was introduced by Dipper-James-Mathas [DJM], where H N,l is the Ariki-Koike algebra associated to the complex reflection group S N ⋉ ( Z /l Z ) N .Let Λ + be the set of λ ∈ Π l such that | λ | = N . S ( Λ ) is a cellular algebra in thesense of Graham-Lehrer [GL], and the Weyl module W λ and its irreducible quotient L λ are defined for each λ ∈ Λ + . The main problem in the representation theory of S ( Λ ) is the determination of the decomposition numbers [ W λ : L µ ] for λ, µ ∈ Λ + .By making use of the Jantzen filtration of W λ , the q -decomposition number d λµ ( q )is defined, which is a polynomial analogue of the decomposition number, and wehave d λµ (1) = [ W λ : L µ ].Let p = ( l , . . . , l g ) be a g -tuple of positive integers such that l + · · · + l g = l .For each µ = ( µ (1) , . . . , µ ( l ) ) ∈ Λ + , one can associate a g -tuple of multi-partitions( µ [1] , . . . , µ [ g ] ) by using p , where µ [1] = ( µ (1) , . . . , µ ( l ) ) , µ [2] = ( µ ( l +1) , . . . , µ ( l + l ) )and so on. We define α p ( µ ) = ( N , . . . , N g ), where N i = | µ [ i ] | . Hence for λ, µ ∈ Λ + , α p ( λ ) = α p ( µ ) means that | λ [ i ] | = | µ [ i ] | for i = 1 , . . . , g . Let S ( Λ N i ) be the cyclo-tomic v -Schur algebra associated to H N i ,l i with parameters v, Q [ i ]1 , . . . , Q [ i ] l i , where Q [1]1 = Q , . . . , Q [1] l = Q l , Q [2]1 = Q l +1 , Q [2]2 = Q l +2 , . . . . In [SW], the product for-mula for the decomposition numbers of S ( Λ ) was proved, and it was extended in[W] to the product formula for q -decomposition numbers, which is given as follows;for λ, µ ∈ Λ + such that α p ( λ ) = α p ( µ ), we have(*) d λµ ( q ) = g Y i =1 d λ [ i ] µ [ i ] ( q ) , where d λ [ i ] µ [ i ] ( q ) is the q -decomposition number for S ( Λ N i ).We assume that the parameters are given by ( v ; Q , . . . , Q l ) = ( ξ, ξ s , . . . , ξ s l ),where ξ is a primitive n -th root of unity in C and s = ( s , . . . , s l ) is a multi-charge.For an integer M , we say that | λ, s i is M -dominant if s i − s i +1 > | λ | + M for i = 1 , . . . , l −
1. Yvonne [Y] gave a conjecture that d λµ ( q ) coincides with ∆ + µ † λ † ( q ) if | λ, s i is 0-dominant, where λ † , µ † are certain elements in Λ + induced from λ, µ (seeRemark 2.7).In view of Yvonne’s conjecture, it is natural to expect a formula for ∆ + λµ ( q )as a counter-part for the Fock space of the product formula for d λµ ( q ). In fact,our result shows that it is certainly the case. We write s = ( s [1] , . . . , s [ g ] ) with s [1] = ( s , . . . , s l ) , s [2] = ( s l +1 , . . . , s l + l ), and so on. Let F q [ s [ i ] ] be the q -deformedFock space of level l i with multi-charge s [ i ] for i = 1 , . . . , g . Then one can definepolynomials ∆ ± λ [ i ] µ [ i ] ( q ) for F q [ s [ i ] ] similar to F q [ s ]. The main result in this paper isthe following product formula (Theorem 2.9); assume that | λ, s i is M -dominant for M > n . Then for λ, µ ∈ Λ + such that α p ( λ ) = α p ( µ ), we have(**) ∆ ± λµ ( q ) = g Y i =1 ∆ ± λ [ i ] µ [ i ] ( q ) . RODUCT FORMULAS 3
Yvonne’s conjecture implies, by substituting q = 1, that [ W λ : L µ ] = ∆ + µ † λ † (1),which we call LLT-type conjecture. In the case where l = 1, i.e., the case where S ( Λ ) is the v -Schur algebra of type A , it is known by Varagnolo-Vasserot [VV] thatLLT-type conjecture holds. By applying the formulas (*), (**) to the case where p = (1 , . . . , S ( Λ ), but assume that | λ ( i ) | = | µ ( i ) | for i = 1 , . . . , l . Then LLT-typeconjecture holds (under a stronger dominance condition) for [ W λ : L µ ].This paper is organized as follows; in Section 1, we give a brief survey on theproduct formula for S ( Λ ) based on [SW], [W], which is a part of the first author’stalk at the conference in Canberra, 2007. In Section 2 and 3, we prove the productformula (**) for the canonical bases of the Fock space.1. Product formula for the cyclotomic v -Schur algebra Let H = H N,l be the Ariki-Koike algebra over an integral domain R associated to the complex reflection group W N,l = S N ⋉ ( Z /l Z ) N with parameters v, Q , . . . , Q l ∈ R such that v is invertible, which is an associative algebra withgenerators T , T , . . . , T N − and relations( T − Q ) · · · ( T − Q l ) = 0( T i − v )( T i + v − ) = 0 for i = 1 , . . . , N − T , . . . , T N − is isomorphicto the Iwahori-Hecke algebra associated to the symmetric group S N , and has a basis { T w | w ∈ S N } , where T w = T i . . . T i r for a reduced expression w = s i . . . s i r . of w .An element µ = ( µ , . . . , µ m ) ∈ Z m ≥ is called a composition of | µ | consisting of m -parts, where | µ | = P mi =1 µ i . A composition µ is called a partition if µ ≥ · · · ≥ µ m ≥
0. An l -tuple of compositions (resp. partitions) µ = ( µ (1) , . . . , µ ( l ) ) is calledan l -composition (resp. an l -partition) of | µ | , where | µ | = P i | µ ( i ) | . We define, following [DJM], a cyclotomic v -Schur algebra S ( Λ ) associatedto H . Fix m = ( m , . . . , m l ) ∈ Z l> and let Λ = Λ N,l ( m ) (resp. Λ + = Λ + N,l ( m ))be the set of l -compositions (resp. l -partitions) µ = ( µ (1) , . . . , µ ( l ) ) of N such that µ ( i ) has m i -parts. For each µ ∈ Λ , we define an element m µ ∈ H as follows; define L k ∈ H (1 ≤ k ≤ N ) by L = T and by L k = T k − L k − T k − for k = 2 , . . . , N . Then L , . . . , L N commute each other. For each µ ∈ Λ , we define a = a ( µ ) = ( a , . . . , a l )by a k = P k − i =1 | µ ( i ) | for k = 2 , . . . , l , and by a = 0. Put u + a = l Y k =1 a k Y i =1 ( L i − Q k ) , x µ = X w ∈ S µ v l ( w ) T w , where l ( w ) is the length of w ∈ S n , and S µ is the Young subgroup of S n corre-sponding to µ . Then u + a commutes with x µ , and we put m µ = u + a x µ . SHOJI
For each µ ∈ Λ , we define a right H -module M µ by M µ = m µ H , and put M = L µ ∈ Λ M µ . Then the cyclotomic q -Schur algebra S ( Λ ) is defined as S ( Λ ) = End H M = M ν,µ ∈ Λ Hom H ( M µ , M ν ) . For each λ ∈ Λ + and µ ∈ Λ , the notion of semistandard tableau of shape λ and type µ was introduced by [DJM], extending the case of a partition λ and acomposition µ . We denote by T ( λ, µ ) the set of semistandard tableau of shape λ and type µ for λ ∈ Λ + , µ ∈ Λ . The notion of dominance order for partitions is alsogeneralized for Λ , which we denote by µ ⊳ ν . Note that T ( λ, µ ) is empty unless λ D µ . We put T ( λ ) = S µ ∈ Λ T ( λ, µ ).For each S ∈ T ( λ, µ ) , T ∈ T ( λ, ν ), Dipper-James-Mathas [DJM] constructedan H -equivariant map ϕ ST : M ν → M µ . They showed that S ( Λ ) is a cellularalgebra, in the sense of Graham-Lehrer [GL] with cellular basis C ( Λ ) = { ϕ ST | S, T ∈ T ( λ ) for some λ ∈ Λ + } . Let p = ( l , . . . , l g ) be a g -tuple of positive integers such that P gi =1 l i = l . For each µ = ( µ (1) , . . . , µ ( l ) ), one can associate a g -tuple of multi-compositions( µ [1] , . . . , µ [ g ] ) by making use of p , where µ [1] = ( µ (1) , . . . , µ ( l ) ) , µ [2] = ( µ ( l +1) , . . . , µ ( l + l ) ) , · · · . For example assume that N = 20 , l = 5 and p = (2 , , µ = (21; 121; 32; 1 ; 41).Then µ is written as µ = ( µ [1] , µ [2] , µ [3] ) with µ [1] = (21; 121) , µ [2] = (32; 1 ) , µ [3] = (41) . For µ = ( µ (1) , . . . , µ ( l ) ) = ( µ [1] , . . . , µ [ g ] ) ∈ Λ , put α p ( µ ) = ( N , . . . , N g ) , a p ( µ ) = ( a , . . . , a g ) , where N k = | µ [ k ] | , and a k = P k − i =1 N i for k = 1 , . . . , g with a = 0.Note that we often use the following relation. Take λ = ( λ [1] , . . . , λ [ g ] ) , µ =( µ [1] , . . . , µ [ g ] ) ∈ Λ . Then α p ( λ ) = α p ( µ ) if and only if | λ [ k ] | = | µ [ k ] | for k = 1 , . . . , g . Put C p = { ϕ ST ∈ C ( Λ ) | S ∈ T ( λ, µ ) , T ∈ T ( λ, ν ) , a p ( λ ) > a p ( µ ) if α p ( µ ) = α p ( ν ) , µ, ν ∈ Λ, λ ∈ Λ + } , where a p ( λ ) = ( a , . . . , a g ) ≥ a p ( µ ) = ( b , . . . , b g ) if a k ≥ b k for k = 1 , . . . , g , and a p ( λ ) > a p ( µ ) if a p ( λ ) ≥ a p ( µ ) and a p ( λ ) = a p ( µ ). Let S p be the R -span of C p .Then by using the cellular structure, one can show that S p is a subalgebra of S ( Λ )containing the identity in S ( Λ ). The algebra S p turns out to be a standardly based RODUCT FORMULAS 5 algebra in the sense of Du-Rui [DR] with respect to the poset Σ p , where Σ p is asubset of Λ + × { , } given by Σ p = ( Λ + × { , } ) \{ ( λ, | T ( λ, µ ) = ∅ for any µ ∈ Λ such that a p ( λ ) > a p ( µ ) } , and the partial order ≥ on Σ p is defined as ( λ , ε ) > ( λ , ε ) if λ ⊲ λ or λ = λ and ε > ε . For each λ ∈ Λ + , µ ∈ Λ , we define a set T p ( λ, µ ) by T p ( λ, µ ) = ( T ( λ, µ ) if a p ( λ ) = a p ( µ ) , ∅ otherwise,and put T p ( λ ) = S µ ∈ Λ T p ( λ, µ ).Let b S p be the R -submodule of S p spanned by b C p = C p \{ ϕ ST | S, T ∈ T p ( λ ) for some λ ∈ Λ + } . Then b S p turns out to be a two-sided ideal of S p . We denote by S p = S p ( Λ ) thequotient algebra S p / b S p . Let π : S p → S p be the natural projection, and put ϕ = π ( ϕ ) for ϕ ∈ S p . One can show that S p is a cellular algebra with cellular basis C p = { ϕ ST | S, T ∈ T p ( λ ) for λ ∈ Λ + } . Thus we have constructed, for each p , a subalgebra S p of S ( Λ ) and its quotientalgebra S p . So we are in the following situation, S p ι −−−→ S ( Λ ) π y S p where ι is an injection and π is a surjection. Remark 1.7.
In the special case where p = (1 , . . . , g = l , and µ [ k ] = µ ( k ) for k = 1 , . . . , g = l . Moreover in this case S p and S p coincide with thesubalgebra S ( m , N ) and its quotient S ( m , N ) considered in [SawS] in connectionwith the Schur-Weyl duality between H and the quantum group U v ( g ), where g = gl m ⊕ · · · ⊕ gl m l (cf. [SakS], [HS]). The other extreme case is p = ( l ), in which case S p and S p coincide with S ( Λ ). Thus in general S p are regarded as intermediateobjects between S ( Λ ) and S ( m , N ). In the rest of this section, unless otherwise stated we assume that R is afield. By a general theory of cellular algebras, one can define, for each λ ∈ Λ + , theWeyl module W λ and its irreducible quotient L λ for S ( Λ ). Similarly, since S p is a SHOJI cellular algebra, we have the Weyl module Z λ and its irreducible quotient L λ . Notethat S ( Λ ) (resp. S p ) is a quasi-hereditary algebra, and so the set { L λ | λ ∈ Λ + } (resp. { L λ | λ ∈ Λ + } ) gives a complete set of representatives of irreducible S ( Λ )-modules (resp. irreducible S p -modules).On the other hand, by using the general theory of standardly based algebras,one can construct, for each η = ( λ, ε ) ∈ Σ p , the Weyl module Z η , and its irreduciblequotient L η (if it is non-zero). Thus the set { L η | η ∈ Σ p , L η = 0 } gives a completeset of representatives of irreducible S p -modules. In the case where η = ( λ, L ( λ, is always non-zero for λ ∈ Λ + , and the composition factors of Z ( λ, are all of the form L ( µ, for some µ ∈ Λ + . We shall discuss the relationsamong the decomposition numbers[ W λ : L µ ] S ( Λ ) , [ Z ( λ, : L ( µ, ] S p , [ Z λ : L µ ] S p . (In order to distinguish the decomposition numbers for S ( Λ ) , S p and S p , we use thesubscripts such as [ W λ : L µ ] S ( Λ ) ). In the case where p = (1 , . . . , p . First we consider the relation between the decomposition numbers of S p and that of S p . Under the surjection π : S p → S p , we regard an S p -module as an S p -module. We have the following lemma. Lemma 1.10.
For λ, µ ∈ Λ + , we have (i) Z λ ≃ Z ( λ, as S p -modules. (ii) L µ ≃ L ( µ, as S p -modules. (iii) [ Z λ : L µ ] S p = [ Z ( λ, : L ( µ, ] S p . (iv) Assume that α p ( λ ) = α p ( µ ) . Then we have [ Z λ : L µ ] S p = 0 . Next we consider the relationship between the decomposition numbersof S p and that of S ( Λ ). Under the injection ι : S p ֒ → S ( Λ ), we regard an S ( Λ )-module as an S p -module. The following proposition was first proved in [Sa] in thecase where p = (1 , , . . . , p . Proposition 1.12.
For each λ ∈ Λ + , there exists an isomorphism of S ( Λ ) -modules Z ( λ, ⊗ S p S ( Λ ) ≃ W λ . By using Lemma 1.10 and Proposition 1.12, we have the following theorem,which is a generalization of [Sa, Th. 5.7]. In fact in the theorem, the inequality[ Z ( λ, : L ( µ, ] S p ≤ [ W λ : L µ ] S ( Λ ) always holds, and the converse inequality holds only when α p ( λ ) = α p ( µ ). RODUCT FORMULAS 7
Theorem 1.13 ([SW, Th. 3.13]) . For any λ, µ ∈ Λ + such that α p ( λ ) = α p ( µ ) , wehave [ Z λ : L µ ] S p = [ Z ( λ, : L ( µ, ] S p = [ W λ : L µ ] S ( Λ ) . In view of Theorem 1.13, the determination of the decomposition num-bers [ W λ : L µ ] is reduced to that of the decomposition numbers [ Z λ : L µ ] S p for S p asfar as the case where α p ( λ ) = α p ( µ ). The algebra S p has a remarkable structure asthe following formula shows. In order to state our result, we prepare some notation.For each N k ∈ Z ≥ , put Λ N k = Λ N k ,l k ( m [ k ] ) and Λ + N k = Λ + N k ,l k ( m [ k ] ). ( Λ N k or Λ N + k isregarded as the empty set if N k = 0.) For each µ [ k ] ∈ Λ N k , the H N k ,l k -module M µ [ k ] is defined as in the case of the H -module M µ , and the cyclotomic q -Schur algebra S ( Λ N k ) associated to the Ariki-Koike algebra H N k ,l k is defined. The following the-orem was first proved in [SawS] for p = (1 , , . . . ,
1) under a certain condition onparameters. Here we don’t need any assumption on parameters.
Theorem 1.15 ([SW, Th. 4.15]) . There exists an isomorphism of R -algebras S p ≃ M ( N ,...,N g ) N + ··· + N g = N S ( Λ N ) ⊗ · · · ⊗ S ( Λ N g ) . For λ [ k ] , µ [ k ] ∈ Λ + N k , let W λ [ k ] be the Weyl module, and L µ [ k ] be the irreduciblemodule with respect to S ( Λ N k ). As a corollary to the theorem, we have Corollary 1.16.
Let λ, µ ∈ Λ + . Then under the isomorphism in Theorem 1.15, wehave the following. (i) Z λ ≃ W λ [1] ⊗ · · · ⊗ W λ [ g ] . (ii) L µ ≃ L µ [1] ⊗ · · · ⊗ L µ [ g ] . (iii) [ Z λ : L µ ] S p = (Q gi =1 [ W λ [ i ] : L µ [ i ] ] S ( Λ Ni ) if α p ( λ ) = α p ( µ ) , otherwise. Combining this with Theorem 1.13, we have the following product formula forthe decomposition numbers of S ( Λ ). The special case where p = (1 , . . . ,
1) is dueto [Sa, Cor. 5.10], (still under a certain condition on parameters).
Theorem 1.17 ([SW, Theorem 4.17]) . For λ, µ ∈ Λ + such that α p ( λ ) = α p ( µ ) , wehave [ W λ : L µ ] S ( Λ ) = g Y i =1 [ W λ [ i ] : L µ [ i ] ] S ( Λ Ni ) . By making use of the Jantzen filtration, we shall define a polynomialanalogue of the decomposition numbers, namely for each λ, µ ∈ Λ + , we define apolynomial d λµ ( q ) ∈ Z [ q ] with indeterminate q such that d λµ (1) coincides with thedecomposition number [ W λ : L µ ] S ( Λ ) . We define similar polynomials also in the casefor S p and S p .We assume that R is a discrete valuation ring with the maximal ideal p , andlet F = R/ p be the quotient field. We fix parameters b v, b Q , . . . , b Q l in R , and let SHOJI v, Q , . . . , Q l ∈ F be their images under the natural map R → R/ p = F . Let S R = S R ( Λ ) be the cyclotomic b v -Schur algebra over R with parameters b v, b Q , . . . , b Q l , and S = S ( Λ ) be the cyclotomic v -Schur algebra over F with parameters v, Q , . . . , Q l .Thus S ≃ ( S R + p S R ) / p S R . The algebras S p R , S p R over R , and the algebras S p , S p over F are defined as before. Let W λR be the Weyl module of S R , and let h , i bethe canonical bilinear form on W λR arising from the cellular structure of S ( Λ ) R . For i = 0 , , . . . , put W λR ( i ) = { x ∈ W λR | h x, y i ∈ p i for any y ∈ W λR } and define an F -vector space W λ ( i ) = ( W λR ( i ) + p W λR ) / p W λR . Then W λ (0) = W λ is the Weyl module of S , and we have a filtration W λ = W λ (0) ⊃ W λ (1) ⊃ W λ (2) ⊃ · · · of W λ , which is the Jantzen filtration of W λ .Similarly, by using the cellular structure of S p , and by using the property of thestandardly based algebra of S p , one can define the Jantzen filtrations, Z λ = Z λ (0) ⊃ Z λ (1) ⊃ Z λ (2) ⊃ · · · ,Z ( λ, = Z ( λ, (0) ⊃ Z ( λ, (1) ⊃ Z ( λ, (2) ⊃ · · · . Since W λ (resp. Z ( λ, , Z λ ) is a finite dimensional F -vector space, the Jantzenfiltration gives a finite sequence. One sees that W λ ( i ) is an S -submodule of W λ by the associativity of the bilinear form, and similarly for Z ( λ, and Z λ . Thus wedefine a polynomial d λµ ( q ) by d λµ ( v ) = X i ≥ [ W λ ( i ) /W λ ( i + 1) : L µ ] q i , where [ M : L µ ] = [ M : L µ ] S denotes the multiplicity of L µ in the composition seriesof the S -module M as before. (In the notation below, we omit the subscripts S ,etc.) Similarly, we define, for Z ( λ, and Z λ , d ( λ, λµ ( q ) = X i ≥ [ Z ( λ, ( i ) /Z ( λ, ( i + 1) : L ( µ, ] q i ,d λµ ( q ) = X i ≥ [ Z λ ( i ) /Z λ ( i + 1) : L µ ] q i . RODUCT FORMULAS 9 d λµ ( q ) , d ( λ, λµ ( q ) and d λµ ( q ) are polynomials in Z ≥ [ q ] and we call them q -decompositionnumbers. Note that d λµ (1) coincides with [ W λ : L µ ], and similarly, we have d ( λ, λµ (1) =[ Z ( λ, : L ( λ, ], d λµ (1) = [ Z λ : L µ ].As a q -analogue of Theorem 1.13 and Theorem 1.17, we have the followingproduct formula for q -decomposition numbers. Theorem 1.19 ([W, Th. 2.8, Th. 2.14]) . For λ, µ ∈ Λ + such that α p ( λ ) = α p ( µ ) ,we have d λµ ( q ) = d λµ ( q ) = g Y i =1 d λ [ i ] µ [ i ] ( q ) . Product formula for the canonical bases of the Fock space
In the remainder of this paper, we basically follow the notation in Uglov[U]. First we review some notations. Fix positive integers n , l . Let Π l = { λ =( λ (1) , · · · , λ ( l ) ) } be the set of l -partitions. Take an l -tuple s = ( s , · · · , s l ) ∈ Z l ,and call it a multi-charge. Let U q ( b sl n ) be the quantum group of type A (1) n − . The q -deformed Fock space F q [ s ] of level l with multi-charge s is defined as a vector spaceover Q ( q ) with a basis {| λ, s i | λ ∈ Π l } , equipped with a U q ( b sl n )-module structure.The U q ( b sl n )-module structure is defined as in [U, Th. 2.1], which depends on thechoice of s . Put s = s + · · · + s l for a multi-charge s = ( s , · · · , s l ). Let P ( s ) be theset of semi-infinite sequences k = ( k , k , · · · ) ∈ Z ∞ such that k i = s − i + 1 for all i ≫
1, and P ++ ( s ) = { k = ( k , k , · · · ) ∈ P ( s ) | k > k > · · · } . For k ∈ P ( s ), put u k = u k ∧ u k ∧ · · · , and call it a semi-infinite wedge. In the case where k ∈ P ++ ( s )we call it an ordered semi-infinite wedge.Let Λ s + ∞ be a vector space over Q ( q ) spanned by { u k | k ∈ P ( s ) } satisfyingthe ordering rule [U, Prop. 3.16]. By the ordering rule any semi-infinite wedge u k can be written as a linear combination of some ordered semi-infinite wedges. It isknown (cf. [U, Prop. 4.1]) that Λ s + ∞ has a basis { u k | k ∈ P ++ ( s ) } .The vector space Λ s + ∞ is called a semi-infinite wedge product. By [U, 4.2], Λ s + ∞ has a structure of a U q ( b sl n )-module. Let Z l ( s ) = { s = ( s , · · · , s l ) ∈ Z l | s = P s i } .Then we have(2.2.1) Λ s + ∞ ≃ M s ∈ Z l ( s ) F q [ s ] as U q ( b sl n )-modules.Thus we can regard F q [ s ] as a U q ( b sl n )-submodule of Λ s + ∞ . The isomorphism in(2.2.1) is given through a bijection between two basis { u k | k ∈ P ++ ( s ) } and {| λ, s i | λ ∈ Π l , s ∈ Z l ( s ) } as in [U, 4.1]. Identifying these bases, we write u k = | λ, s i if | λ, s i corresponds to u k . For later use, we explain the explicit correspondence u k ↔ | λ, s i givenin [U, 4.1]. Assume given u k . Then for each i ∈ Z ≥ , k i is written as k i = a i + n ( b i − − nlm i , where a i ∈ { , . . . , n } , b i ∈ { , . . . , l } and m i ∈ Z are determined uniquely. For b ∈ { , . . . , l } , let k ( b )1 be equal a i − nm i where i is the smallest numbersuch that b i = b , and let k ( b )2 be equal a j − nm j where j is the next smallest numbersuch that b j = b , and so on. In this way, we obtain a strictly decreasing sequence k ( b ) = ( k ( b )1 , k ( b )2 , . . . ) such that k ( b ) i = s b − i +1 for i ≫ s b . Thus k ( b ) ∈ P ++ ( s b ), and one can define a partition λ ( b ) = ( λ ( b )1 , λ ( b )2 , . . . )by λ ( b ) i = k ( b ) i − s b + i −
1. We see that P b s b = s , and we obtain λ = ( λ (1) , . . . , λ ( l ) )and s = ( s , . . . , s l ). u k → | λ, s i gives the required bijection.Note that the correspondence u k ( b ) ↔ | λ ( b ) , s b i for each b is nothing but thecorrespondence Λ s b + ∞ ≃ F q [ s b ] in the case where F q [ s b ] is a level 1 Fock space withcharge s b . In [U], Uglov defined a bar-involution on Λ s + ∞ by making use of therealization of the semi-infinite wedge product in terms of the affine Hecke algebra,which is semi-linear with respect to the involution q q − on Q ( q ), and commuteswith the action of U q ( b sl n ), i.e., u · x = u · x for u ∈ U q ( b sl n ) , x ∈ Λ s + ∞ (here u isthe usual bar-involution on U q ( b sl n ) ). We give a property of the bar-involution on Λ s + ∞ , which makes it possible to compute explicitly the bar-involution.For k ∈ P ++ ( s ), we have(2.4.1) u k = u k ∧ u k ∧ · · · ∧ u k r ∧ u k r +1 ∧ u k r +2 ∧ · · · for any r ≫
1. Moreover, for any ( k , k , . . . , k r ), not necessarily ordered, we have(2.4.2) u k ∧ u k ∧ · · · ∧ u k r = α ( q ) u k r ∧ · · · ∧ u k ∧ u k with some α ( q ) ∈ Q ( q ) of the form ± q a . The quantity α ( q ) is given explicitly as in[U, Prop. 3.23]. Thus one can express u k by the ordering rule as a linear combinationof ordered semi-infinite wedges.The bar-involution on Λ s + ∞ leaves the subspace F q [ s ] invariant, and so definesa bar-involution on the Fock space F q [ s ]. Let us define L + (resp. L − ) as the Q [ q ]-lattice (resp. Q [ q − ]-lattice) of Λ s + ∞ generated by {| λ, s i | λ ∈ Π l , s ∈ Z l ( s ) } . Underthis setting, Uglov constructed the canonical bases on F q [ s ]. Proposition 2.5 ([U, Prop. 4.11] ) . There exist unique bases {G + ( λ, s ) } , {G − ( λ, s ) } of F q [ s ] satisfying the following properties; (i) G + ( λ, s ) = G + ( λ, s ) , G − ( λ, s ) = G − ( λ, s ) , (ii) G + ( λ, s ) ≡ | λ, s i mod q L + , G − ( λ, s ) ≡ | λ, s i mod q − L − , We define ∆ ± λ,µ ( q ) ∈ Q [ q ± ], for λ, µ ∈ Π l , by the formula G ± ( λ, s ) = X µ ∈ Π l ∆ ± λ,µ ( q ) | µ, s i . Note that ∆ ± λ,µ ( q ) = 0 unless | λ | = | µ | .For λ ∈ Π l , s = ( s , . . . , s l ), and M ∈ Z , we say that | λ, s i is M -dominant if s i − s i +1 > M + | λ | for i = 1 , . . . , l − RODUCT FORMULAS 11
Remark 2.7.
Let S ( Λ ) be the cyclotomic v -Schur algebra over R with parameters v, Q , · · · , Q l . We consider the special setting for parameters as follows; R = C and( v ; Q , . . . , Q l ) = ( ξ ; ξ s , . . . , ξ s l ), where ξ = exp(2 πi/n ) ∈ C and s = ( s , . . . , s l ) isa multi-charge. For λ = ( λ (1) , . . . , λ ( l ) ) ∈ Π l , we define an l -partition λ † by λ † = (( λ ( l ) ) ′ , ( λ ( l − ) ′ , . . . , ( λ (1) ) ′ )) , where ( λ ( i ) ) ′ denotes the dual partition of the partition λ ( i ) . Recall that d λµ ( q ) ∈ Z [ q ]is the v -decomposition number defined in 1.18. In [Y], Yvonne gave the followingconjecture; Conjecture I:
Assume that | λ, s i is 0-dominant. Then we have d λµ ( q ) = ∆ + µ † λ † ( q ) . By specializing q = 1, Conjecture I implies an LLT-type conjecture for decomposi-tion numbers of S ( Λ ), Conjecture II:
Under the same setting as in Conjecture I, we have[ W λ : L µ ] S ( Λ ) = ∆ + µ † λ † (1) . In the case where l = 1, i.e., the case where S ( Λ ) is the v -Schur algebra associ-ated to the Iwahori-Hecke algebra of type A , Conjecture II was proved by Varagnolo-Vasserot [VV]. It is open for the general case, l >
1. Concerning Conjecture I, it isnot yet verified even in the case where l = 1. Fix p = ( l , . . . , l g ) ∈ Z g> such that P gi =1 l i = l as in 1.4. For i = 1 , . . . , g ,define s [ i ] by s [1] = ( s , . . . , s l ) , s [2] = ( s l +1 , . . . , s l + l ), and so on. Thus we canwrite s = ( s [1] , . . . , s [ g ] ). For each λ ∈ Π l , we express it as λ = ( λ [1] , . . . , λ [ g ] ) as in1.4. Recall the integer α p ( λ ) in 1.4. We have α p ( λ ) = α p ( µ ) if and only if | λ | = | µ | and | λ [ i ] | = | µ [ i ] | for i = 1 , . . . , g .Let F q [ s [ i ] ] be the q -deformed Fock space of level l i with multi-charge s [ i ] , withbasis {| λ [ i ] , s [ i ] i | λ [ i ] ∈ Π l i } . We consider the canonical bases {G ± ( λ [ i ] , s [ i ] ) | λ [ i ] ∈ Π l i } of F q [ s [ i ] ]. Put G ± ( λ [ i ] , s [ i ] ) = X µ ∈ Π li ∆ ± λ [ i ] ,µ [ i ] ( q ) | µ [ i ] , s [ i ] i with ∆ ± λ [ i ] ,µ [ i ] ( q ) ∈ Q [ q ± ]. The following product formula is our main theorem,which is a counter-part of Theorem 1.19 to the case of the Fock space, in view ofConjecture I. Theorem 2.9.
Let λ, µ ∈ Π l be such that | λ, s i is M -dominant for M > n , andthat α p ( λ ) = α p ( µ ) . Then we have ∆ ± λ,µ ( q ) = g Y i =1 ∆ ± λ [ i ] ,µ [ i ] ( q ) . As a corollary, we obtain a special case of Conjecture II (though we require astronger dominance condition for | λ, s i ). Corollary 2.10.
Let λ, µ ∈ Π l be such that | λ ( i ) | = | µ ( i ) | for i = 1 , . . . , l . Assumethat | λ, s i is M -dominant for M > n . Then we have [ W λ : L µ ] S ( Λ ) = ∆ + µ † λ † (1) . Proof.
Take p = (1 , . . . , λ [ i ] = λ ( i ) for i = 1 , . . . , g = l , and S ( Λ N i )coincides with the v -Schur algebra of type A . By applying Theorem 1.17 or Theorem1.19, we have [ W λ : L µ ] S ( Λ ) = l Y i =1 [ W λ ( i ) : L µ ( i ) ] S ( Λ Ni ) . Also, by applying Theorem 2.9, we have ∆ + µ † λ † (1) = l Y i =1 ∆ +( µ ( i ) ) ′ ( λ ( i ) ) ′ (1) . On the other hand, we know [ W ( λ ( i ) ) : L µ ( i ) ] S ( Λ Ni ) = ∆ +( µ ( i ) ) ′ ( λ ( i ) ) ′ (1) by a result ofValagnolo-Vasserot (see Remark 2.7). The corollary follows from these formulas. (cid:3) Clearly the proof of the theorem is reduced to the case where g = 2,i.e., the case where λ = ( λ [1] , λ [2] ), etc. So, we assume that p = ( l , l ) = ( t, l − t )for some t ∈ Z > . We write the multi-charge s as s = ( s [1] , s [2] ), and consider the q -deformed Fock spaces F q [ s [ i ] ] of level l i with multi-charge s [ i ] for i = 1 ,
2. We havean isomorphism F q [ s ] ≃ F q [ s [1] ] ⊗ F q [ s [2] ] of vector spaces via the bijection of thebases | λ, s i ↔ | λ [1] , s [1] i⊗| λ [2] , s [2] i for each λ = ( λ [1] , λ [2] ) ∈ Π l . For s ∈ Z l ( s ), put s ′ = s + · · · + s t , s ′′ = s t +1 + · · · + s l . Under the isomorphism in (2.2.1) we have Λ s ′ + ∞ ≃ M s [1] ∈ Z l ( s ′ ) F q [ s [1] ] , Λ s ′′ + ∞ ≃ M s [2] ∈ Z l ( s ′′ ) F q [ s [2] ] . Then we have an injective Q ( q )-linear map(2.11.1) Λ s ′ + ∞ ⊗ Λ s ′′ + ∞ ≃ M s [1] ∈ Z l s ′ ) s [2] ∈ Z l s ′′ ) F q [ s [1] ] ⊗ F q [ s [2] ] → M s ∈ Z l ( s ) F q [ s ] ≃ Λ s + ∞ RODUCT FORMULAS 13 via | λ [1] , s [1] i⊗| λ [2] , s [2] i 7→ | λ, s i . We denote the embedding in (2.11.1) by Φ. For λ, µ ∈ Π l , we define a ( λ ) > a ( µ ) if | λ | = | µ | and | λ [1] | > | µ [1] | . Notethat this is the same as the partial order a p ( λ ) > a p ( µ ) defined in 1.5 for the casewhere p = ( l , l ). We have the following proposition. Proposition 2.13.
Assume that u k = | λ, s i is M -dominant for M > n . Under theembedding Φ : Λ s ′ + ∞ ⊗ Λ s ′′ + ∞ → Λ s + ∞ in 2.11, we have | λ, s i = | λ [1] , s [1] i⊗ | λ [2] , s [2] i + X µ ∈ Π l a ( λ ) > a ( µ ) α λ,µ | µ, s i with α λ,µ ∈ Q [ q, q − ] . Proposition 2.13 will be proved in 3.11 in the next section. Here assumingthe proposition, we continue the proof of the theorem. We have the following result.
Theorem 2.15.
Assume that | λ, s i = | λ [1] , s [1] i⊗| λ [2] , s [2] i is M -dominant for M > n . Then we have G ± ( λ, s ) = G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] )+ X µ ∈ Π l a ( λ ) > a ( µ ) b ± λ,µ G ± ( µ [1] , s [1] ) ⊗ G ± ( µ [2] , s [2] ) with b ± λ,µ ∈ Q [ q ± ] .Proof. Throughout the proof, we write ∆ ± λ [ i ] ,µ [ i ] ( q ) as ∆ ± λ [ i ] ,µ [ i ] for simplicity. Since G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] )= X µ [1] ∈ Π l ∆ ± λ [1] ,µ [1] | µ [1] , s [1] i⊗ X µ [2] ∈ Π l ∆ ± λ [2] ,µ [2] | µ [2] , s [2] i = X µ ∈ Π l ∆ ± λ [1] ,µ [1] ∆ ± λ [2] ,µ [2] | µ, s i , we have, by Proposition 2.13, G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] )= X µ ∈ Π l ∆ ± λ [1] ,µ [1] ∆ ± λ [2] ,µ [2] | µ, s i = X µ ∈ Π l ∆ ± λ [1] ,µ [1] ∆ ± λ [2] ,µ [2] n | µ [1] , s [1] i⊗ | µ [2] , s [2] i + X ν ∈ Π l a ( µ ) > a ( ν ) α µ,ν | ν, s i o = G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] ) + X µ ∈ Π l ∆ ± λ [1] ,µ [1] ∆ ± λ [2] ,µ [2] n X ν ∈ Π l a ( µ ) > a ( ν ) α µ,ν | ν [1] , s [1] i⊗| ν [2] , s [2] i o . By the property of the canonical bases, we have G ± ( λ [ i ] , s [ i ] ) = G ± ( λ [ i ] , s [ i ] ) for i =1 ,
2. Note that, | λ [ i ] | = | µ [ i ] | if ∆ ± λ [ i ] ,µ [ i ] = 0 for i = 1 ,
2. Moreover, a vector | ν [ i ] , s [ i ] i can be written as a linear combination of the canonical bases G ± ( κ [ i ] , s [ i ] ) such that | κ [ i ] | = | ν [ i ] | for i = 1 ,
2. Hence we have G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] )= G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] ) + X µ ∈ Π l a ( λ ) > a ( µ ) b ′± λ,µ G ± ( µ [1] , s [1] ) ⊗ G ± ( µ [2] , s [2] )(2.15.1)with b ′± λ,µ ∈ Q [ q, q − ]. Thus one can write as G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] ) = X µ ∈ Π l | µ | = | λ | R ± λ,µ G ± ( µ [1] , s [1] ) ⊗ G ± ( µ [2] , s [2] ) , with R ± λ,µ ∈ Q [ q, q − ], where the matrix (cid:0) R ± λ,µ (cid:1) | λ | = | µ | is unitriangular with respect tothe order compatible with a ( λ ) > a ( µ ) by (2.15.1). Thus, by a standard argumentfor constructing the canonical bases, we have G ± ( λ, s ) = G ± ( λ [1] , s [1] ) ⊗ G ± ( λ [2] , s [2] )+ X µ ∈ Π l a ( λ ) > a ( µ ) b ± λ,µ G ± ( µ [1] , s [1] ) ⊗ G ± ( µ [2] , s [2] ) , with b ± λ,µ ∈ Q [ q ± ]. This proves Theorem 2.15. (cid:3) We now prove Theorem 2.9 in the case where g = 2, assuming thatProposition 2.13 holds. Assume that | λ, s i is M -dominant for M > n . Since G ( λ [ i ] , s [ i ] ) = P µ [ i ] ∈ Π li ∆ ± λ [ i ] ,µ [ i ] | µ [ i ] , s [ i ] i , it follows from Theorem 2.15 that G ± ( λ, s ) = X µ ∈ Π lα p ( λ )= α p ( µ ) ∆ ± λ [1] ,µ [1] ∆ ± λ [2] ,µ [2] | µ, s i + X µ ∈ Π l a ( λ ) > a ( µ ) e b ± λ,µ | µ, s i . Since α p ( λ ) = α p ( µ ) is equivalent to a ( λ ) = a ( µ ), this implies that for any µ ∈ Π l such that α p ( λ ) = α p ( µ ),(2.16.1) ∆ ± λ,µ ( q ) = ∆ ± λ [1] ,µ [1] ( q ) ∆ ± λ [2] ,µ [2] ( q ) . This proves Theorem 2.9 modulo Proposition 2.13.
RODUCT FORMULAS 15
Remark 2.17.
By [U, Th. 3.26], ∆ ± λµ ( q ) can be interpreted by parabolic Kazhdan-Lusztig polynomials of an affine Weyl group. So, Theorem 2.9 gives a productformula for parabolic Kazhdan-Lusztig polynomials. It would be interesting to givea geometric interpretation of this formula.3. Tensor product of the Fock spaces
In this section, we prove Proposition 2.13 after some preliminaries. Theproof will be given in 3.11. For a given u k = | λ, s i , we associate semi-infinitesequences k ( b ) = ( k ( b )1 , k ( b )2 , . . . ) ∈ P ++ ( s b ) for b = 1 , . . . , l as in 2.3. For b ∈{ , · · · , l } and k ∈ Z , we put(3.1.1) u ( b ) k = u a + n ( b − − nlm , where a ∈ { , · · · , n } and m ∈ Z are uniquely determined by k = a − nm . Fora positive integer r , put k r = ( k , k , . . . , k r ). Then u k can be written as u k = u k r ∧ u k r +1 ∧ u k r +2 ∧ · · · , where u k r = u k ∧ · · · ∧ u k r ∈ Λ r , which is called a finitewedge of length r . We also define a finite wedge u k + r ∈ Λ r for a sufficiently large r by(3.1.2) u k + r = u (1) k (1) r ∧ u (2) k (2) r ∧ · · · ∧ u ( l ) k ( l ) rl with u ( i ) k ( i ) ri = u ( i ) k ( i )1 ∧ · · · ∧ u ( i ) k ( i ) ri for i = 1 , . . . , l , where each r i is sufficiently large and r = r + · · · + r l . Then in view of 2.3, we see that u k + r is obtained from u k r bypermuting the sequence k r . Moreover, u k ( b )1 ∧ · · · ∧ u k ( b ) rb is the first r b -part of thewedge u k ( b ) corresponding to | λ ( b ) , s b i (under the correspondence for the case l = 1in 2.3) for each b = 1 , . . . , l .Note that u k + r is not necessarily ordered in general, and it is written as a linearcombination of ordered wedges. But the situation becomes drastically simple underthe assumption on M -dominance. We have following two lemmas due to Uglov. Lemma 3.2 ([U, Lemma 5.18]) . Let b , b ∈ { , . . . , l } and a , a ∈ { , . . . , n } , andassume that b < b , a ≥ a . For any m ∈ Z , t ∈ Z ≥ , put X = u ( b ) a − nm ∧ u ( b ) a − nm − ∧· · · ∧ u ( b ) a − nm − t . Then there exists c ∈ Z such that the following relation holds. X ∧ u ( b ) a − nm = q c u ( b ) a − nm ∧ X. Lemma 3.3 ([U, Lemma 5.19]) . Take λ ∈ Π l and s ∈ Z l ( s ) , and assume that | λ, s i is -dominant. Then under the notation of 2.3, we have | λ, s i = u k = ( u k ∧ u k ∧ · · · ∧ u k r ) ∧ u k r +1 ∧ u k r +2 ∧ · · · = q − c r ( k ) u k + r ∧ u k r +1 ∧ u k r +2 · · · where c r ( k ) = ♯ { ≦ i < j ≦ r | b i > b j , a i = a j } . Returning to the setting in 2.11, we describe the map Φ in terms of thewedges. Take u k = | λ, s i , and assume that | λ, s i is 0-dominant. Then by Lemma3.3, we can write as(3.4.1) u k = q − c r ( k ) (cid:0) u (1) k (1) r ∧ · · · ∧ u ( t ) k ( t ) rt (cid:1) ∧ (cid:0) u ( t +1) k ( t +1) rt +1 ∧ · · · ∧ u ( l ) k ( l ) rl (cid:1) ∧ u r +1 ∧ · · · , where r , . . . , r l are sufficiently large. Let u ′ k ′ (resp. u ′′ k ′′ ) be the ordered wedge in Λ s ′ + ∞ (resp. in Λ s ′′ + ∞ ) corresponding to | λ [1] , s [1] i (resp. | λ [2] , s [2] i ). Since | λ, s i is0-dominant, | λ [ i ] , s [ i ] i is 0-dominant for i = 1 ,
2. Hence again by Lemma 3.3, we have(3.4.2) u ′ k ′ = q − c r ′ ( k ′ ) (cid:0) u ′ (1) k (1) r ∧ · · · ∧ u ′ ( t ) k ( t ) rt (cid:1) ∧ u ′ k r ′ +1 ∧ · · · u ′′ k ′′ = q − c r ′′ ( k ′′ ) (cid:0) u ′′ (1) k ( t +1) rt +1 ∧ · · · ∧ u ′′ ( l − t ) k ( l ) rl (cid:1) ∧ u ′′ k r ′′ +1 ∧ · · · , where r ′ = r + · · · + r t and r ′′ = r t +1 + · · · r l . Thus, in the case where | λ, s i is0-dominant, the map Φ : u ′ k ′ ⊗ u ′′ k ′′ u k is obtained by attaching u ′ (1) k (1) r ∧ · · · ∧ u ′ ( t ) k ( t ) rt u (1) k (1) r ∧ · · · ∧ u ( t ) k ( t ) rt u ′′ (1) k ( t +1) rt +1 ∧ · · · ∧ u ′′ ( l − t ) k ( l ) rl u ( t +1) k ( t +1) rt +1 ∧ · · · ∧ u ( l ) k ( l ) rl and by adjusting the power of q . We are mainly concerned with the expression as in the right hand side of(3.4.1), instead of treating u k directly. So we will modify the ordering rule so as tofit the expression by u ( b ) k . Recall that u ( b ) k = u a + n ( b − − nlm in (3.1.1). We define atotal order on the set { u ( b ) k | b ∈ { , . . . , l } , k ∈ Z } by inheriting the total order onthe set { u k | k ∈ Z } ≃ Z . The following property is easily verified.(3.5.1) Assume that u ( b i ) k i = u a i + n ( b i − − nlm i for i = 1 ,
2. Then u ( b ) k < u ( b ) k if andonly if one of the following three cases occurs;(i) m < m ,(ii) m = m and b > b ,(iii) m = m , b = b and a > a .Under this setting, the ordering rule in [U, Prop. 3.16] can be rewritten as follows. Proposition 3.6. (i)
Suppose that u ( b ) k ≤ u ( b ) k for b , b ∈ { , · · · , l } , k , k ∈ Z . Let γ be the residue of k − k modulo n . Then we have the followingformulas. (R1) the case where γ = 0 and b = b , u ( b ) k ∧ u ( b ) k = − u ( b ) k ∧ u ( b ) k , RODUCT FORMULAS 17 (R2) the case where γ = 0 and b = b , u ( b ) k ∧ u ( b ) k = − q − u ( b ) k ∧ u ( b ) k + ( q − − X m ≥ q − m u ( b ) k − γ − nm ∧ u ( b ) k + γ + nm − ( q − − X m ≥ q − m +1 u ( b ) k − nm ∧ u ( b ) k + nm , (R3) the case where γ = 0 and b = b , u ( b ) k ∧ u ( b ) k = qu ( b ) k ∧ u ( b ) k + ( q − X m ≥ ε q m u ( b ) k − nm ∧ u ( b ) k + nm + ( q − X m ≥ q − m +1 u ( b ) k − nm ∧ u ( b ) k + nm , (R4) the case where γ = 0 and b = b , u ( b ) k ∧ u ( b ) k = u ( b ) k ∧ u ( b ) k + ( q − q − ) X m ≥ ( q m +1 + q − m − )( q + q − ) u ( b ) k − γ − nm ∧ u ( b ) k + γ + nm + ( q − q − ) X m ≥ ε ( q m +1 + q − m − )( q + q − ) u ( b ) k − nm ∧ u ( b ) k + nm + ( q − q − ) X m ≥ ε ( q m − q − m )( q + q − ) u ( b ) k − γ − nm ∧ u ( b ) k + γ + nm + ( q − q − ) X m ≥ ( q m − q − m )( q + q − ) u ( b ) k − nm ∧ u ( b ) k + nm , where in the formula (R3) and (R4), ε = ( if b < b , if b > b . The sums are taken over all m such that the wedges in the sum remainordered. (ii) For a wedge u ( b ) k ∧ u ( b ) k ∧ · · · ∧ u ( b r ) k r , above relations hold in every pair ofadjacent factors. Remark 3.7.
The ordering rule in the proposition does not depend on the choiceof l . It depends only on k , k and whether b = b or not. This implies the following.Assume that p = ( t, l − t ) , s = s ′ + s ′′ and u ′ ( b ) k ∈ Λ s ′ + ∞ , u ′′ ( b ) k ∈ Λ s ′′ + ∞ as before. Then if b , b ∈ { , , . . . , t } , the ordering rule for u ( b ) k ≤ u ( b ) k is the same as therule for u ′ ( b ) k ≤ u ′ ( b ) k . Similarly, if b , b ∈ { t + 1 , . . . , l } the rule for u ( b ) k ≤ u ( b ) k isthe same as the rule for u ′′ ( b − t ) k ≤ u ′′ ( b − t ) k .We show the following three lemmas. Lemma 3.8.
Let M be an integer such that M > n . Assume that u k = | λ, s i is M -dominant. For b ∈ { , · · · , l } and i ∈ Z , put k ( b ) i = s b − i + 1 + λ ( b ) i = a ( b ) i − nm ( b ) i ( a ( b ) i ∈ { , · · · , n } , m ( b ) i ∈ Z ) . Fix b , b ∈ { , · · · , l } such that b < b .For k ( b ) i , let σ ( i ) be the smallest j such that u ( b ) k ( b i > u ( b ) k ( b j . Then we have (i) λ ( b ) σ ( i ) = 0 . (ii) k ( b ) σ ( i ) = n − nm ( b ) i .Proof. Let ℓ ( µ ) be the number of non-zero parts of a partition µ . We put p = ℓ ( λ ( b ) ) + 1. In order to show (i), it is enough to see(3.8.1) a ( b ) p + n ( b − − nlm ( b ) p > a ( b )1 + n ( b − − nlm ( b )1 In fact, by (3.8.1), we have u ( b ) k ( b p > u ( b ) k ( b σ ( i ) since u ( b ) k ( b ≥ u ( b ) k ( b i > u ( b ) k ( b σ ( i ) . This impliesthat p < σ ( i ). Since λ ( b ) p = 0, we have λ ( b ) σ ( i ) = 0.We show (3.8.1). If we put X = ( a ( b ) p + n ( b − − nlm ( b ) p ) − ( a ( b )1 + n ( b − − nlm ( b )1 ) , we have X = l { ( a ( b ) p − nm ( b ) p ) − ( a ( b )1 − nm ( b )1 ) } (3.8.2) − ( l − a ( b ) p − a ( b )1 ) − n ( b − b ) . Since k ( b )1 = s b + λ ( b )1 = a ( b )1 − nm ( b )1 ,k ( b ) p = s b − ℓ ( λ ( b ) ) = a ( b ) p − nm ( b ) p , by replacing a ( b )1 − nm ( b )1 by s b + λ ( b )1 , and similarly for a ( b ) p − nm ( b ) p in (3.8.2),we see that X = l (cid:8)(cid:0) s b − ℓ ( λ ( b ) ) (cid:1) − (cid:0) s b + λ ( b )1 (cid:1)(cid:9) − ( l − a ( b ) p − a ( b )1 ) − n ( b − b )= l ( s b − s b ) − l (cid:0) ℓ ( λ ( b ) ) − λ ( b )1 ) − ( l − a ( b ) p − a ( b )1 ) − n ( b − b ) ≥ l ( s b − s b ) − l | λ | − ln − nl = l (cid:8) ( s b − s b ) − ( | λ | + 2 n ) (cid:9) . RODUCT FORMULAS 19
Since
M > n , we have s b − s b ≥ | λ | + M > | λ | + 2 n , and so X >
0. This proves(3.8.1) and (i) follows.Next we show (ii). By definition, σ ( i ) is the smallest integer j such that a ( b ) i + n ( b − − nlm ( b ) i > a ( b ) j + n ( b − − nlm ( b ) j . If for a ∈ { , . . . , n } amd m ∈ Z ,(3.8.3) a ( b ) i + n ( b − − nlm ( b ) i > a + n ( b − − nlm, then we have k ( b ) p > a − nm by (3.8.1) and (3.5.1). Note that k ( b ) j = a ( b ) j − nm ( b ) j = s b − j + 1 + λ ( b ) j and λ ( b ) j = 0 for any j ≥ p . It follows that k ( b ) p + j = k ( b ) p − j forany j ≥
1. Hence there exists an integer j ≥ k ( b ) p + j = a − nm . Thismeans that k ( b ) σ ( i ) is the largest integer a − nm for a ∈ { , . . . , m } , m ∈ Z satisfyingthe inequality (3.8.3). Clearly, a − nm = n − nm ( b ) i is the largest, and we obtain(ii). (cid:3) Lemma 3.9.
For b , b ∈ { , · · · , l } such that b < b , and k , k ∈ Z such that u ( b ) k < u ( b ) k , we have (3.9.1) u ( b ) k ∧ u ( b ) k = α ( k , k ) u ( b ) k ∧ u ( b ) k + X ( k ′ ,k ′ ∈ Z k >k ′ ,k ′ >k α ( k ′ , k ′ ) u ( b ) k ′ ∧ u ( b ) k ′ with α ( k , k ) , α ( k ′ , k ′ ) ∈ Q [ q, q − ] .Proof. Put k i = a i − nm i for i = 1 ,
2, where a i ∈ { , · · · , n } , m i ∈ Z . First assumethat a = a . By the ordering rule (R3) in Proposition 3.6, we have(3.9.2) u ( b ) k ∧ u ( b ) k = qu ( b ) k ∧ u ( b ) k + ( q − X m ≥ q m u ( b ) k − nm ∧ u ( b ) k + nm + ( q − X m ≥ q − m +1 u ( b ) k − nm ∧ u ( b ) k + nm , and the only ordered wedges appear in the sums. Note that a = a , b < b and m ≥
1. Then in the second sum, the condition u ( b ) k − nm > u ( b ) k + nm implies that k > k − nm > k + nm > k by (3.5.1). It follows that the terms in the secondsum are all of the form u ( b ) k ′ ∧ u ( b ) k ′ as in (3.9.1). On the other hand, in the firstsum, the condition u ( b ) k − nm > u ( b ) k + nm implies that k > k − nm ≥ k + nm > k by(3.5.1). Hence if k − k < n , the terms u ( b ) k − nm ∧ u ( b ) k + nm do not appear in the sum.So assume that k − k ≥ n . We apply the ordering rule (R3) to u ( b ) k + nm ∧ u ( b ) k − nm , and we obtain(3.9.3) u ( b ) k − nm ∧ u ( b ) k + nm = q − u ( b ) k + nm ∧ u ( b ) k − nm + X + X , where X (resp. X ) is a linear combination of the wedges u ( b ) k − nm ′ ∧ u ( b ) k + nm ′ (resp. u ( b ) k − nm ′ ∧ u ( b ) k + nm ′ ) with m ′ > m . Note that k > k + nm and k − nm > k , andso u ( b ) k + nm ∧ u ( b ) k − nm is of the form u ( b ) k ′ ∧ u ( b ) k ′ in (3.9.1). We can apply the sameprocedure as above for replacing u ( b ) k − nm ′ ∧ u ( b ) k + nm ′ in X by the terms u ( b ) k ′ ∧ u ( b ) k ′ and other terms. Repeating this procedure, finally we obtain the expression as in(3.9.1). Note that since 2 n ≤ k ′ − k ′ < k − k for k ′ = k − nm and k ′ = k + nm ,this procedure will end up after finitely many steps.Next consider the case where a = a . In this case we apply the ordering rule(R4). Then one can write as u ( b ) k ∧ u ( b ) k = u ( b ) k ∧ u ( b ) k + X + X + X + X , where X , . . . , X are the corresponding sums in (R4) with ε = 1. For X , X , X ,similar arguments as above can be applied. So we have only to consider the sum X which contains the terms of the form u ( b ) k − γ − nm ∧ u ( b ) k + γ + nm . In this case, thecondition u ( b ) k − γ − nm > u ( b ) k + γ + nm implies that ( k − nm ) − ( k + nm ) ≥ a − a > − n ,and so k − k > n (2 m − k − k ≤ n , the terms u ( b ) k − γ − nm ∧ u ( b ) k + γ + nm do not appear in the sum. If k − k > n , we can apply a similar argument as beforeby using the ordering rule (R4). Thus we obtain the expression in (3.9.1) in thiscase also. The lemma is proved. (cid:3) Lemma 3.10.
Assume that u k = | λ, s i is M -dominant for M > n . For b ∈{ , · · · , l } and i ∈ Z , put k ( b ) i = s b − i + 1 + λ ( b ) i . Then for d ∈ { t + 1 , t + 2 , · · · , l } and i ∈ Z , we have u ( d ) k ( d ) i ∧ (cid:0) u ( t ) k ( t ) rt ∧ u ( t ) k ( t ) rt − ∧ · · · ∧ u ( t ) k ( t )1 ∧ u ( t − k ( t − rt − ∧ · · · ∧ u (1) k (1)2 ∧ u (1) k (1)1 (cid:1) = α (cid:0) u ( t ) k ( t ) rt ∧ u ( t ) k ( t ) rt − ∧ · · · ∧ u ( t ) k ( t )1 ∧ u ( t − k ( t − rt − ∧ · · · ∧ u (1) k (1)2 ∧ u (1) k (1)1 (cid:1) ∧ u ( d ) k ( d ) i + Y , where α ∈ Q [ q, q − ] and Y is a Q [ q, q − ] -linear combination of the wedges of theform (cid:0) u ( t ) e k ( t ) rt ∧ u ( t ) e k ( t ) rt − ∧ · · · ∧ u (1) e k (1)1 (cid:1) ∧ u e k ( d ) i for ( e k ( t ) r t , . . . , e k (1)1 ; e k ( d ) i ) ∈ Z r ′ × Z under the condition k ( t ) r t + · · · + k (1)1 > e k ( t ) r t + · · · + e k (1)1 , k ( d ) i < e k ( d ) i . RODUCT FORMULAS 21
Proof.
Put k ( d ) i = a i − nm i . Let σ ( i ) be the smallest j such that u ( d ) k ( d ) i > u ( t ) k ( t ) j . Since d > t , by applying Lemma 3.8, we have k ( t ) σ ( i ) = n − nm i and λ ( t ) σ ( i ) = 0. Thus, wehave u ( d ) k ( d ) i ∧ (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u ( t ) k ( t ) σ ( i )+1 ∧ u ( t ) k ( t ) σ ( i ) (cid:1) = u ( d ) a i − nm i ∧ (cid:0) u ( t ) n − nm i − ( r t − σ ( i )) ∧ · · · ∧ u ( t ) n − nm i − ∧ u ( t ) n − nm i (cid:1) . Using the formula obtained by applying the bar-involution on the formula in Lemma3.2, we have u ( d ) k ( d ) i ∧ (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u ( t ) k ( t ) ξ ( i )+1 ∧ u ( t ) k ( t ) σ ( i ) (cid:1) = β (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u ( t ) k ( t ) σ ( i )+1 ∧ u ( t ) k ( t ) σ ( i ) (cid:1) ∧ u ( d ) k ( d ) i with β ∈ Q [ q, q − ]. Since u ( d ) k ( d ) i < u ( t ) k ( t ) σ ( i ) − < · · · < u ( t ) k ( t )1 , using Lemma 3.9 repeatedly,we have u ( d ) k ( d ) i ∧ (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u ( t ) k ( t ) σ ( i )+1 ∧ u ( t ) k ( t ) σ ( i ) ∧ u ( t ) k ( t ) σ ( i ) − ∧ · · · ∧ u ( t ) k ( t )1 (cid:1) = β (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u ( t ) k ( t ) σ ( i )+1 ∧ u ( t ) k ( t ) σ ( i ) (cid:1) ∧ u ( d ) k ( d ) i ∧ (cid:0) u ( t ) k ( t ) σ ( i ) − ∧ · · · ∧ u ( t ) k ( t )1 (cid:1) = e β (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u ( t ) k ( t )1 (cid:1) ∧ u ( d ) k ( d ) i + Y ′ , where e β ∈ Q [ q, q − ] and Y ′ is a Q [ q, q − ]-linear combination of the wedges of theform (cid:0) u ( t ) e k ( t ) rt ∧ · · · ∧ u ( t ) e k ( t )1 ) ∧ u ( d ) e k i ( d ) for ( e k ( t ) r t , . . . , e k ( t )1 ; e k ( d ) i ) ∈ Z r t × Z , under the condition k ( t ) r t + · · · + k ( t )1 > e k ( t ) r t + · · · + e k ( t )1 , k ( d ) i < e k ( d ) i . Thus repeating this procedure for t − , . . . ,
1, we obtain the lemma. (cid:3)
We now give a proof of Proposition 2.13. Since | λ, s i is M -dominant, wecan write, as in 3.4, that | λ, s i = (cid:0) u k ∧ · · · ∧ u k r (cid:1) ∧ u k r +1 ∧ u k r +2 ∧ · · · = β (cid:0) u (1) k (1) r ∧ · · · ∧ u ( l ) k ( l ) rl (cid:1) ∧ u k r +1 ∧ u k r +2 ∧ · · · , where β ∈ Q [ q, q − ] and r is sufficient large. By (2.4.1), we have | λ, s i = (cid:0) u k ∧ · · · ∧ u k r (cid:1) ∧ u k r +1 ∧ u k r +2 ∧ · · · = β (cid:0) u (1) k (1) r ∧ · · · ∧ u ( l ) k ( l ) rl (cid:1) ∧ u k r +1 ∧ u k r +2 ∧ · · · . By (2.4.2), we have u (1) k (1) r ∧ · · · ∧ u ( l ) k ( l ) rl = β ′ (cid:0) u ( l ) k ( l ) rl ∧ · · · ∧ u ( t +1) k ( t +1)1 (cid:1) ∧ (cid:0) u ( t ) k ( t ) rt ∧ · · · u (1) k (1)2 ∧ u (1) k (1)1 (cid:1) with β ′ ∈ Q [ q, q − ]. By using Lemma 3.10 repeatedly, we have (cid:0) u ( l ) k ( l ) rl ∧ · · · ∧ u ( t +1) k ( t +1)2 ∧ u ( t +1) k ( t +1)1 (cid:1) ∧ (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u (1) k (1)2 ∧ u (1) k (1)1 (cid:1) = β ′′ (cid:0) u ( t ) k ( t ) rg ∧ · · · u (1) k (1)2 ∧ u (1) k (1)1 (cid:1) ∧ (cid:0) u ( l ) k ( l ) rl ∧ · · · ∧ u ( t +1) k ( t +1)2 ∧ u ( t +1) k ( t +1)1 (cid:1) + Y, where β ′′ ∈ Q [ q, q − ], and Y is a Q [ q, q − ]-linear combination of the wedges of theform (cid:0) u ( t ) e k ( t ) rt ∧ · · · ∧ u (1) e k (1)1 (cid:1) ∧ (cid:0) u ( l ) e k ( l ) rl ∧ · · · ∧ u ( t +1) e k ( t +1)1 (cid:1) for ( ( e k ( t ) r t , . . . , e k (1)1 ) ∈ Z r ′ , ( e k ( l ) r l , . . . , e k ( t +1)1 ) ∈ Z r ′′ under the condition(3.11.1) k (1)1 + · · · + k ( t ) r t > e k (1)1 + · · · + e k ( t ) r t ,k ( t +1)1 + · · · + k ( l ) r l < e k ( t +1)1 + · · · + e k ( l ) r l . We claim that(3.11.2) The wedges appearing in Y is written as a linear combination of the wedges u h = | µ, s i such that a ( λ ) > a ( µ ) for µ ∈ Π l .We show (3.11.2). By using the ordering rule, u ( t ) e k ( t ) rt ∧ · · · ∧ u (1) e k (1)1 (resp. u ( l ) e k ( l ) rl ∧ · · · ∧ u ( t +1) e k ( t +1)1 ) can be written as a linear combination of the wedges u (1) h (1) r ∧ · · · ∧ u ( t ) h ( t ) rt (resp. u (1) h (1) rt +1 ∧ · · · ∧ u ( l ) h ( l ) rl ) with u ( i ) h ( i ) ri = u ( i ) h ( i )1 ∧ · · · ∧ u ( i ) h ( i ) ri . Proposition 3.6 says that the sumof the indices is stable in applying the ordering rule. Hence we have(3.11.3) e k (1)1 + · · · + e k ( t ) r t = h (1)1 + · · · + h ( t ) r t , e k ( t +1)1 + · · · + e k ( l ) r l = h ( t +1)1 + · · · + h ( l ) r l . RODUCT FORMULAS 23
Recall that k ( b ) i = s b − i + 1 + λ ( b ) i . We define µ ∈ Π l by setting h ( b ) i = s b − i + 1 + µ ( b ) i for any i, b , and write it as µ = ( µ [1] , µ [2] ). Then (3.11.1) and (3.11.3) imply that | λ [1] | > | µ [1] | . Also we note that | λ | = | µ | since k (1)1 + · · · + k ( l ) r l = e k (1)1 + · · · + e k ( l ) r l = h (1)1 + · · · + h ( l ) r l . Hence we have a ( λ ) > a ( µ ). Moreover, by using (3.4.1), we see that ( u (1) h (1) r ∧ · · · ∧ u ( l ) h ( l ) rl ) ∧ u k r +1 ∧ · · · coincides with u h = | µ, s i up to scalar. Thus (3.11.2) is proved.Now as noted in Remark 3.7, the ordering rule for u ( t ) k ( t ) rt ∧ · · · ∧ u (1) k (1)1 , regarded asan element in Λ s + ∞ or as an element in Λ s ′ + ∞ , is the same. Hence under the mapΦ, u ′ ( t ) k ( t ) rt ∧ · · · ∧ u ′ (1) k (1)1 (resp. u ′′ ( l − t ) k ( l ) rl ∧ · · · ∧ u ′′ ( t +1) k (1)1 ) corresponds to u ( t ) k ( t ) rt ∧ · · · ∧ u (1) k (1)1 (resp. u ( l ) k ( l ) rl ∧ · · · ∧ u ( t +1) k ( t +1)1 ). It follows that (cid:0) u ( t ) k ( t ) rt ∧ · · · ∧ u (1) k (1)1 (cid:1) ∧ (cid:0) u ( l ) k ( l ) rl ∧ · · · ∧ u ( t +1) k ( t +1)1 (cid:1) ∧ u r +1 ∧ u r +2 ∧ · · · = α | λ [1] , s [1] i⊗ | λ [2] , s [2] i with some α ∈ Q [ q, q − ]. Summing up the above arguments, we have(3.11.4) | λ, s i = α | λ [1] , s [1] i⊗ | λ [2] , s [2] i + X µ ∈ Π l a ( λ ) > a ( µ ) α λ,µ | µ, s i with α, α λ,µ ∈ Q [ q, q − ]. Since the coefficient of | λ, s i in the expansion of | λ, s i interms of the ordered wedges is equal to 1, and similarly for | λ [1] , s [1] i , | λ [2] , s [2] i , wesee that α = 1 by comparing the coefficient of | λ, s i = | λ [1] , s [1] i⊗| λ [2] , s [2] i in theboth sides of (3.11.4). This proves the proposition. References [DJM] R. Dipper, G. James and A. Mathas; Cyclotomic q -Schur algebras, Math. Z. , (1998)385 - 416.[DR] J. Du and H. Rui; Based algebras and standard bases for quasi-hereditary algebras,Trans. Amer. Math. Soc. (1998), 3207-3235.[GL] J.J. Graham and G.I. Lehrer; Cellular algebras, Invent. Math., (1996), 1 - 34.[HS] J. Hu and F. Stoll; On double centralizer properties between quantum groups and Ariki-Koike algebras, preprint.[Sa] N. Sawada; On decomposition numbers of the cyclotomic q -Schur algebras, to appear inJ. Algebra.[SakS] M. Sakamoto and T. Shoji; Schur-Weyl reciprocity for Ariki-Koike algebras, J. Algebra (1999), 293 - 314.[SawS] N. Sawada and T. Shoji; Modified Ariki-Koike algebras and cyclotomic q -Schur algebras,Math. Z. (2005), 829 - 867.[SW] T. Shoji and K. Wada; Cyclotomic q -Schur algebras associated to the Ariki-Koike alge-bra, preprint. [U] D. Uglov; Canonical bases of higher-level q -deformed Fock spaces and Kazhdan-Lusztigpolynomials, in “Physical combinatorics (Kyoto 1999)”, Prog. Math. Vol. , Birkha¨userBoston, 2000, pp.249 - 299.[VV] M. Varagnolo and E. Vasserot; On the decomposition matrices of the quantized Scuralgebras, Duke Math. J., (1999), 267 - 297.[W] K. Wada; On decomposition numbers with Jantzen filtration of cyclotomic q -Schur al-gebras, preprint.[Y] X. Yvonne; A conjecture for q -decomposition matrices of cyclotomic v -Schur algebras,J. Algebra,304