On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras

Let $\Sc(\vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\Sc(\vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $\Sc(\vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $\He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $\He_{n,r}$ by using a Schur functor.


Introduction
Let H = H n,r be the Ariki-Koike algebra over an integral domain R associated to the complex reflection group S n ⋉ (Z/rZ) n . Dipper, James and Mathas [DJM] introduced the cyclotomic q-Schur algebra S (Λ) associated to the Ariki-Koike algebra H , and they showed that H and S (Λ) are cellular algebras in the sence of Graham and Lehrer [GL], by constructing the cellular basis respectively. It is a fundamental problem for the representation theory to determine the decomposition numbers of H and S (Λ). It is well-known that the decomposition matrix of H coincides with the submatrix of that of S (Λ) by the Schur functor.
In the case where H is the Iwahori-Hecke algebra H n of type A, Lascoux, Leclerc and Thibon [LLT] conjectured that the decomposition numbers of H n can be described by using the canonical basis of a certain irreducible U v ( sl e )-module, and gave the algorithm to compute this canonical basis. The cojecture has been solved by Ariki [A1], by extending to the case of Ariki-Koike algebras.
In the case of the q-Schur algebra associated to H n , Leclerc and Thibon [LT] conjectured that the decomposition matrix coincides with the transition matrix between the canonical basis and the standard basis of the Fock space of level 1 equipped with the U v ( sl e )-module structure, and gave the algorithm to compute the transition matrix. This conjecture has been solved by Varagnolo and Vasserot in [VV].
More generally, in the case of the cyclotomic q-Schur algebra S , Yvonne [Y] has conjectured that the decomposition matrix coincides with the transition matrix between the canonical basis and the standard basis of the higher-level Fock space. This canonical basis was constructed by Uglov [U] and the algorithm to compute the transition matrix was also given there. Yvonne's conjecture is still open. We remark that Ariki's theorem, Varagnolo-Vasserot's theorem and Yvonne's conjecture are concerned with the situation where R is a complex number field and parameters are roots of unity.
In order to study the decomposition numbers of S , we constructed in [SW] some subalgebras S p of S (Λ) and their quotients S p , and showed that S p is a standardly based algebra in the sence of Du and Rui [DR], and that S p is a cellular algebra. Hence, one can consider the decomposition numbers of S p and S p also.
We denote the decomposition numbers of S , S p and S p by d λµ , d (λ,0) λµ and d λµ respectively, where d λµ is a decomposition number of the irreducible module L µ in the Weyl module W λ of S for r-partitions λ, µ, and d (λ,0) λµ , d λµ are defined similarly for S p and S p (see Section 1 for details). It is proved in [SW,Theorem 3.13] that (1) d λµ = d (λ,0) λµ = d λµ whenever λ, µ satisfy a certain condition α p (λ) = α p (µ). Moreover for such λ, µ, the product formula for d λµ , (2) d λµ = g k=1 d λ [k] µ [k] , was proved in [SW,Theorem 4.17], where d λ [k] µ [k] for k = 1, · · · , g is the decomposition number of the cyclotomic q-Schur algebra associated to a certain Ariki-Koike algebra H n k ,r k . Related to the above conjectures on Fock spaces, Leclerc-Thibon and Yvonne give a more precise conjecture concerning the v-decomposition numbers defined by using Jantzen filtrations of Weyl modules. (For definition of v-decomposition numbers, see §2.) We remark that decomposition numbers coincide with v-decomposition numbers at v = 1. Thus we regard v-decomposition numbers as a v-analogue of decomposition numbers. The conjecture for v-decomposition numbers has been still open even in the case of the q-Schur algebra of type A.
We note that our result is a v-analogue of (1),(2), and it reduces to them by taking v → 1. Moreover, for a certain v-decomposition number d H λµ (v) of the Ariki-Koike algebra, we also have the following product formula (Theorem 3.5).
We remark that our results hold for any parameters and any modular system, even for the case where the base field has non-zero characteristic, though Yvonne's conjecture is formulated under certain restrictions for parameters and modular systems.

A review of known results
1.1. Througout the paper, we follow the notation in [SW]. Here we review some of them. We fix positive integers r, n and an r-tuple m = (m 1 , · · · , m r ) ∈ Z r >0 . A composition λ = (λ 1 , λ 2 , · · · ) is a finite sequence of non-negative integers, and |λ| = i λ i is called the size of λ. If λ l = 0 and λ k = 0 for any k > l, then l is called the length of λ. If the composition λ is a weakly decreasing sequence, λ is called a partition. An r-tuple µ = (µ (1) , · · · , µ (r) ) of compositions is called the r-composition, and size |µ| of µ is defined by r i=1 |µ (i) |. In particular, if all µ (i) are partitions, µ is called an r-partition. We denote by Λ = P n,r (m) the set of r-compositions µ = (µ (1) , · · · , µ (r) ) such that |µ| = n and that the length of µ (k) is smaller than m k for k = 1, · · · , r. We define Λ + = P n,r (m) as the subset of Λ consisting of r-partitions .
We define the partial order, the so-called "dominance order", on Λ by µ ν if and only if for any 1 ≤ l ≤ r, 1 ≤ k ≤ m l . If µ ν and µ = ν, we write it as µ ⊲ ν.
1.2. Let H = H n,r be the Ariki-Koike algebra over an integral domain R with parameters q, Q 1 , · · · , Q r with defining relations in [SW,§1.1]. It is known by [DJM] that H has a structure of the cellular algebra with a cellular basis {m st | s, t ∈ Std(λ) for some λ ∈ Λ + }. Then the general theory of a cellular algebra by [GL] implies the following results. There exists an anti-automorphism h → h * of H such that m * st = m ts . For λ ∈ Λ + , let H ∨λ be the R-submodule of H spaned by m st , where s, t ∈ Std(µ) for some µ ∈ Λ + such that µ ⊲ λ. Then H ∨λ is an ideal of H . One can construct the standard (right) H -module S λ , called a Specht module, with the R-free basis {m t | t ∈ Std(λ)}. We define the bilinear form , H on S λ by where u, v ∈ Std(λ), and the scalar m s , m t H does not depend on the choice of u, v ∈ Std(λ). The bilinear form , H is associative, namely we have Let rad S λ = {x ∈ S λ | x, y H = 0 for any y ∈ S λ }. Then rad S λ is the Hsubmodule of S λ by the associativity of the bilinear form. Put D λ = S λ / rad S λ . Assume that R is a field. Then D λ is an absolutely irreducible module or zero, and the set {D λ | λ ∈ Λ + such that D λ = 0} gives a complete set of non-isomorphic irreducible H -modules.
1.3. Let S = S (Λ) be the cyclotomic q-Schur algebra introduced by [DJM], associated to the Ariki-Koike algebra H with respect to the set Λ. It is known by [DJM] that S is a cellular algebra with a cellular basis {ϕ ST | S, T ∈ T 0 (λ) for some λ ∈ Λ + }. Again by the general theory of a cellular algebra, the following results hold. There exists the anti-automorphism x → x * of S such that ϕ * ST = ϕ T S . For λ ∈ Λ + , let S ∨λ be the R-submodule spaned by ϕ ST , where S, T ∈ T 0 (µ) for some µ ∈ Λ + such that µ ⊲ λ. Then S ∨λ is an ideal of S . One can construct the standard (right) S -module W λ (λ ∈ Λ + ), called a Weyl module, with the R-free basis {ϕ T | T ∈ T 0 (λ)}. We define a bilinear form , on W λ by where U, V ∈ T 0 (λ), and the scalar ϕ S , ϕ T does not depend on a choice of U, V ∈ T 0 (λ). The bilinear form , is associative, namely we have Let rad W λ = {x ∈ W λ | x, y = 0 for any y ∈ W λ }, Then rad W λ is the Ssubmodule of W λ . Put L λ = W λ / rad W λ . Then it is known by [DJM] that L λ = 0 for any λ ∈ Λ + . Assume that R is a field. Then L λ is an absolutely irreducible module, and the set {L λ | λ ∈ Λ + } gives a complete set of non-isomorphic irreducible S -modules.
By the general theory of standardly based algebra due to [DR], we have the following results. For η ∈ Σ p , one can consider the standard left S p -modules ♦ Z η with the basis ϕ η T T ∈ I(η) and the standard right S p -module Z η with the basis ϕ η T T ∈ J(η) . We call them Weyl modules of S p . We define the bilinear form β η : where β η is determined independent of the choice of U ∈ I(η) and V ∈ J(η). The bilinear form β η is associative, namely we have Assume that R is a filed. Then L η is an absolutely irreducible module or zero, and the set L η η ∈ Σ p such that β η = 0 is a complete set of non-isomorphic irreducible (right) S p -modules.
Later we shall only consider the Weyl modules Z η and irreducible modules L η of S p for η of the form (λ, 0). Note that the composition fuctors of Z (λ,0) are isomorphic to L (µ,0) for some µ ∈ Λ + by [SW,Proposition 3 It is known by [SW] that S p is a two-sided ideal of S p . Thus, we can define the quotient algebra We denote by ϕ the image of ϕ ∈ S p under the natural surjection π : S p → S p , and set Then C p is a free R-basis of S p . By [SW,Theorem 2.13], S p turns out to be a cellular algebra with the cellular basis C p . Hence by the general theory of cellular algebra, the following results hold. For λ ∈ Λ + , we can consider the standard (right) S p -module Z λ with the free R-basis ϕ T T ∈ T p 0 (λ) . We call it a Weyl module of S p . We define the bilinear form , p : The bilinear form , p is associative, namely we have Assume that R is a field. Then L λ is an absolutely irreducible module, and the set L λ λ ∈ Λ + is a complete set of non-isomorphic irreducible (right) S p -modules.
1.6. Assuming that R is a field, we set, for λ, µ ∈ Λ + , where W λ : L µ is the decomposition number of L µ in W λ , and similarly for S p and S p . The following theorem was proved in [SW].
, · · · , µ (p i +r i ) ). According to the expression of µ as above, n k is regarded as the empty set if n k = 0.) Let S (Λ n k ) be the cyclotomic q-Schur algebra associated to the Ariki-Koike algebra H n k ,r k with parameters q, Q p k +1 , · · · , Q p k +r k . Let ∆ n,g be the set of (n 1 , · · · , n g ) ∈ Z g ≥0 such that n 1 + · · · + n g = n. Then we have the following decomposition theorem of S p by [SW,Theorem 4.15].
under the isomorphism given by Assuming that R is a field, for be the irreducible module. By [SW,Corollary 4.16 ], the following properties hold. Under the isomorphism in (1.8.1), we have, for Under the isomorphism in (1.8.3), a bilinear form , P on Z λ decomposes to a product of bilinear forms on W λ [k] for k = 1, · · · , g, namely we have the following lemma.
Proof. Fix U, V ∈ T p 0 (λ). Then by (1.8.2) and the definition of the bilinear form on The lemma is proved.
Remark 1.10. For the isomorphism in (1.8.3), we do not need to assume that R is a field. But for (1.8.4) and (1.8.5), we need that R is a field.

Decomposition numbers with Jantzen filtration
2.1. In the rest of this paper, we assume that R is a discrete valuation ring. Let ℘ be a unique maximal ideal of R and F = R/℘ be the residue filed. Fix q, Q 1 , · · · , Q r in R and let q = q + ℘ , Q 1 = Q 1 + ℘ , · · · , Q r = Q r + ℘ be their canonical images in F . Moreover let K be the quotient field of R. Then (K, R, F ) is a modular system with parameters. Let S R = S R (Λ) be the cyclotomic q-Schur algebra over R with parameters q, Q 1 , · · · , Q r and S = S (Λ) be the cyclotomic q-Schur algebra over F with parameters q, Q 1 , · · · , Q r . Then S = S R + ℘S R ℘S R . We consider the subalgebra S p R (resp. S p ) of S R (resp. S ) and its quotient S p R (resp. S p ) as in the previous section with the notation there. Note that the subscript R is used to indicate the objects related to R. (0) is the Weyl module of S , and we have the Jantzen filtration of Similarly, by using the bilinear form , p on Z λ R , one can define the Jantzen filtration of Moreover for the Weyl module Z Then we have the Jantzen filtration of Z (λ,0) Since W λ is a finite dimentional F -vector space, one can find a positive integer k such that W λ (k ′ ) = W λ (k) for any k ′ > k. We choose a minimal k in such numbers and set W λ (k +1) = 0. Then the Jantzen filtration of W λ becomes a finite sequence. Similarly, Jantzen filtrations of Z (λ,0) and Z λ also become finite sequences.
2.4. Next, we consider the relation between d (λ,0) λµ (v) and d λµ (v). In order to see this we prepare two lemmas. Recall that there exists an injective S p -homomorphism [SW,Lemma 3.5] and that Z (λ,0) ⊗ S p S ∼ = W λ as S -module by [SW,Proposition 3.6]. Let ι i : Z (λ,0) (i) ֒→ Z (λ,0) be an inclusion map. Then (ι i ⊗ id S ) Z (λ,0) (i) ⊗ S p S is the S -submodule of Z (λ,0) ⊗ S p S . Similar results hold also for R. We have the following.
Lemma 2.6. Let λ ∈ Λ + . For any i ≥ 0, we have Proof. Recall that any element of Z (λ,0) R can be written in the form ϕ (λ,0) T λ ·ψ with ψ ∈ S p R . Moreover it follows from [SW,Proposition 3.6] that, under the isomorphism T λ ·ψ ⊗ϕ = ϕ T λ ·ψϕ for ψ ∈ S p R , ϕ ∈ S R . This is true also for Z (λ,0) and W λ . Thus in order to show the lemma, it is enough to prove the following.
(2.6.1) Suppose that ϕ (λ,0) . This implies that ϕ T λ ψ , y ∈ ℘ i for any y ∈ W λ R by a similar argument as the proof of Lemma 2.5.
Combining Proposition 2.3 and Proposition 2.7, we have the following theorem.
2.9. For later use, we shall consider the basis of W λ R (i), following the computation in the proof of [M2,Lemma 5.30]. Let π be the generator of ℘, namely ℘ = (π), and ν ℘ be the valuation map on R. Let G λ = ϕ S , ϕ T S,T ∈T 0 (λ) be the Gram matrix of Since both P and Q are regular matrices, {f S | S ∈ T 0 (λ)} and {g T | T ∈ T 0 (λ)} are basis of W λ R respectively. Moreover we have diag(d S 1 , · · · , d S N ) = P G λ Q = f S , g T S,T ∈T 0 (λ) by definition. Thus we have It follows from this that W λ R (i) is a free R-module with basis 2.10. We consider the Jantzen filtration of W λ [k] (1 ≤ k ≤ g) as in the case of W λ and use the notation similar to the case of W λ . Since we see that R as of W λ R in 2.9. For S, T ∈ T p 0 (λ), we define f S := f S [1] ⊗ · · · ⊗ f S [g] and g T := g T [1] ⊗ · · · ⊗ g T [g] . Then f S S ∈ T p 0 (λ) and g T T ∈ T p 0 (λ) turn out to be the bases of Z λ R . By Lemma 1.9 and (2.9.1), we have for S, T ∈ T p 0 (λ).
. Then we have the following result by a similar argument as in 2.9. Z λ R (i) is a free R-module with basis Recall that ∆ i,g is the set of (i 1 , · · · , i g ) ∈ Z g ≥0 such that i 1 + · · · + i g = i. Then we have the following proposition.
Proposition 2.11. Let λ ∈ Λ + and i ≥ 0. Under the isomorphism Z Proof. First we show that the right hand side is contained in the left hand side.
Thus we have x ∈ Z λ R (i). Then in order to show the equality, we have only to show that the basis element of Z λ R (i) is contained in the right hand side of (2.11.1). First, we consider f T such that ), one can find (i 1 , · · · , i g ) ∈ Z g ≥0 such that i 1 + · · · + i g = i and that ν ℘ (d T [k] ) ≥ i k for k = 1, · · · , g. Then and so f T is contained in the right hand side of (2.11.1).
Next we consider f T such that ν ℘ (d T ) < i. Then one can find (i 1 , · · · , i g ) such that i 1 + · · · + i g = i and that ν ℘ (d T [k] ) ≤ i k for k = 1, · · · , g. Therefore g] is also contained in the right hand side of (2.11.1). The proposition is proved.
We have the following corollary.
Corollary 2.12. For λ ∈ Λ + and i ≥ 0, under the isomorphism Proof. By definition, we have By Proposition 2.11, we have a surjective map We claim that Ker Φ = Z λ R (i) ∩ ℘Z λ . Then the claim implies the corollary. So we shall show the claim. By definition, it is clear that . By the proof of Proposition 2.11, for some (i 1 , · · · , i g ) R (i g ). Moreover one can find at least one k such that ) is zero. Hence for T ∈ T p 0 such that ν ℘ (d T ) < i, r T f T is also contained in Ker Φ. Now the claim is proved, and the corollary follows.
By using the corollary, we show the following lemma.
Theorem 2.14. For λ, µ ∈ Λ + such that α p (λ) = α p (µ), we have Proof. The first equality follows from Theorem 2.8. So we prove the second equality. By Lemma 2.13, we have This proves the theorem.

v-Decomposition numbers for Ariki-Koike algebras
We keep the notation in the previous section. We consider the v-decomposition numbers of the Ariki-Koike algebra H , and show that similar results hold as in the previous section.

3.1.
Let ω = (−, · · · , −, (1 n )) be the r-partition and T ω be the ω-tableau of type ω. Since ϕ T ω T ω is an identity map on M ω and a zero map M µ for µ ∈ Λ such that µ = ω, ϕ T ω T ω is an idempotent in S . Moreover we see that ϕ T ω T ω S ϕ T ω T ω = Hom H (M ω , M ω ) = Hom H (H , H ) ∼ = H . It is well known that, for an S -module M, Mϕ T ω T ω becomes a H -module through the isomorphism ϕ T ω T ω S ϕ T ω T ω ∼ = H . Then we can define a functor, the so-called "Schur functor", from the category of right S -modules to the category of right H -modules by M → Mϕ T ω T ω . The following facts are known by [JM,Proposition 2.17].
where [S λ : D µ ] is the decomposition number of D µ in S λ .
3.2. One can define the Jantzen filtration of the Specht module S λ in a similar way as in the case of W λ , and we use a similar notation for this case. Then one can define the v-decomposition number of H , for λ, µ ∈ Λ + such taht D µ = 0, by We have the following lemma.

This shows that
x · ϕ T ω T ω , m t H ∈ ℘ i for any t ∈ Std(λ) by (3.3.1) and (3.3.2). Hence x · ϕ T ω T ω ∈ S λ R (i), and the claim follows by taking the quotient.
Write y = t∈Std(λ) r t m t , and put x = T∈T 0 (λ,ω) r t ϕ T ∈ W λ , where T is the λtableau of type ω corresponding to t. Then we have y = x · ϕ T ω T ω , and x, ϕ S ∈ ℘ i for any S ∈ T 0 (λ, ω) by (3.3.1), (3.3.2) and (3.3.3). Since ϕ T , ϕ S = 0 if the type of T is not the same as the type of S, we have x, ϕ S ∈ ℘ i for any S ∈ T 0 (λ).
This shows that x ∈ W λ R (i), and the claim follows. The lemma is proved.
This lemma implies the following proposition.
In particular, we have d λµ (v) = d H λµ (v). Proof. We consider the S -module filtration W λ (i) = W 0 W 1 · · · W k = W λ (i + 1) such that W j /W j+1 ∼ = L µ j . By applying the Schur functor, together with Lemma 3.3, we have The proposition follows from this.