MODIFIED ARIKI-KOIKE ALGEBRAS AND CYCLOTOMIC q-SCHUR ALGEBRAS

NOBUHARU SAWADA AND TOSHIAKI SHOJI

Department of Mathematics Science University of Tokyo Noda, Chiba 278-8510, Japan

Abstract. The modified Ariki-Koike algebra H is a variation of the original Ariki-Koike algebra H̃ over an integral domain R. When R is a rational function field over the independent parameters, H ≃ H̃. But for general R, H is not isomorphic to H̃, and has a simpler structure than H̃. In this paper, we construct a cellular basis of H which has a similar property as the cellular basis of H̃ introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of H̃ in terms of the decomposition numbers of H. We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_{q} and H on the tensor space V^{⊗n}.

0. Introduction

Let H̃ be the Ariki-Koike algebra over an integral domain R with parameters q, q^{−1}, Q_{1}, …, Q_{r} ∈ R, associated to the complex reflection group 𝔖_{n} ⋉ (ℤ/rℤ)^{n}. In [S], the second author constructed an algebra H over R, in the case where R is a field K_{0} of rational functions with variables q, Q_{1}, …, Q_{r}, and showed that H is isomorphic to H̃, giving a new presentation of H̃ (see [M2], where the survey on H, and related topics are found). The construction of H works for general R as far as the parameters satisfy the condition that (*) Q_{i} −Q_{j} are invertible for any i ≠ j. H and H̃ contain the Iwahori-Hecke algebra H_{n} of type A_{n−1}. The algebras H and H̃ are not isomorphic in general. However there exists a natural homomorphism ρ_{0} : H̃ → H which is identity on H_{n}. Although the map is not surjective nor injective, these two algebras seem to be closely related by it. The algebra H has a simple (degenerate) structure compared to H̃, which belong essentially to an object of type A. Even so, it is expected that H gives some informations on the representation theory of H̃, and we call H the modified Ariki-Koike algebra.

The purpose of this paper is to show that many results for H̃ established by Dipper-James-Mathas [DJM], and results for the Iwahori-Hecke algebra H_{n} written in the textbook of Mathas [M1], obtained by using the properties of cellular basis, holds as well for our algebra H, in many cases in a simpler form. First we construct the cellular basis of H, which enjoys many properties in common with the cellular basis of H̃ constructed in [DJM], though there is no direct relation between them. The construction of our basis is carried out by making use of the representation of H on the tensor space V^{⊗n}, on which H̃ also acts. This action of H̃ was defined in [SS], and plays a crucial role in defining H. In [H], Jun Hu also constructed an action of H̃ on V^{⊗n} by making use of the cellular basis of H̃. We show that these two actions of H̃ are essentially the same. (This fact was also proved by F. Stoll independently, see [HS].) This result gives a link between the cellular basis of H̃ and of H. We can deduce from this that the Specht modules of H̃ and that of H (regarded as H̃-modules via ρ_{0}) are isomorphic, and that there are some relations for irreducible modules also. This enables us to estimate the decomposition numbers of H̃ in terms of the decomposition numbers of H (Theorem 4.13).

Assume that V = ⨁_{i=1}^{m} V_{i} where V_{i} is a free R-module of rank m_{i}. We put m = (m_{1}, …, m_{r}). Then the quantum algebra U_{q}(𝔤) (its Kostant-Lusztig integral form over R) acts naturally on V^{⊗n}, which commutes with the action of H̃ (and of H), where 𝔤 = 𝔤𝔩_{m_{1}} ⊕ ⋯ ⊕ 𝔤𝔩_{m_{r}}. In the generic case, (i.e., the case where R = K_{0}), the Schur-Weyl reciprocity concerning the action of U_{q}(𝔤) and H̃ on V^{⊗n} was proved by [SS] (and independently by [Hu]). In [A], S. Ariki considered the centralizer algebra EndH̃ V^{⊗n} as an analogy of the cyclotomic q-Schur algebra defined in [DJM], and showed, under a certain condition on parameters (the separation condition, see (5.4.2)), that EndH̃ V^{⊗n} coincides with the image of the representation τ : U_{q}(𝔤) → End V^{⊗n}, and that it is isomorphic to the direct sum of the tensor product of the q-Schur algebras associated to H_{n_{i}} for various n_{i}. In this paper, we define a cyclotomic q-Schur algebra S(m, n) associated to H by End_{H} V^{⊗n}, and show that a similar result as in the case of Ariki holds for our case. (Note that our condition (*) is weaker than the separation condition.) We can also construct a cellular basis of S(m, n) which has similar properties as the cellular basis of the cyclotomic q-Schur algebra S̃(m, n) defined in [DJM]. We discuss the relationship between these two algebras.

By making use of the Schur functors, the above results are converted to the case of H̃, i.e, we obtain some estimate of the decomposition numbers of H̃ in terms of the decomposition numbers of H_{n_{i}} for various n_{i} (Corollary 6.3). The Schur functor is also used to show the integral form of the Schur-Weyl reciprocity concerning the action of U_{q}(𝔤) and H on V^{⊗n}.

Table of contents

0. Introduction

1. Preliminaries on Ariki-Koike algebras

2. A cellular basis of H

3. Hu’s action of H̃ on V^{⊗n}

4. Specht filtrations for M_{♮}^{μ} and M^{μ}

5. Cyclotomic q-Schur algebras

6. A comparison of cyclotomic q-Schur algebras

7. Schur functors

8. Schur-Weyl reciprocity

1. Preliminaries on Ariki-Koike algebras

1.1. Let R be an integral domain, q, Q_{1}, …, Q_{r} be elements in R, with q invertible. The Ariki-Koike algebra, i.e., the cyclotomic Hecke algebra associated to the complex reflection group W_{n,r} = G(r, 1, n), is the algebra H̃ = H̃_{n,r} over R with generators T_{1}, …, T_{n} subject to the following conditions,

(T_{1} − Q_{1}) ⋯ (T_{1} − Q_{r}) = 0,
(T_{i} − q)(T_{i} + q^{−1}) = 0  (i ≥ 2),
T_{1}T_{2}T_{1}T_{2} = T_{2}T_{1}T_{2}T_{1},
T_{i}T_{j} = T_{j}T_{i}  (|i − j| ≥ 2),
T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}  (2 ≤ i ≤ n − 1).

It is known that H̃ is a free R-module of rank n!r^{n}. The subalgebra H̃(𝔖_{n}) of H̃ generated by T_{2}, …, T_{n} is isomorphic to the Iwahori-Hecke algebra H_{n} of the symmetric group 𝔖_{n}.

1.2. We consider the following condition on parameters Q_{1}, …, Q_{r} in R.

(1.2.1) Q_{i} − Q_{j} are invertible in R for any i ≠ j.

In [S], a variation H of the Ariki-Koike algebra H̃ was definend subject to the condition (1.2.1). In this paper, we call H the modified Ariki-Koike algebra. It is known that if R = ℚ(q̄, Q̄_{1}, …, Q̄_{r}), the field of rational functions with variables q̄, Q̄_{1}, …, Q̄, H is isomorphic to H̃, and it gives a new presentation of H̃ apart from 1.1. Following [S], we recall the definition of H. Let A be a square matrix of degree r whose ij-entry is given by Q_{j}^{i−1} for 1 ≤ i, j ≤ r. Thus A is the Vandermonde matrix, and Δ = det A = ∏_{i>j}(Q_{i} − Q_{j}) is invertible by (1.2.1). We express the inverse of A as A^{−1} = Δ^{−1}B with B = (h_{ij}), and define a polynomial F_{i}(X) ∈ R[X], for 1 ≤ i ≤ r, by

(1.2.2)  F_{i}(X) = ∑_{1≤j≤r} h_{ij}X^{j−1}.

Then H = H_{n,r} is generated by T_{2}, …, T_{n} and ξ_{1}, …, ξ_{n} with relations

(1.2.3)  \begin{matrix}(T_{i} − q)(T_{i} + q^{−1}) = 0 & (2 ≤ i ≤ n), \\ (ξ_{i} − Q_{1}) ⋯ (ξ_{i} − Q_{r}) = 0 & (1 ≤ i ≤ n), \\ T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1} & (2 ≤ i ≤ n), \\ T_{i}T_{j} = T_{j}T_{i} & (|i − j| ≥ 2), \\ ξ_{i}ξ_{j} = ξ_{j}ξ_{i} & (1 ≤ i, j ≤ n) \end{matrix}

(1.2.4)  T_{j}ξ_{j} = ξ_{j−1}T_{j} + Δ^{−2} ∑_{c_{1}<c_{2}}(u_{c_{2}} − u_{c_{1}})(q − q^{−1})F_{c_{1}}(ξ_{j−1})F_{c_{2}}(ξ_{j})
(1.2.5)  T_{j}ξ_{j−1} = ξ_{j}T_{j} − Δ^{−2} ∑_{c_{1}<c_{2}}(u_{c_{2}} − u_{c_{1}})(q − q^{−1})F_{c_{1}}(ξ_{j−1})F_{c_{2}}(ξ_{j})
(1.2.6)  T_{j}ξ_{k} = ξ_{k}T_{j} (k ≠ j − 1, j),

The subalgebra H(𝔖_{n}) of H generated by T_{2}, …, T_{n} is isomorphic to H_{n}, hence it can be naturally identified with the corresponding subalgebra H̃(𝔖_{n}) of H̃. We denote by s_{2}, … s_{n} the simple reflections of 𝔖_{n} corresponding to T_{2}, …, T_{n}. The set {T_{w} | w ∈ 𝔖_{n}} forms a basis of H_{n}, where T_{w} = T_{i_{1}} ⋯ T_{i_{r}} for a reduced expression w = s_{i_{1}} ⋯ s_{i_{r}}. Moreover, it is known by [S] that the set

{ξ_{1}^{c_{1}} ⋯ ξ_{n}^{c_{n}}T_{w} | w ∈ 𝔖_{n}, 0 ≤ c_{i} ≤ r − 1}

gives rise to a basis of H.

Remark 1.3. In this paper, all the H-modules and H̃-modules are assumed to be right modules. In [SS], [S], the left H-module structure is discussed. The left module structure is converted to the right module structure if one changes the generators T_{i}, ξ_{j} by T_{i}^{−1}, ξ_{j}^{−1}, i.e., by replacing the parameters q, Q_{j} by q^{−1}, Q_{j}^{−1}. However, it is also possible to exchange the left module structure and the right module structure without changing the parameters since H is isomorphic to the opposite algebra H^{∘}. Namely, if the left H-module M is given, we define a right action of H on M by vh = h^{∗}v (v ∈ M, h ∈ H), where ∗ is an anti-algebra automorphism of H (see 2.7 below). The situation is the same for H̃ also. In this paper, we adopt this right action, and use freely the results in [S] accordingly.

1.4. Let V = ⨁_{i=1}^{r} V_{i} be a free R-module, with rank V_{i} = m_{i}. We put m = ∑ m_{i}. Let us fix a basis {v_{1}^{(i)}, …, v_{m_{i}}^{(i)}} of V_{i} for i = 1, …, r. The vectors

v_{1}^{(1)}, …, v_{m_{1}}^{(1)}, v_{1}^{(2)}, …, v_{m_{2}}^{(2)}, …, v_{1}^{(r)}, …, v_{m_{r}}^{(r)}

forms a basis of V, which we denote also by v_{1}, …, v_{m} in this order. We write v_{i} ∼ v_{j} if v_{i} and v_{j} are contained in a common subspace V_{k}. Let us define linear operators T, S on V ⊗ V and wt : V → V by

(1.4.1)  (v_{i} ⊗ v_{j})T = \begin{cases}qv_{i} ⊗ v_{j} & if i = j, \\ v_{j} ⊗ v_{i} & if i > j, \\ v_{j} ⊗ v_{i} + (q − q^{−1})v_{i} ⊗ v_{j} & if i < j, \end{cases}

(1.4.2)  (v_{i} ⊗ v_{j})S = \begin{cases}(v_{i} ⊗ v_{j})T & if v_{i} ∼ v_{j}, \\ v_{j} ⊗ v_{i} & otherwise. \end{cases}

(1.4.3)  (v_{i})wt = Q_{j}v_{i} if v_{i} ∈ V_{j}.

Then define T_{i}, S_{j}, ξ_{j} ∈ End_{R} V^{⊗n} (2 ≤ i ≤ n, 1 ≤ j ≤ n) by

T_{i} = id_{V}^{⊗(i−2)} ⊗T ⊗ id_{V}^{⊗(n−i)},
S_{i} = id_{V}^{⊗(i−2)} ⊗S ⊗ id_{V}^{⊗(n−i)},
ξ_{j} = id_{V}^{⊗(j−1)} ⊗wt ⊗ id_{V}^{⊗(n−j)} .

We define an operator T_{1} on V^{⊗n} by

(1.4.4)  T_{1} = ξ_{1}S_{2} ⋯ S_{n}T_{n}^{−1} ⋯ T_{2}^{−1}.

It is known by [SS] that the actions of T_{i} (1 ≤ i ≤ n) define a right H̃-module structure on V^{⊗n}. We denote this representation by ρ̃: H̃ → End V^{⊗n}. Note that this construction works under the general setting in 1.1, without (1.2.1). Also it is shown in [S] that, under the assumption (1.2.1), the actions of T_{i} (2 ≤ i ≤ n), ξ_{j}(1 ≤ j ≤ n) define a right action of H on V^{⊗n}. We denote this representation by ρ : H → End V^{⊗n}. By [S, Lemma 3.5], we know that S_{i} ∈ Im ρ. It follows that Im ρ̃ ⊂ Im ρ by (1.4.4).

Assuming that m_{i} ≥ n for i = 1, …, r, we consider the R-submodule U of V^{⊗n} generated by

v_{w(1)}^{(c_{1})} ⊗ ⋯ ⊗ v_{w(n)}^{(c_{n})} (w ∈ 𝔖_{n}, 1 ≤ c_{i} ≤ r).

Then U turns out to be an H-submodule of V^{⊗n}. Put

v_{0} = (∑_{c_{1}=1}^{r} v_{n}^{(c_{1})})⊗, (∑_{c_{2}=1}^{r} v_{n−1}^{(c_{2})}) ⊗ ⋯ ⊗ (∑_{c_{n}=1}^{r} v_{1}^{(c_{n})}).

It is easy to see, by using the actions of ξ_{j}, that the H-module v_{0}H contains the elements of the form v_{n}^{(c_{1})} ⊗ ⋯ ⊗ v_{1}^{(c_{1})}. It follows, by (1.4.1), that

(1.4.5) U is generated by v_{0} as an H-module. The map h ↦ v_{0}h gives an isomorphism H ≃ U as right H-modules. In particular, ρ is a faithful representation of H.

We have a lemma.

Lemma 1.5. Assume that m_{i} ≥ n for i = 1, …, r. Then there exists an R-algebra homomorphism ρ_{0} : H̃ → H such that ρ_{0} induces the identity on H_{n}. (Here we regard H_{n} ⊂ H̃, H_{n} ⊂ H under the previous identifications.) If Im ρ = Im ρ̃ and R is a field, then H̃ ≃ H.

Proof. By (1.4.5), we have an isomorphism H ≃ Im ρ. Since Im ρ̃ ⊂ Im ρ, we have a homomorphism ρ_{0} : H̃ → Im ρ̃ → H. If Im ρ = Im ρ̃, then ρ_{0} is surjective. Since H̃ and H are both free R-modules of the same rank, they are isomorphic if R is a field. □

1.6. Let R_{0} = ℤ[q̄, q̄^{−1}, Q̄_{1}, …, Q̄_{r}] be the polynomial ring with variables q̄, Q̄_{j}, and K_{0} be the quotient field of R_{0}. Let V̄ = ⨁_{i=1}^{r} V̄_{i} be the vector space over K_{0} with dim V̄_{i} = m_{i}. Then the quantum algebra U_{q,K_{0}}(𝔤𝔩_{m}) associated to the Lie algebra 𝔤𝔩_{m} over K_{0} acts from the left on the tensor space V̄^{⊗n} via the natural action of U_{q,K_{0}}(𝔤𝔩_{m}) on V̄. Let 𝔤 = 𝔤𝔩_{m_{1}} ⊕ ⋯ ⊕ 𝔤𝔩_{m_{r}} be the subalgebra of 𝔤𝔩_{m}. Then U_{q,K_{0}}(𝔤) ≃ U_{q,K_{0}}(𝔤𝔩_{m_{1}}) ⊗ ⋯ ⊗ U_{q,K_{0}}(𝔤𝔩_{m_{r}}) is a subalgebra of U_{q,K_{0}}(𝔤𝔩_{m}), and so U_{q,K_{0}}(𝔤) acts on V̄^{⊗n} as the restriction of the action of U_{q,K_{0}}(𝔤𝔩_{m}).

Let U_{q,R_{0}}(𝔤) be the Kostant-Lusztig form of U_{q,K_{0}}(𝔤). We denote by U_{q,R}(𝔤) the specialized algebra R ⊗_{R_{0}} U_{q,R_{0}}(𝔤) by the specialization R_{0} → R via q̄ ↦ q, Q̄_{j} ↦ Q_{j}. If there is no fear of confusion, we simply write U_{q,R}(𝔤) by U_{q} or U_{q}(𝔤). Now U_{q,R_{0}} stabilizes the R_{0}-form V̄_{R_{0}} of V̄, and so the space V̄_{R_{0}}^{⊗n}. Via the specialization V = R ⊗_{R_{0}} V̄_{R_{0}}, U_{q} acts on the R-module V^{⊗n}. It is known by [SS] that the thus obtained action of U_{q} on V^{⊗n} commutes with the left action of H, and that in the generic case, i.e., the case where R = K_{0}, the Schur-Weyl reciprocity holds between U_{q} and H. By Remark 1.3, the left action of U_{q} on V^{⊗n} commutes with the right action of H on V^{⊗n}, and a similar results hold also for this case.

2. A cellular basis of H

2.1. From now on, throughout the paper, we fix an r-tuple m = (m_{1}, …, m_{r}) of integers such that m_{i} ≥ n.

An ordered r-turple μ = (μ^{(1)}, …, μ^{(r)}) of μ^{(i)} = (μ_{1}^{(i)}, …, μ_{m_{i}}^{(i)}) ∈ ℤ_{≥0}^{m_{i}} is called an r-composition. For an r-composition μ, we define the type α = α(μ) of μ by α = (n_{1}, …, n_{r}) with n_{i} = |μ^{(i)}|, and the size of μ by n = ∑_{i=1}^{r} n_{i}. We also define a sequence a = a(μ) = (a_{1}, …, a_{r}) by a_{i} = ∑_{k=1}^{i−1} n_{i}. If each μ^{(i)} is a partition, μ is called an r-partition. We denote by P̃_{n,r} = P̃_{n,r}(m) (resp. P_{n,r} = P_{n,r}(m)) the set of r-compositions (resp. r-partitions) of size n. (Note that P_{n,r}(m) are naturally identified with each other for any m such that m_{i} ≥ n. However, P̃_{n,r} depends on the choice of m.)

Any μ ∈ P̃_{n,r} may be regarded as an element in P̃_{n,1} (i.e., 1-composition) of n by arranging the entries of μ = (μ_{j}^{(i)}) in the order

μ_{1}^{(1)}, …, μ_{m_{1}}^{(1)}, μ_{1}^{(2)}, …, μ_{m_{2}}^{(2)}, …, μ_{1}^{(r)}, …, μ_{m_{r}}^{(r)},

which we denote by {μ}.

The diagram of μ is [μ] = {(i, j, s) | 1 ≤ j ≤ μ_{i}^{(s)}, 1 ≤ s ≤ r}. The elements of [μ] are called nodes. A μ-tableau 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}) is a bijection 𝔱 : [μ] → {1, 2, …, n}, where 𝔱^{(i)} is a tableau of Shape(t^{(i)}) = μ^{(i)}. We write Shape(𝔱) = μ if 𝔱 is a μ-tableau. A μ-tableau 𝔱 is called standard (resp. row standard) if all 𝔱^{(i)} are standard (resp. row standard). For an r-composition μ, we denote by r-Std(μ) the set of row standard tableaux of shape μ. If λ is an r-partition, we denote by Std(λ) the set of standard tableaux of shape λ.

For each μ ∈ P̃_{n,r}, we denote by 𝔱^{μ} the tableau obtained from μ by associating numbers 1, 2, …, n to nodes of [μ], from left to right, and from top to bottom, regarding μ as a 1-composition {μ}. 𝔖_{n} acts on the set of tableaux of shape μ from the right, and we denote by 𝔖_{μ} the row stabilizer of 𝔱^{μ} in 𝔖_{n}. Hence

𝔖_{μ} = 𝔖_{μ_{1}^{(1)}} × 𝔖_{μ_{2}^{(1)}} × ⋯ × 𝔖_{μ_{mr}^{(r)}} ,

which coincides with the Young subgroup 𝔖_{{μ}} of 𝔖_{n}.

2.2. We consider the representation of H on V^{⊗n} as in 1.4. From now on we always assume that m_{i} ≥ n for i = 1, …, r whenever we consider V^{⊗n}. Let us prepare some notation. We write v_{i_{1}}^{(b_{1})} ⊗ ⋯ ⊗ v_{i_{n}}^{(b_{n})} ∈ V^{⊗n} by v_{I}^{b}, where I = (i_{1}, …, i_{n}) and b = (b_{1}, …, b_{n}). The permutation action of w ∈ 𝔖_{n} on V^{⊗n} maps v_{I}^{b} to v_{Iw}^{bw}, where Iw = (j_{w(1)}, …, j_{w(n)}) and bw = (b_{w(1)}, …, b_{w(n)}), respectively. We also write as v_{I} = v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{n}} with I = (i_{1}, …, i_{n}). The action of 𝔖_{n} on v_{I} is given similarly.

For α = (n_{1}, …, n_{r}) ∈ ℤ_{≥0}^{r} such that ∑ n_{i} = n, we define v_{α} ∈ V^{⊗n} by v_{α} = v_{J}^{c}, where

J = (n, n − 1, …, 2, 1), c = (\underbrace{r, …, r}_{n_{1}−times}, \underbrace{r − 1, …, r − 1}_{n_{2}−times}, …, \underbrace{1, …, 1}_{n_{r}−times}).

We note that v_{α} ∈ U. For μ ∈ P̃_{n,r} of type α, we define v_{μ} ∈ U by

(2.2.1)  v_{μ} = v_{α}m_{{μ}},

where m_{{μ}} = ∑_{w∈𝔖_{{μ}}} q^{l(w)}T_{w} ∈ H_{n}. Let H_{{μ}} be the parabolic subalgebra of H_{n} corresponding to 𝔖_{μ}, and θ_{q} the linear character of H_{{μ}} defined by θ_{q}(T_{w}) = q^{l(w)} for w ∈ 𝔖_{μ}. It is clear that v_{μ} ≠ 0, and since m_{{μ}} is an element affording θ_{q}, v_{μ} also satisfies the same property, i.e., we have

(2.2.2)  v_{μ}T_{w} = q^{l(w)}v_{μ} for w ∈ 𝔖_{μ}.

We define m_{μ} ∈ H as an element satisfying the formula v_{μ} = v_{0}m_{μ}, which is determined uniquely by (1.4.5). We shall describe the element m_{μ} ∈ H more explicitly. Let c = (c_{1}, …, c_{n}) be such that v_{α} = v_{J}^{c}. We define F_{α} ∈ H by

(2.2.3)  F_{α} = Δ^{−n}F_{c_{1}}(ξ_{1})F_{c_{2}}(ξ_{2}) ⋯ F_{c_{n}}(ξ_{n}).

Then as in the proof of Corollary 3.9 in [S], we see that v_{α} = v_{0}F_{α}. Again by using (1.4.5), we have the following.

Lemma 2.3. Assume that μ is of type α. Then we have m_{μ} = F_{α}m_{{μ}}.

2.4. For a 1-composition ν = (ν_{1}, …, ν_{k}), we set
H_{ν,r} = H_{ν_{1},r} ⊗ H_{ν_{2},r} ⊗ ⋯ ⊗ H_{ν_{k},r}.

For an r-composition μ of n with α(μ) = (n_{1}, …, n_{r}), let H_{μ} be the subalgebra of H generated by H_{{μ}} ⊂ H_{n} and ξ_{1}, …, ξ_{n}. Then by the discussion in [S,4], H_{μ} is isomorphic to the algebra

H_{μ^{(1)},r} ⊗ H_{μ^{(2)},r} ⊗ ⋯ ⊗ H_{μ^{(r)},r}

according to the natural embedding W_{μ^{(1)}} × W_{μ^{(2)}} × ⋯ × W_{μ^{(r)}} ⊂ W_{n,r}, where W_{μ^{(i)}} ≃ 𝔖_{μ^{(i)}} ⋉ (ℤ/rℤ)^{n_{i}}. Let θ_{q,μ}^{(s)} (1 ≤ s ≤ r) be the linear character of H_{μ,r} given by

\begin{aligned}θ_{q,μ}^{(s)}(T_{j}) & = q & for T_{j} ∈ H_{\{μ\}}, \\ θ_{q,μ}^{(s)}(ξ_{j}) & = Q_{s} & (1 ≤ j ≤ n) \end{aligned}

(cf. [S, 5.2]), which is an extension of θ_{q} on H_{{μ}}. We define a linear character θ_{q}^{μ} of H_{μ} by

θ_{q}^{μ} = θ_{q,μ^{(1)}}^{(r)} ⊗ θ_{q,μ^{(2)}}^{(r−1)} ⊗ ⋯ ⊗ θ_{q,μ^{(r)}}^{(1)}.

Then we have

(2.4.1) v_{μ} spans a one-dimensional H_{μ}-module affording the linear character θ_{q}^{μ}.

In fact it is enough to show that

v_{μ}T_{w} = q^{l(w)}v_{μ} for w ∈ 𝔖_{μ}
v_{μ}ξ_{j} = Q_{r−s+1}v_{μ} if a_{s} + 1 ≤ j ≤ a_{s+1},

where a_{s} = ∑_{k=1}^{s−1} n_{k}. But the first formula follows from (2.2.2). The second one follows easily, if we notice that v_{μ} is a linear combination of the vectors v_{I}^{c}, where c is the same one appeared in v_{α} = v_{J}^{c}. Hence (2.4.1) holds.

For a row standard μ-tableau 𝔱, we denote by d(𝔱) the unique element in 𝔖_{n} such that 𝔱 = 𝔱^{μ}d(𝔱). Then the map 𝔱 ↦ d(𝔱) gives a bijection from the set r-Std(μ) of row standard μ-tableaux to the set D_{μ} of distinguished coset representatives of 𝔖_{μ}\𝔖_{n}.

Let M^{μ} = v_{μ}H be the H-submodule of V^{⊗n} generated by v_{μ}. We have the following lemma.

Lemma 2.5. (i) M^{μ} is isomorphic to to Ind_{H_{μ}}^{H} θ_{q}^{μ} (resp to Ind_{H_{{μ}}}^{H_{n}} θ_{q} ) as an H-

module (resp. an H_{n}-module). (ii) M^{μ} is a free R-module with basis {v_{μ}T_{d(𝔱)} | 𝔱 ∈ r-Std(μ)}.

Proof. By using the commutation relation between T_{i} and ξ_{j}, together with (2.4.1), we see that M^{μ} = v_{μ}H_{n}. It is easy to see that v_{μ}H_{n} is spanned by {v_{μ}T_{d} | d ∈ D_{μ}}. Now v_{μ} can be written as a linear combination of v_{I}^{c} with the common c as before. Then one can check that v_{μ}T_{d} are linearly independent, and so {v_{μ}T_{d}} gives a basis of v_{μ}H_{n} = M^{μ}. Thus (ii) holds. Also we see that v_{μ}H_{n} ≃ Ind_{H_{{μ}}}^{H_{n}} θ_{q}. as an H_{n}-module.

Since H_{μ} is a free R-module with basis {ξ_{1}^{c_{1}} ⋯ ξ_{n}^{c_{n}}T_{σ} | σ ∈ 𝔖_{μ}}, H is a free H_{μ}- module with basis {T_{d} | d ∈ D_{μ}}. Now M^{μ} is generated by a vector v_{μ}, and v_{μ}T_{d} corresponds to a basis of θ_{q}^{μ} ⊗ T_{d} ⊂ θ_{q}^{μ} ⊗_{H_{λ}} H with the action of H. Thus we obtain (i). □

Lemma 2.6. Let 𝔱 ∈ r-Std(μ).

(i) If 𝔰 = 𝔱s_{i} for some i (2 ≤ i ≤ n), we have

v_{μ}T_{d(𝔱)}T_{i} = \begin{cases}qv_{μ}T_{d(𝔱)} & if 𝔰 ∉ r-Std(μ), \\ v_{μ}T_{d(𝔱)} & if 𝔰 ∈ r-Std(μ) and l(d(𝔰)) > l(d(𝔱)), \\ v_{μ}T_{d(𝔰)} + (q − q^{−1})v_{μ}T_{d(𝔱)} & if 𝔰 ∈ r-Std(μ) and l(d(𝔰)) < l(d(𝔱)). \end{cases}

(ii) For ξ_{j} (1 ≤ j ≤ n), we have

v_{μ}T_{d(𝔱)}ξ_{j} = Q_{ε(j)}v_{μ}T_{d(𝔱)},

where ε(j) = ε(j; 𝔱) ∈ ℤ_{>0} is determined by the condition that

v_{α}T_{d(𝔱)} = v_{j_{1}}^{(ε(1))} ⊗ v_{j_{2}}^{(ε(2))} ⊗ ⋯ ⊗ v_{j_{n}}^{(ε(n))}.

Proof. We note that d(𝔱) ∈ 𝔖_{n} coincides with the corresponding element in 𝔖_{n} obtained from 𝔱 and 𝔱^{{μ}}, where 𝔱 is regarded as a row standard tableau of shape {μ}. Now Lemma 2.5 shows that M^{μ} ≃ m_{{μ}}H_{n} as H_{n}-modules, via v_{μ}h ↔ m_{{μ}}h for h ∈ H_{n}. Hence (i) follows from the known formula for m_{{μ}}H_{n} (see, e.g., [M1, Corollary 3.4]) with respect to the action of T_{i} on m_{{μ}}T_{d(𝔱)}.

Next we show (ii). First we note the following general fact.

(2.6.1) Let v_{I} = v_{j_{1}} ⊗ ⋯ ⊗ v_{j_{n}} with j_{1} > j_{2} > ⋯ > j_{n}. Then for w ∈ 𝔖_{n}, v_{I}T_{w} coincides with v_{Iw}.

In fact, for w = w′s_{i} with l(w) > l(w′), we have w′(i − 1) < w′(i) and w(i − 1) > w(i). Assume that v_{I}T_{w′} is of the form v_{j_{w′(1)}} ⊗ ⋯ ⊗ v_{j_{w′(n)}}. Then we have j_{w′(i−1)} > j_{w′(i)}, and by (1.4.1), v_{I}T_{w} is obtained from v_{I}T_{w′} by permuting i − 1 and i factors. This shows (2.6.1) by induction on l(w).

Now v_{μ}T_{d(𝔱)} = ∑_{σ∈𝔖_{n}} q^{l(σ)}v_{α}T_{σd(𝔱)}. By applying (2.6.1) for v_{I} = v_{α}, we see that v_{α}T_{wd(𝔱)} = v_{J′}^{c′}, where J′ = Jwd(𝔱) and c′ = cwd(𝔱). Since cw = c for w ∈ 𝔖_{α}, we see that v_{μ}T_{d(𝔱)} is a linear combination of v_{I}^{cd(𝔱)}. Since cd(𝔱) = (ε(1), …, ε(n)), the assertion (ii) follows. □

2.7. Let M^{μ} = m_{μ}H. Then the map in (1.3.3) gives an isomorphism M^{μ} ≃ M^{μ} as H-modules, which maps m_{μ} to v_{μ}. Thus the statements in Lemma 2.5 and Lemma 2.6 hold without change for M^{μ} by replacing v_{μ} by m_{μ}.

We now define an R-linear anti-automorphism h ↦ h^{∗} on H by the condition that ∗ fixes the generators T_{i} (2 ≤ i ≤ n) and ξ_{j} (1 ≤ j ≤ n). We note that this condition induces a well-defined anti-automorphism on H since the map ∗ permutes (1.2.4) and (1.2.5) in the defining relations in 1.2, and leaves other formulas fixed.

Before going further, we remark the following property.

Lemma 2.8. The algebra R[ξ_{1}, …, ξ_{n}]^{𝔖_{n}} of symmetric polynomials of ξ_{1}, …, ξ_{n} is contained in the center of H.

Proof. It is enough to show that the elementary symmetric function e_{i}(ξ_{1}, …, ξ_{n}) commutes with T_{j} for 1 ≤ i ≤ n and 2 ≤ j ≤ n. Since T_{j}ξ_{k} = ξ_{k}T_{j} for k ≠ j − 1, j, we have only to verify the formulas

(ξ_{j−1} + ξ_{j})T_{j} = T_{j}(ξ_{j−1} + ξ_{j}),
(ξ_{j−1}ξ_{j})T_{j} = T_{j}(ξ_{j−1}ξ_{j}).

Now the first one follows by adding (1.2.4) and (1.2.5) in the defining relation in 1.2. The second one follows by multiplying ξ_{j−1} on (1.2.4) from the right, on (1.2.5) from the left, and then by adding these two formulas. □

By making use of Lemma 2.8, we show

Lemma 2.9. Let μ be an r-composition of type α = (n_{1}, …, n_{r}). Then

(i) F_{α} is contained in the center of H_{n_{1},r} ⊗ ⋯ ⊗ H_{n_{r},r}. In particular, we have F_{α} ∈ Z(H_{μ}).

(ii) We have m_{μ}^{∗} = m_{μ}.

Proof. By (2.2.3), F_{α} is a polynomial in ξ_{1}, …, ξ_{n}, and is written as

F_{α} = Δ^{−n}F_{r}(ξ_{1}) ⋯ F_{r}(ξ_{n_{1}})F_{r−1}(ξ_{n_{1}+1}) ⋯ F_{r−1}(ξ_{n_{1}+n_{2}})F_{r−2}(ξ_{n_{1}+n_{2}+1}) ⋯

It follows that F_{α} ∈ R[ξ_{1}, …, ξ_{n}]^{𝔖_{α}} with 𝔖_{α} = 𝔖_{n_{1}} × ⋯ × 𝔖_{n_{r}}. By applying Lemma 2.8 to H_{α,r} = H_{n_{1},r} ⊗ ⋯ ⊗ H_{n_{r},r}, we see that F_{α} is contained in the center of H_{α,r}. Since H_{μ} ⊂ H_{α,r}, we have F_{α} ∈ Z(H_{μ}). Hence (i) holds.

Since m_{μ} = F_{α}m_{{μ}} by Lemma 2.3, we have m_{μ}^{∗} = m_{{μ}}^{∗}F_{α}^{∗}. The map ∗ induces an R-linear anti-automorphism on H_{n} which coincides with the standard map ∗ defined by T_{i} ↦ T_{i} for i = 2, …, n. Then it is easy to see that m_{{μ}}^{∗} = m_{{μ}}. Clearly we have F_{α}^{∗} = F_{α}. Since m_{{μ}} ∈ H_{μ} and F_{α} ∈ Z(H_{μ}) by (i), we have m_{{μ}}F_{α} = F_{α}m_{{μ}} = m_{μ}. This shows (ii). □

2.10. For 𝔰, 𝔱 ∈ r-Std(μ), we define an element m_{𝔰𝔱} ∈ H by m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{μ}T_{d(𝔱)}. By Lemma 2.9, we have m_{𝔰𝔱}^{∗} = m_{𝔱𝔰}. We want to show that such m_{𝔰𝔱} for various μ ∈ P̃_{n,r} span the space H. First, we note the following lemmas.

Lemma 2.11. Let F_{α} = Δ^{−n}F_{c_{1}}(ξ_{1})F_{c_{2}}(ξ_{2}) ⋯ F_{c_{n}}(ξ_{n}) with c = (c_{1}, …, c_{n}). Then for any vector v_{I}^{b} ∈ V^{⊗n}, we have

v_{I}^{b}F_{α} = \begin{cases}v_{I}^{c} & if b = c, \\ 0 & otherwise. \end{cases}

Proof. Let b = (b_{1}, …, b_{n}). Since v_{I}^{b}ξ_{k} = Q_{b_{k}} · v_{I}^{b}, we have

v_{I}^{b}F_{α} = Δ^{−n}F_{c_{1}}(Q_{b_{1}})F_{c_{2}}(Q_{b_{2}}) ⋯ F_{c_{n}}(Q_{b_{n}})v_{I}^{b}.

The lemma then follows from the formula F_{i}(Q_{j}) = Δδ_{ij} (cf. [S, (3.3.3)]). □

Lemma 2.12. For d = (d_{1}, …, d_{n}) ∈ [1, r]^{n}, let w = w(d) ∈ 𝔖_{n} be the unique element such that d_{w(1)} ≥ d_{w(2)} ≥ ⋯ ≥ d_{w(n)} and that l(w) is minimal among those elements. Let α = α(d) = (n_{1}, …, n_{r}) with n_{i} = ♯{j | d_{j} = r − i + 1}. Then for v_{J}^{b} with J = (n, n − 1, …, 1) we have

v_{J}^{b}T_{w}F_{α}T_{w^{−1}} = \begin{cases}v_{J}^{d} & if b = d, \\ v_{J}^{b}h (h ∈ H_{n}) & if α(d) = α(b), l(w(d)) > l(w(b)), \\ 0 & otherwise. \end{cases}

Proof. By (1.4.1), we see easily describe the effect of T_{w} on v_{J}^{b} as

(2.12.1)  v_{J}^{b}T_{w} = v_{Jw}^{bw} + ∑_{x<w} r_{x}v_{Jx}^{bx} (r_{x} ∈ R),

where x < w means the Bruhat order on 𝔖_{n}. First we consider the case where b = d, and we apply F_{α} on both sides of (2.12.1). Note that c = (c_{1}, …, c_{n}) associated to F_{α} coincides with dw in this case. Also note that dx ≠ dw for any x < w by our assumption on w. It follows from Lemma 2.11 that

(2.12.2)  v_{J}^{d}T_{w}F_{α} = v_{Jw}^{dw}.

Here we note that

(2.12.3) Write v_{Jw}^{dw} as v_{k_{1}} ⊗ ⋯ ⊗ v_{k_{n}}. Then we have k_{1} > k_{2} > ⋯ > k_{n}.

In fact, let w_{0} be the longest element of 𝔖_{n}. Then v_{Jw}^{dw} can be written as

v_{Jw}^{dw} = v_{w_{0}w(1)}^{d_{w(1)}} ⊗ ⋯ ⊗ v_{w_{0}w(n)}^{d_{w(n)}}.

Hence we have only to show that if d_{w(i−1)} = d_{w(i)}, then w_{0}w(i − 1) > w_{0}w(i), in other words, w(i − 1) < w(i). Suppose that this fails for some i, i.e., w(i − 1) > w(i). Then we have l(ws_{i}) < l(w). Since d_{ws_{i}(1)} ≥ d_{ws_{i}(2)} ≥ ⋯ ≥ d_{ws_{i}(n)}, this contradicts the choice of w. So (2.12.3) holds.

Now by applying (2.6.1), we see that v_{Jw}^{dw}T_{w^{−1}} coincides with v_{J}^{d}. It follows that v_{J}^{d}T_{w}F_{α}T_{w^{−1}} = v_{J}^{d} by (2.12.2).

Next we consider the case where b ≠ d. If bx is not equal to dw for any x ≤ w, then v_{J}^{b}T_{w}F_{α} = 0 by (2.12.1). Assume that there exists x < w such that bx = dw. Then α(d) = α(b). Clearly we have l(w(b)) ≤ l(x) < l(w(d)). By (2.12.1), v_{J}^{b}T_{w}F_{α} is written as a linear combination of v_{Jx}^{bx} for x < w. Hence v_{J}^{b}T_{w}F_{α} is contained in v_{J}^{b}H_{n}. This proves the second equality, and the lemma follows. □

By making use of Lemma 2.12, we show the following.

Proposition 2.13. The set C̃ = {m_{𝔰𝔱} | 𝔰, 𝔱 ∈ r-Std(μ), μ ∈ P̃_{n,r}} spans the R-module H.

Proof. Let μ ∈ P̃_{n,r}. By Lemma 2.5, and its counter parts for M^{μ} = m_{μ}H, any elements in m_{μ}H can be written as a linear combination of m_{μ}T_{d(𝔱)} with 𝔱 ∈ r-Std(μ). Similarly Hm_{μ} = (m_{μ}H)^{∗} is spanned by the elements (m_{μ}T_{d(𝔰)})^{∗} = T_{d(𝔰)}^{∗}m_{μ} with 𝔰 ∈ r-Std(μ). Thus Hm_{μ}H is spanned by elements T_{d(𝔰)}^{∗}m_{μ}T_{d(𝔱)} = m_{𝔰𝔱} ∈ C̃. It follows that the R-span of C̃ coincides with the set ⋃_{μ} Hm_{μ}H. In particular it is a two sided ideal of H. Hence the proposition is proved if one shows that 1 ∈ H is expressed as a linear combination of elements in C̃.

Let L be the R-span of all the elements m_{𝔰𝔱}, 𝔰, 𝔱 ∈ Std(λ), where λ runs over the r-partitions of the form λ = (1^{n_{1}}, 2^{n_{2}}, …, r^{n_{r}}) with α(λ) = (n_{1}, …, n_{r}). We show that

(2.13.1) The R-space v_{0}L contains v_{0}.

Note that the proposition follows from (2.13.1), since it implies that 1 ∈ L by (1.4.5). We shall show (2.13.1) by applying Lemma 2.12. First note that if λ ∈ P_{n,r} is as above, then m_{{λ}} = 1 and D_{λ} = 𝔖_{n}. It follows that any element of the form T_{w′}F_{α}T_{w} (w, w′ ∈ 𝔖_{n}) with α = α(λ) is expressed as m_{𝔰𝔱} for some 𝔰, 𝔱 ∈ Std(λ). In particular, L is a right H_{n}-module.

Now v_{0} is written as v_{0} = ∑_{d∈[1,r]^{n}} v_{J}^{d} with J = (n, n − 1, …, 1). We shall show that v_{J}^{d} ∈ v_{0}L for any d ∈ [1, r]^{n}. Take d ∈ [1, r]^{n}, and α = α(d). Then by Lemma 2.12 we have

v_{0}T_{w}F_{α}T_{w^{−1}} = v_{J}^{d} + ∑_{b} v_{J}^{b}h_{b}

with h_{b} ∈ H_{n}, where b runs over the elements in [1, r]^{n} under the condition that α(b) = α(d) and that l(w(b)) < l(w(d)). By induction on l(w(d)), we may assume that v_{J}^{v} ∈ v_{0}L. Since v_{0}L is a right H_{n}-module, then v_{j}^{b}h_{b} ∈ v_{0}L also. Since v_{0}T_{w}F_{α}T_{w^{−1}} = v_{0}m_{𝔰𝔱} ∈ v_{0}L, we see that v_{J}^{d} ∈ v_{0}L. Thus all the v_{J}^{d} are contained in v_{0}L, and so v_{0} ∈ v_{0}L. (2.13.1) is now proved, and the proposition follows. □

2.14. We define a dominance order λ ⊵ μ on P̃_{n,r} by {λ} ⊵ {μ}, where the latter is the usual dominance order for 1-compositions of n, i.e., for {λ} = (λ_{1}, …, λ_{m}), {μ} = (μ_{1}, …, μ_{m}) ∈ P̃_{n,1}, we require that ∑_{i=1}^{k} λ_{i} ≥ ∑_{i=1}^{k} μ_{i} for k = 1, …, m. Let μ be an r-composition. As in the case of 1-composition, for 𝔱 ∈ r-Std(μ) and for a positive integer k ≤ n, we define 𝔱 ↓ k as the row standard tableau obtained from 𝔱 by deleting all of the entries containing k+1, …, n. For row standard tableaux 𝔱 and 𝔰 (of possibly different shape), we write 𝔱 ⊵ 𝔰 if Shape(𝔱 ↓ k) ⊵ Shape(𝔰 ↓ k) for k = 1, …, n. Then it is clear that 𝔱^{μ} ⊵ 𝔱 for any 𝔱 ∈ r-Std(μ). We have the following lemma.

Lemma 2.15. Let μ be an r-composition of n and 𝔰, 𝔱 ∈ r-Std(μ). Then m_{𝔰𝔱} can be written as an R-linear combination of m_{𝔲𝔳}, where 𝔲, 𝔳 ∈ Std(μ) with μ ∈ P_{n,r} such that 𝔲 ⊵ 𝔰, 𝔳 ⊵ 𝔱 (hence λ ⊵ μ).

Proof. This lemma can be proved in a completely similar way as [DJM, Proposition 3.18]. However, this fact (the similarity to [DJM]) itself is important for later discussions, we give here an outline of the proof.

Let μ = (μ^{(1)}, …, μ^{(r)}) be of type α = (n_{1}, …, n_{r}). We write 𝔰 = (𝔰^{(1)}, …, 𝔰^{(r)}), where 𝔰^{(i)} ∈ r-Std(μ^{(i)}), and similarly for 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}). Then there exist w_{1}, w_{2} ∈ 𝔖_{n}, and 𝔭 = (𝔭^{(1)}, …, 𝔭^{(r)}), 𝔮 = (𝔮^{(1)}, …, 𝔮^{(r)}) ∈ r-Std(μ) satisfying the following; 𝔰 = 𝔭w_{1}, 𝔱 = 𝔮w_{2}, the entries of 𝔭^{(i)}, 𝔮^{(i)} consist of numbers a_{i} + 1, …, a_{i+1}, where a(μ) = (a_{1}, …, a_{r}). Those w_{1}, w_{2}, 𝔭, 𝔮 are unique under the condition that w_{1}, w_{2} ∈ D_{α}. We define d(𝔭^{(i)}), d(𝔮^{(q)}) ∈ 𝔖_{n_{i}} by

𝔱^{μ^{(i)}}d(𝔭^{(i)}) = 𝔭^{(i)}, 𝔱^{μ^{(i)}}d(𝔮^{(i)}) = 𝔮^{(i)} (1 ≤ i ≤ r),

where 𝔱^{μ} = (𝔱^{μ^{(1)}}, …, 𝔱^{μ^{(r)}}). Then we have d(𝔰) = d(𝔭)w_{1}, d(𝔱) = d(𝔮)w_{2} with

d(𝔭) = d(𝔭^{(1)}) ⋯ d(𝔭^{(r)}), d(𝔮) = d(𝔮^{(1)}) ⋯ d(𝔮^{(r)}).

Since m_{{μ}} can be written as m_{{μ}} = m_{μ^{(1)}} ⋯ m_{μ^{(r)}}, and F_{α} commutes with d(𝔭^{(i)}), d(𝔮^{(i)}) for any i, we can write

m_{𝔰𝔱} = T_{w_{1}}^{∗}(x_{1}x_{2} ⋯ x_{r})F_{α}T_{w_{2}}

with x_{i} = T_{d(𝔭^{(i)})}^{∗}m_{μ^{(i)}}T_{d(𝔮^{(i)})}.

Now by applying Murphy’s result [Mu, 4.18] on H_{n_{i}} for x_{i}, we see that x_{i} can be written as a linear combination of y_{i} = T_{d(𝔲^{(i)})}^{∗}m_{μ^{(i)}}T_{d(𝔳^{(i)})}, where μ^{(i)} is a partition of n_{i} and 𝔲^{(i)}, 𝔳^{(i)} ∈ Std(μ^{(i)}) with 𝔲^{(i)} ⊵ 𝔭^{(i)}, 𝔳^{(i)} ⊵ 𝔮^{(i)}. It follows that m_{𝔰𝔱} can be written as a linear combination of

(2.15.1)  T_{w_{1}}^{∗}(y_{1}y_{2} ⋯ y_{r})F_{α}T_{w_{2}}.

Set λ = (λ^{(1)}, …, λ^{(r)}) and 𝔲 = (𝔲^{(1)}, …, 𝔲^{(r)}), 𝔳 = (𝔳^{(1)}, …, 𝔳^{(r)}). Then we see that λ ∈ P_{n,r} with type α, and 𝔲, 𝔳 ∈ Std(λ) with 𝔲 ⊵ 𝔭, 𝔳 ⊵ 𝔮. The formula in (2.15.1) coincides with

T_{w_{1}}^{∗}T_{d(𝔲)}^{∗}m_{{λ}}T_{d(𝔳)}F_{α}T_{w_{2}} = T_{w_{1}}^{∗}T_{d(𝔲)}^{∗}m_{λ}T_{d(𝔳)}T_{w_{2}}.

If we put 𝔲′ = 𝔲w_{1}, 𝔳′ = 𝔳w_{2}, we have 𝔲′, 𝔳′ ∈ Std(λ). Thus the lemma follows. □

Summing up the above arguments, we have

Proposition 2.16. Let C = {m_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ), λ ∈ P_{n,r}}. Then the set C forms a free basis of the R-module H.

Proof. By Proposition 2.13, together with Lemma 2.15, we see that the set C spans the R-module H. The linearly independence follows from the fact that the cardinality of C coincides with |W_{n,r}| = n!r^{n}, which is equal to the rank of H. The proposition follows. □

For μ ∈ P_{n,r}, let Ň^{μ} be the R-submodule of H spanned by m_{𝔲𝔳} with 𝔲, 𝔳 ∈ Std(μ) for various μ ∈ P_{n,r} such that μ ▷ μ. We have the following lemma.

Lemma 2.17. Let μ ∈ P_{n,r} and 𝔱 ∈ Std(μ). Then for w ∈ 𝔖_{n}, there exist r_{𝔳} ∈ R such that

(2.17.1)  m_{𝔱^{μ}𝔱}T_{w} ≡ ∑_{𝔳∈Std(μ)}r_{𝔳}m_{𝔱^{μ}𝔳} mod Ň^{μ}.

Proof. By Lemma 2.6, m_{𝔱^{μ}𝔱}T_{w} can be written as a linear combination of m_{𝔱^{μ}𝔮} with 𝔮 ∈ r-Std(μ). Then by Lemma 2.15, m_{𝔱^{μ}𝔱}T_{w} can be written as a linear combination of m_{𝔲𝔳} for 𝔲, 𝔳 ∈ Std(μ) for various r-partitions μ such that 𝔲 ⊵ 𝔱^{μ}, 𝔳 ⊵ 𝔱. We note that μ ⊵ μ since 𝔲 ⊵ 𝔱^{μ}. To prove the lemma, it is enough to show that μ ▷ μ if 𝔲 ≠ 𝔱^{μ}. Now assume that 𝔲 ≠ 𝔱^{μ}. Then 𝔲 ▷ 𝔱^{μ}. If μ = μ, we have 𝔱^{μ} ⊵ 𝔲 by the remark in 2.14. This is a contradiction, and the lemma follows. □

Theorem 2.18. Let C be the R-free basis of H given above. Then C gives rise to a cellular basis of H in the sense of [GL], i.e., the following properties hold.

(i) The R-linear map determined by m_{𝔰𝔱} ↦ m_{𝔱𝔰} (m_{𝔰𝔱} ∈ C) is an anti-linear automorphism on H.

(ii) For a given h ∈ H and 𝔱 ∈ Std(μ), there exist r_{𝔳} ∈ R such that for all 𝔰 ∈ Std(μ), we have

(2.18.1)  m_{𝔰𝔱}h ≡ ∑_{𝔳∈Std(μ)}r_{𝔳}m_{𝔰𝔳} mod Ň^{μ}.

Proof. (i) is clear since m_{𝔰𝔱}^{∗} = m_{𝔱𝔰} and ∗ is an anti-automorphism of the R-algebra H. We show (ii). First we show the formula (2.18.1) in the case where h = T_{w} ∈ H_{n} by induction on the dominance order of P_{n,r}. Assume that μ = (n; −; …; −). Then 𝔖_{μ} = 𝔖_{n} and 𝔱^{μ} is the unique standard tableau. Hence m_{𝔰𝔱} = m_{𝔱^{μ}𝔱^{μ}} = m_{μ}, and m_{μ}T_{w} = q^{l(w)}m_{μ}. So (2.18.1) holds. Now assume that (2.18.1) holds for h = T_{w} for any λ such that λ ▷ μ. By noticing that m_{𝔰𝔳} = T_{d(𝔰)}^{∗}m_{𝔱^{μ}𝔳}, we multiply T_{d(𝔰)}^{∗} from the left on both sides of (2.17.1). If one shows that T_{d(𝔰)}^{∗}Ň^{μ} ⊂ Ň^{μ}, we obtain (2.18.1) for h = T_{w}. Take λ such that λ ▷ μ, and define N^{λ} as the R-submodule of H spanned by m_{𝔲𝔳} for 𝔲, 𝔳 ∈ Std(ν) such that ν ⊵ λ. Note that, for 𝔰 ∈ Std(μ) and for 𝔲, 𝔳 ∈ Std(λ), we have

T_{d(𝔰)}^{∗}m_{𝔲𝔳} = (m_{𝔲𝔳}T_{d(𝔰)})^{∗} ∈ N^{λ}

since induction works for m_{𝔲𝔳}T_{d(𝔰)}, and N^{μ} is stable by ∗. Clearly we have N^{λ} ⊆ Ň^{μ}, and so T_{d(𝔰)}^{∗}m_{𝔲𝔳} ∈ Ň^{μ}. This proves that T_{d(𝔰)}^{∗}Ň^{μ} ⊆ Ň^{μ}, and so (2.18.1) holds also for μ.

Now by Lemma 2.6 (ii), ξ_{j} acts on m_{𝔰𝔱} as a scalar multiplication by Q_{ε(j)}. Since any element of H is written as a linear combination of ξ_{1}^{c_{1}} ⋯ ξ_{n}^{c_{n}}T_{w}, (2.18.1) holds for any h ∈ H. This proves (ii). □

2.19. For λ ∈ P_{n,r}, Ň^{λ} is a two sided ideal of H by (2.18.1). Let \overline{m}_{λ} be the image of m_{λ} under the algebra homomorphism H → H/Ň^{λ}. We denote by S^{λ} the right Hsubmodule of H/Ň^{λ} generated by \overline{m}_{λ}, which is called the Specht module associated to λ. By Theorem 2.18, S^{λ} is a free R-module with basis \{\overline{m}_{λ}T_{d(𝔱)} | 𝔱 ∈ Std(λ)\}. By a general theory of cellular basis [GL], one can define an H-module D^{λ} of S^{λ} for each λ ∈ P_{n,r} as the quotient of S^{λ} by the radical of certain bilinear form on S^{λ}. If R is a filed, the set {D^{λ} ≠ 0 | λ ∈ P_{n,r}} gives a complete set of non-isomorphic irreducible H-modules.

3. Hu’s action of H̃ on V^{⊗n}

3.1. In [H], J. Hu constructed a representation of H̃ on V^{⊗n}, independently from [SS]. He claims in [H, Remark 3.5] that his H̃-module structure on V^{⊗n} is different from ours. In fact, he gave a counter example in the case where n = r = 2, m_{1} = m_{2} = 1, by comparing the result of [ATY]. In turn, our representation coincides with the one in [ATY] in the special case where m_{1} = m_{2} = ⋯ = m_{r} = 1. Nevertheless, in this section we shall show that Hu’s representation essentially coincides with ours, i.e., we show that a modified Hu’s representation actually coincides with ours. (The same result was proved by F. Stoll independently, and is announced in [HS] without proof.)

In this section, we consider the general setting as in the beginning of 1.4, without any restriction on m, and only assuming that q, Q_{1}, …, Q_{r} are in R, with q invertible. In order to explain Hu’s construction, we review some results in [DJM] concerning a cellular basis of H̃. We define L_{i} = T_{i}T_{i−1} ⋯ T_{2}T_{1}T_{2} ⋯ T_{i−1}T_{i} for i = 1, 2, …, n. For an r-tuple a = (a_{1}, … a_{r}) of integers such that 0 = a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{r} ≤ n, let u_{a}^{+} = u_{a,1}u_{a,2} ⋯ u_{a,r}, where

u_{a,k} = ∏_{i=1}^{a_{k}}(L_{i} − Q_{k})

for 1 ≤ k ≤ r. For μ ∈ P̃_{n,r}, we define an element m_{μ}^{♮} ∈ H by

m_{μ}^{♮} = u_{a}^{+}m_{{μ}},

where m_{{μ}} is as in 2.2, and a = a(μ) is as in 2.1.

Now, for 𝔰, 𝔱 ∈ r-Std(μ) with μ ∈ P̃_{n,r}, put

(3.1.1)  m_{𝔰𝔱}^{♮} = T_{d(𝔰)}^{∗}m_{μ}^{♮}T_{d(𝔱)}.

Then it is known by [DJM, 3.26] that

(3.1.2) The set C^{♮} = {m_{𝔰𝔱}^{♮} | 𝔰, 𝔱 ∈ Std(λ), λ ∈ P_{n,r}} forms a cellular basis of H̃.

Remark 3.2. There are some discrepancies in the notation in [DJM], [H] and ours. The algebra H̃ in [DJM] is obtained from ours by putting T_{0} = T_{1} and T_{i} = qT_{i+1} (1 ≤ i ≤ n − 1) and by replacing q by q^{1/2}. While the algebra in [H] is obtained from ours by putting T_{i} = T_{i+1} for (0 ≤ i ≤ n − 1) and by replacing q by q^{−1}. Thus, L_{i} in 3.1 corresponds to L_{i}, and our m_{{μ}} (resp. m_{μ}^{♮}) corresponds to x_{μ} (resp. m_{μ}) in [DJM] under the above identification. Moreover, m_{{μ}} corresponds to y_{μ} in [H]. Thus our m_{μ}^{♮} also corresponds to m\underline{μ} in [H].

Also we note that the element m_{𝔰𝔱}^{♮} in (3.1.1) corresponds to q^{−l(d(𝔰))−l(d(𝔱))}m_{𝔰𝔱} (and replacing q by q^{1/2}) in [DJM].

3.3. We define a bijection ϕ : P̃_{n,r} → P̃_{n,r} by ϕ(μ) = ν for μ ∈ P̃_{n,r}, where ν = (ν^{(1)}, …, ν^{(r)}) is given by

ν^{(k)} = (μ_{m_{k}}^{(r−k+1)}, μ_{m_{k−1}}^{(r−k+1)}, …, μ_{1}^{(r−k+1)}) ∈ ℤ_{≥0}^{m_{k}}

if μ^{(k)} = (μ_{1}^{(k)}, μ_{2}^{(k)}, …, μ_{m_{k}}^{(k)}) ∈ ℤ_{≥0}^{m_{k}}. Hence ν is of type (n_{r}, …, n_{1}). We often write μ′ = ϕ(μ). Don’t confuse μ′ with the dual partition of μ.

We consider the action of U_{q} on V^{⊗n}. The weights of U_{q} are given by r-compositions of n. For μ ∈ P̃_{n,r}, we denote by V_{μ}^{⊗} the weight subspace of V^{⊗n} corresponding to the weight μ. Then V_{μ}^{⊗n} has a basis vector v_{i_{1}} ⊗⋯⊗v_{i_{n}} such that ♯{a | v_{i_{a}} = v_{j}^{(k)}} = μ_{j}^{(k)}. In particular, we have

(3.3.1)  V^{⊗n} = ⨁_{μ∈P̃_{n,r}} V_{μ}^{⊗n}

Clearly the weight space V_{μ}^{⊗n} is stable by H_{n}. Put

(3.3.2)  v_{μ}^{♮} = \underbrace{v_{m_{r}}^{(r)} ⊗ ⋯ ⊗ v_{m_{r}}^{(r)}}_{μ_{mr}^{(r)} factors} ⊗ \underbrace{v_{m_{r}−1}^{(r)} ⊗ ⋯ ⊗ v_{m_{r}−1}^{(r)}}_{μ_{mr−1}^{(r)} factors} ⊗ ⋯ ⊗ \underbrace{v_{1}^{(1)} ⊗ ⋯ ⊗ v_{1}^{(1)}}_{μ_{1}^{(1)} factors} .

Then it is easy to see that V_{μ}^{⊗n} is generated by v_{μ}^{♮} as a right H_{n}-module, and is isomorphic to Ind_{H_{{μ′}}}^{H_{n}} θ_{q} with μ′ = ϕ(μ). Hence we have

(3.3.3)  V_{μ}^{⊗n} = v_{μ}^{♮}H_{n} ≃ v_{μ′}H_{n} ≃ m_{{μ′}}H_{n}

as H_{n}-modules (cf. Lemma 2.5). The isomorphism m_{{μ′}}H_{n} → V_{μ}^{⊗n} is given by m_{{μ′}}h ↦ v_{μ}^{♮}h, (h ∈ H_{n}).

We define a partial order on the set ℤ_{≥0}^{r} by a ≥ a′ for a = (a_{1}, …, a_{r}), a′ = (a′_{1}, …, a′_{r}) ∈ ℤ_{≥0}^{r} if a_{i} ≥ a′_{i} for any i. We write a > a′ if a ≥ a′ and a ≠ a′. It is clear that

(3.3.4) If λ ⊵ μ, then a(λ) ≥ a(μ) for λ, μ ∈ P̃_{n,r}.

For an r-composition μ of n, we define a right ideal of H̃ by M_{♮}^{μ} = m_{μ}^{♮}H̃. Then we define N̂^{a(μ)} as an R-submodule of H̃ generated by m_{𝔰𝔱}^{♮} such that 𝔰, 𝔱 ∈ Std(ν) with a(ν) > a(μ). By Lemma 3.2 in [H], N̂^{a(μ)} is a two sided ideal of H̃, and so if we put M̂_{♮}^{μ} = M_{♮}^{μ} ∩ N̂^{a(μ)}, M_{♮}^{μ}/M̂_{♮}^{μ} has a natural structure of a right H̃-module.

Hu showed the following.

Theorem 3.4 ([H, Theorem 3.3]). Let a = a(μ). The R-linear map

m_{{μ}}H_{n} → M_{♮}^{μ}/M̂_{♮}^{μ}, m_{{μ}}h ↦ u_{a}^{+}m_{{μ}}h + M̂_{♮}^{μ}

gives an isomorphism of right H_{n}-modules. (Note that u_{a}^{+}m_{{μ}} = m_{μ}^{♮}).

3.5. By using the isomorphism in the above theorem, together with (3.3.3), one can define an action of H̃ on V_{μ}^{⊗n} ≃ M_{♮}^{μ′}/M̂_{♮}^{μ′}. Thus by (3.3.1), V^{⊗n} is endowed with an H-module structure. He showed that the thus obtained action of H̃ commutes with the action of U_{q} and that the Schur-Weyl reciprocity holds for U_{q}-H̃-bimodule V^{⊗n} ([H, Theorem 4.3]). We call this action of H̃ on V^{⊗n} as Hu’s action (however see Remark 3.7 below).

We now show the following theorem.

Theorem 3.6. Hu’s action of H̃ on V^{⊗n} coincides with the action given in 1.4.

Proof. In the following proof, we understand that the action of H̃ always means our action, while Hu’s action is denoted as Hu’s action. It is clear from the definition (1.4.4) that V_{μ}^{⊗n} is stable by H̃ for any μ ∈ P̃_{n,r}. Hence we have only to show the following statement.

(3.6.1) For each μ ∈ P̃_{n,r}, Hu’s action of T_{1} on V_{μ}^{⊗n} coincides with our action.

We show (3.6.1). Put μ′ = ϕ(μ) as before. Let α = (n_{1}, …, n_{r}) be the type of μ and α′ = (n_{r}, …, n_{1}), the type of μ′. We write a = a(μ′) = (a_{1}, …, a_{r}). First we describe the Hu’s action of T_{1}. Now V_{μ}^{⊗n} has a basis {v_{μ}^{♮}T_{d} | d ∈ D_{μ′}}. We fix d as above. Then d can be written as d = yc, where y (resp, c) is a distinguished right coset representative of 𝔖_{μ′} in 𝔖_{α′} (resp. of 𝔖_{α′} in 𝔖_{n}). Assume that the letter 1 is in the kth row of the row standard tableau t^{α′}c. Then c can be further decomposed as c = wc′, where w = s_{b_{k}+1}s_{b_{k}} ⋯ s_{3}s_{2} and c′ ∈ 𝔖_{n−1} stabilizing 1, such that l(c) = l(w) + l(c′).

By using the arguments in the proof of [H, Theorem 3.3], (see also [DJM, 3.4]), we see that Hu’s action of T_{1} is given as

(3.6.2)  v_{μ}^{♮}T_{d}T_{1} = Q_{k}v_{μ}^{♮}T_{y}T_{b_{k}+1}^{−1} ⋯ T_{2}^{−1}T_{c′}.

Next we compute our action of T_{1} on the vector v_{μ}^{♮}T_{d} following the definition in (1.4.4). Since T_{c′} is a product of T_{3}, …, T_{n}, we see that T_{1} commutes with T_{c′}. Hence

(3.6.3)  v_{μ}^{♮}T_{d}T_{1} = v_{μ}^{♮}T_{y}T_{w}T_{1}T_{c′}.

We can express v_{μ}^{♮} = v_{I}^{p} with

p = (p_{1}, …, p_{n}) = (\underbrace{r, …, r}_{n_{r}times}, \underbrace{r − 1, …, r − 1}_{n_{r−1}times}, …, \underbrace{1, …, 1}_{n_{1}times})

for some I. Since y ∈ 𝔖_{α′}, v_{λ}^{♮}T_{y} is a linear combination of basis vectors of the form v_{J}^{p} (p is unchanged) with various J. We simply write one of them as v_{J} and compute v_{J}T_{w}. It is easy to see that v_{J}T_{w} = v_{J}w = v′_{J}, where v′_{J} is obtained from v_{J} by permuting the factors according to the cyclic permutation 1 → 2 → ⋯ → b_{k} + 1 → 1. We apply T_{1} on v′_{J}. Since the first factor of v′_{J} is in V_{k}, we have v′_{J}ξ_{1} = Q_{k}v′_{J}. Then we apply S_{2} ⋯ S_{n} on it. The actions of S_{2}, … S_{b_{k}+1} are just given by permutations s_{2}, …, s_{b_{k}+1} since in each step, the corresponding factors are in different V_{j}. Hence we see that

(3.6.4)  v_{J}T_{w}ξ_{1}S_{2} ⋯ S_{b_{k}+1} = Q_{k}v_{J}.

In the next step, the successive actions of S_{j} for j = b_{k} + 2, …, b_{k+1} are given by T_{j} since the corresponding factors are in the same subspace V_{k}. Here note that Q_{k}v_{J}T_{b_{k}+2} ⋯ T_{b_{k+1}} is expressed as a linear combination of basis vectors such that its b_{k+1} − 1 factor belongs to V_{k}, and the factors in the latter part from there belong to V_{l} with l < k. This implies that the subsequent actions of S_{j} for j = b_{k+1}, …, n on these vectors coincide with s_{j}, and which also coincide with T_{j}. It follows from this, together with (3.6.4) that

v_{J}T_{w}ξ_{1}S_{2} ⋯ S_{n} = Q_{k}v_{J}T_{b_{k}+2} ⋯ T_{n}

and we have

v_{J}T_{w}T_{1} = v_{J}T_{w}ξ_{1}S_{2} ⋯ S_{n}T_{n}^{−1} ⋯ T_{2}^{−1} = Q_{k}v_{J}T_{b_{k}+1}^{−1} ⋯ T_{2}^{−1}.

This shows that our action of T_{1} on v_{λ}^{♮}T_{d} is also given by the right hand side of (3.6.2). Hence (3.6.1) holds, and the theorem is proved. □

Remark 3.7. The action of H̃ on V^{⊗n} given in 3.5 is not exactly the same as Hu’s original one. He uses the right action of H̃ on V^{⊗n}, induced from the left action by changing the generators T_{i} by T_{i}^{−1}, i.e., changing the parameters q, Q_{i} by q^{−1}, Q_{i}^{−1} . He also uses a different type of cellular basis of H̃, a modified one from the basis of [DJM], which does not coincide with ours even if the parameters are changed to their inverses.

We now return to the setting in Section 2. As a corollary to Theorem 3.6, combined with Theorem 3.4, we have the following result.

Corollary 3.8. Assume that (1.2.1) holds for H. We regard M^{μ} as an H̃ module via the map ρ_{0} : H̃ → H. Then for each μ ∈ P̃_{n,r}, there exists an exact sequence of H̃-modules

0 ⟶ M̂_{♮}^{μ} ⟶ M_{♮}^{μ} ⟶^{f} M^{μ} ⟶ 0,

where the map f is given by m_{μ}^{♮}h ↦ m_{μ}h for h ∈ H̃.

Proof. It is easy to see that V_{μ′}^{⊗n} = v_{μ′}^{♮}H ≃ Ind_{H_{μ}}^{H} θ_{q}^{μ} as H-modules. Hence by Lemma 2.5, M^{μ} is isomorphic to V_{μ′}^{⊗n}, via m_{μ}h → v_{μ′}^{♮}h (h ∈ H), and is isomorphic to M_{♮}^{μ}/M̂_{♮}^{μ} as H̃-modules by Theorem 3.6. This proves the corollary. □

4. Specht filtrations for M_{♮}^{μ} and M^{μ}

4.1. In [DJM], they constructed a semistandard basis for M_{♮}^{μ} by using their cellular basis, and defined a filtration whose successive quotients are Specht modules. In this section, we construct a similar basis of M^{μ} and its filtration by using our cellular basis, and compare those two filtrations.

First we recall some notation and results on semistandard bases in [DJM, 4]. For an r-composition λ of n, a λ-Tableau S = (S^{(1)}, …, S^{(r)}) is a map S : [λ] → {1, …, n} × {1, …, r}. As in the case of usual λ-tableau, we regard S as the diagram [λ], together with an ordered pair (i, k) (1 ≤ i ≤ n, 1 ≤ k ≤ r) attached to each node. Suppose that λ ∈ P_{n,r} and μ ∈ P̃_{n,r}. A λ-Tableau S is said to be of type μ if the number of (i, k) in the entry of S is equal to μ_{i}^{(k)}. Assume that 𝔰 is a standard λ-tableau. We define a λ-Tableau μ(𝔰) by replacing each entry m in 𝔰 by (i, k) if m is in the i-th row of the k-th component of 𝔱^{μ}. Clearly μ(𝔰) is a λ-Tableau of type μ.

We define a total order on the set of pairs (i, k) as above, by (i_{1}, k_{1}) < (i_{2}, k_{2}) if k_{1} < k_{2}, or k_{1} = k_{2} and i_{1} < i_{2}. Suppose that λ ∈ P_{n,r} and μ ∈ P̃_{n,r}. Let S = (S^{(1)}, …, S^{(r)}) be a λ-Tableau of type μ. S is said to be semistandard if each component S^{(k)} is semistandard with respect to the entries {(i, l)} (i.e., non-decreasing in rows, strictly increasing in columns), and each entry of S^{(k)} consist of (i, l) with l ≥ k. We denote by T_{0}(λ, μ) the set of semistandard λ-Tableaux of type μ.

Assume that S ∈ T_{0}(λ, μ). Then there exists a unique 𝔰_{1} = first(S) ∈ Std(λ) satisfying the property that μ(𝔰_{1}) = S and that 𝔰_{1} ⊵ 𝔰 for any 𝔰 ∈ Std(λ) such that μ(𝔰) = S.

By [DJM, 4.5], the following property of semistandard Tableaux is known. Recall that the order of r-tuples a ∈ ℤ_{≥0}^{r} is given as in 3.3.

(4.1.1) Assume that T_{0}(λ, μ) is non-empty. Then we have a(λ) ≥ a(μ).

4.2. In the special case where r = 1, the above definition coincides with the usual definition of the semistandard λ-Tableaux of type μ for λ ∈ P_{n,1}, μ ∈ P̃_{n,1}. Following [M1, Chap.4], we recall the basic properties of semistandard tableaux. Let λ, μ be 1-compositions of n. A Tableau S of type μ is said be row semistandard if the entries in each row of S are non-decreasing. Moreover, in the case where λ is a 1-partition, S is said to be semistandard if S is row semistandard and the entries in each column are strictly increasing. Assume that λ, μ ∈ P̃_{n,1} and put D_{λμ} = D_{λ} ∩ D_{μ}^{−1}. Then D_{λμ} is the set of minimal length elements in the double cosets 𝔖_{λ}\𝔖_{n}/𝔖_{μ}. The set of row semistandard λ-tableaux of type μ is in one to one correspondence with the set D_{λμ}, by associating λ-tableau μ(𝔱^{λ}d) to d ∈ D_{λμ}.

Let d ∈ D_{λμ} and put S = μ(𝔱^{λ}d), T = λ(𝔱^{μ}d^{−1}). Then S and T are both row semistandard, and we have ([M1, 4.6])

(4.2.1)  ∑_{\begin{matrix}y∈D_{μ} \\ λ(𝔱^{μ}y)=T \end{matrix}} q^{l(y)}T_{y}^{∗}m_{\{μ\}} = ∑_{w∈𝔖_{λ}d𝔖_{μ}} q^{l(w)}T_{w} = ∑_{\begin{matrix}x∈D_{λ} \\ μ(𝔱^{λ}x)=S \end{matrix}} q^{l(x)}m_{\{λ\}}T_{x}.

Note that if λ is a partition and S is semistandard, then the condition μ(𝔱^{λ}x) = S is equivalent to the condition that 𝔱^{λ}x is a standard tableau. Moreover, in this setting, d is given by first(S) = 𝔱^{λ}d.

4.3. Assume that λ ∈ P_{n,r} and μ ∈ P̃_{n,r}. We define a subset T_{0}^{+}(λ, μ) of T_{0}(λ, μ) by

T_{0}^{+}(λ, μ) = {S ∈ T_{0}(λ, μ) | a(λ) = a(μ)}.

Note that the condition a(λ) = a(μ) is also expressed as α(λ) = α(μ). Take S ∈ T_{0}(λ, μ). Then one can check that S ∈ T_{0}^{+}(λ, μ) if and only if each entry of S^{(k)} is of the form (i, k) for some i. Hence in this case S^{(k)} can be identified with a semistandard λ^{(k)}-tableau of type μ^{(k)} as in 4.2, and we have a bijection

(4.3.1)  T_{0}^{+}(λ, μ) ≃ T_{0}(λ^{(1)}, μ^{(1)}) × ⋯ × T_{0}(λ^{(r)}, μ^{(r)})

via S ↔ (S^{(1)}, …, S^{(r)}). Moreover, if 𝔰 ∈ Std(λ) is such that μ(𝔰) = S with S ∈ T_{0}^{+}(λ, μ), then the entries of i-th component of 𝔰 consist of numbers a_{i} + 1, …, a_{i+1} for a(λ) = (a_{1}, …, a_{r}). In particular, d(𝔰) ∈ 𝔖_{α} for α = α(λ).

Take S ∈ T_{0}^{+}(λ, μ) and let 𝔰_{1} = first(S). Let α = α(λ) = α(μ). Then d = d(𝔰_{1}) is given as d = (d_{1}, …, d_{r}) ∈ 𝔖_{α}, with d_{k} a distinguished double coset representative in 𝔖_{λ^{(k)}}\𝔖_{n_{k}}/𝔖_{μ^{(k)}}. Then by combining this with (4.2.1), together with the remark after that, we have

(4.3.2)  ∑_{\begin{matrix}𝔰∈Std(λ) \\ μ(𝔰)=S \end{matrix}}q^{l(d(𝔰))}m_{\{λ\}}T_{d(𝔰)} = ∑_{w∈𝔖_{λ}d𝔖_{μ}} q^{l(w)}T_{w} = hT_{d}m_{\{μ\}},

where h = ∑ T_{v}, the sum running over certain elements v ∈ 𝔖_{λ}.

Let us take S ∈ T_{0}(λ, μ) and 𝔱 ∈ Std(λ) for λ ∈ P_{n,r} and μ ∈ P̃_{n,r}. We define m_{S𝔱}^{♮} ∈ H̃ by

(4.3.3)  m_{S𝔱}^{♮} = ∑_{\begin{matrix}𝔰∈Std(λ) \\ μ(𝔰)=S \end{matrix}}q^{l(d(𝔰))+l(d(𝔱))}m_{𝔰𝔱}^{♮}.

Then we have

Theorem 4.4 ([DJM, Theorem 4.14]). M_{♮}^{μ} is a free R-module with basis

{m_{S𝔱}^{♮} | S ∈ T_{0}(λ, μ), 𝔱 ∈ Std(λ) for λ ∈ P_{n,r}}.

As a corollary, we have

Corollary 4.5. (i) M̂_{♮}^{μ} is a free R-module with basis

{m_{S𝔱}^{♮} | S ∈ T_{0}(λ, μ), 𝔱 ∈ Std(λ), a(λ) > a(μ)}.

(ii) Let f : M_{♮}^{μ} → M^{μ} be the map given in Corollary 3.8. Then the set

{f(m_{S𝔱}^{♮}) | S ∈ T_{0}^{+}(λ, μ), 𝔱 ∈ Std(λ) for λ ∈ P_{n,r}}

gives a basis of M^{μ}.

Proof. (i) is clear from the definition. It is known by [DJM, Corollary 3.24] that for μ ∈ P̃_{n,r}, any element of M_{♮}^{μ} can be written as a linear combination of m_{𝔰𝔱}^{♮}, where 𝔰, 𝔱 ∈ Std(λ) such that λ ⊵ μ. Hence by (3.3.4), we have a(λ) ≥ a(μ) for such m_{𝔰𝔱}^{♮}. Now (ii) follows from this and (i). □

4.6. We now construct a basis of M^{μ} related to the basis m_{𝔰𝔱} as an analogy of m_{S𝔱}^{♮} in M_{♮}^{μ}. Let us define, for S ∈ T_{0}^{+}(λ, μ) and 𝔱 ∈ Std(λ),

(4.6.1)  m_{S𝔱} = ∑_{\begin{matrix}𝔰∈Std(λ) \\ μ(𝔰)=S \end{matrix}}q^{l(d(𝔰))+l(d(𝔱))}m_{𝔰𝔱}.

We have

Lemma 4.7. Let S ∈ T_{0}^{+}(λ, μ) and 𝔱 ∈ Std(λ). Then m_{S𝔱} ∈ M^{μ}.

Proof. The proof is similar to Lemma 4.10 in [DJM] and simpler than it. By (4.3.2) we have

m_{S𝔱} = ∑_{\begin{matrix}𝔰∈Std(λ) \\ μ(𝔰)=S \end{matrix}}q^{l(d(𝔰))+l(d(𝔱))}T_{d(𝔰)}^{∗}m_{\{λ\}}F_{α}T_{d(𝔱)} = q^{l(d(𝔱))}m_{\{μ\}}T_{d}^{∗}h^{∗}F_{α}T_{d(𝔱)},

where α = α(λ). Since h ∈ H(𝔖_{λ}) ⊂ H_{λ}, we have h^{∗}F_{α} = F_{α}h^{∗}. Furthermore, since S ∈ T_{0}^{+}(λ, μ), 𝔰_{1} satisfies the property that the entries of kth component of 𝔰_{1} consist of the same numbers as the entries of kth component of 𝔱^{λ} for each k. It follows that d ∈ 𝔖_{α}, and F_{α} commutes with T_{d} ∈ H(𝔖_{α}) by Lemma 2.9. If we note that α = α(μ), we see that

m_{{μ}}T_{d}^{∗}h^{∗}F_{α} = m_{{μ}}F_{α}T_{d}^{∗}h^{∗} ∈ m_{μ}H = M^{μ},

and m_{S𝔱} ∈ M^{μ} as asserted. □

Proposition 4.8. The set {m_{S𝔱} | S ∈ T_{0}^{+}(λ, μ), 𝔱 ∈ Std(λ) for λ ∈ P_{n,r}} gives rise to an R-basis of M^{μ}.

Proof. The basis elements m_{𝔰𝔱} contained in the expression of m_{S𝔱} are disjoint for different m_{S𝔱}. It follows that m_{S𝔱} are linearly independent. By Corollary 4.5 (ii), M^{μ} is a free R-module, and its rank is equal to the number of {m_{S𝔱}} given in the theorem. Hence, any element in M^{μ} can be written as a linear combination of various m_{S𝔱} with coefficients in the quotient field of R. But since the set {q^{l(d(𝔰))+l(d(𝔱))}m_{𝔰𝔱}} is an R-basis of H, these coefficients are actually in R. This proves the proposition. □

We can now state

Theorem 4.9. Let μ ∈ P̃_{n,r}, and f : M_{♮}^{μ} → M^{μ} be as in Corollary 3.8. Let λ ∈ P_{n,r} and consider m_{S𝔱} for S ∈ T_{0}(λ, μ), 𝔱 ∈ Std(λ). Then we have

f(m_{S𝔱}^{♮}) = \begin{cases}m_{S𝔱} & if S ∈ T_{0}^{+}(λ, μ), \\ 0 & otherwise. \end{cases}

Proof. The second equality is clear from Corollary 4.5 (i). We show the first one. We know that {m_{μ}T_{d(𝔱)} | 𝔱 ∈ r-Std(μ)} is a basis of M^{μ} by Lemma 2.5 and 2.7. Since m_{μ}T_{d(𝔱)} = m_{𝔱^{μ}𝔱}, m_{μ}T_{d(𝔱)} can be written as a linear combination of various m_{𝔲𝔳} with 𝔲, 𝔳 ∈ Std(λ) for λ ⊵ μ, by the procedure described in Lemma 2.15. This procedure also determines to write down m_{μ}T_{d(𝔱)} as a linear combination of the basis m_{S𝔱′} of M^{μ}. On the other hand, by the definition of f, the element m_{μ}^{♮}T_{d(𝔱)} is mapped by f to m_{μ}T_{d(𝔱)}. Since m_{μ}^{♮}T_{d(𝔱)} = m_{𝔱^{μ}𝔱}^{♮}, m_{μ}^{♮}T_{d(𝔱)} can be written as a linear combination of m_{𝔲𝔳}^{♮} with 𝔲, 𝔳 ∈ Std(λ) for λ ⊵ μ, by the procedure given in [DJM, Prop. 3.18]. However, as is seen in the proof of Lemma 2.15, these two procedures are completely parallel. Hence the coefficient of m_{𝔲𝔳} in m_{μ}T_{d(𝔱)} is the same as that of m_{𝔲𝔳}^{♮} in m_{μ}^{♮}T_{d(𝔱)}. The same is true for the coefficients of m_{S𝔱′} and of m_{S𝔱′}^{♮}. It follows that the basis {f(m_{S𝔱}^{♮})} and the basis {m_{S𝔱}} have the same transition matrix for the basis {m_{μ}T_{d(𝔱)}} of M^{μ}. This proves that f(m_{S𝔱}^{♮}) = m_{S𝔱} as asserted. □

4.10. The Specht module S_{♮}^{λ} for λ ∈ P_{n,r} is defined in [DJM] by using the cellular basis C^{♮} of H̃, which is the submodule of H̃/Ň_{♮}^{λ} generated by the image \overline{m}_{λ}^{♮} of m_{λ}^{♮} under the map H̃ → H̃/Ň_{♮}^{λ}, where Ň_{♮}^{λ} is the R-submodule of H̃ spanned by m_{𝔰𝔱}^{♮} with 𝔰, 𝔱 ∈ Std(ν) such that ν ▷ λ. Hence S_{♮}^{λ} is a free R-module with basis \{\overline{m}_{λ}^{♮}T_{d(𝔱)} | 𝔱 ∈ Std(λ)\}.

Let μ ∈ P̃_{n,r}. We define a total order S_{1}, S_{2}, …, S_{l} on the set of semistandard Tableaux of type μ satisfying the following property; Assume that S_{i} ∈ T_{0}(λ_{i}, μ) and S_{j} ∈ T_{0}(λ_{j}, μ). Then i > j if a(λ_{i}) > a(λ_{j}) or λ_{i} ▷ λ_{j}. Hence the first k terms of S_{i} are contained in T_{0}^{+}(λ, μ) for some λ. Let us define M_{i}^{♮} as an R-submodule of M_{♮}^{μ} with basis {m_{S_{j}𝔱}^{♮} | j ≥ i, 𝔱 ∈ Std(λ_{j})}. Similarly, we define M_{i} (1 ≤ i ≤ k) as an R-submodule of M^{μ} with basis {m_{S_{j}𝔱} | j ≥ i, 𝔱 ∈ Std(λ_{j})}. Then by the property of cellular bases, M_{i}^{♮} is a right ideal of H̃, and M_{i} is a right ideal of H. Thus, we have filtrations

M_{♮}^{μ} = M_{1}^{♮} ⊃ M_{2}^{♮} ⊃ ⋯ ⊃ M_{k}^{♮} ⊃ ⋯ ⊃ M_{l}^{♮} ⊃ M_{l+1}^{♮} = 0,
M^{μ} = M_{1} ⊃ M_{2} ⊃ ⋯ ⊃ M_{k} ⊃ M_{k+1} = 0,

which are called the Specht filtrations for M_{♮}^{μ}, M^{μ}, respectively.

As a corollary to Theorem 4.9, we have the following.

Corollary 4.11. Let f : M_{♮}^{μ} → M^{μ} be as in Corollary 3.8. Then

(i) The map f is compatible with the above filtrations, i.e., f(M_{i}^{♮}) = M_{i} for 1 ≤ i ≤ k, and f(M_{i}^{♮}) = 0 otherwise.

(ii) M_{i}^{♮}/M_{i+1}^{♮} is isomorphic to S_{♮}^{λ_{i}} as H̃-modules for 1 ≤ i ≤ l.

(iii) M_{i}/M_{i+1} is isomorphic to S^{λ_{i}} as H-modules for 1 ≤ i ≤ k.

(iv) S_{♮}^{λ_{i}} is isomorphic to S^{λ_{i}} as H̃-modules for 1 ≤ i ≤ k.

Proof. (i) is immediate from Theorem 4.9. (ii) is nothing but Corollary 4.15 in [DJM]. A similar argument as in [DJM] works also for (iii). Now (iv) follows from (i) ∼ (iii) since f^{−1}(M_{i}) = M_{i}^{♮} for 1 ≤ i ≤ k, and f is H̃-equivariant. □

4.12. For each λ ∈ P_{n,r}, the quotient module D_{♮}^{λ} of Sλ_{♮} is defined as in 2.19. When R is a field, the set {D_{♮}^{λ} ≠ 0 | λ ∈ P_{n,r}} gives a complete set of non-isomorphic irreducible H̃-modules. We denote by d_{λμ}^{♮} = [S_{♮}^{λ} : D_{♮}^{μ}] the composition multiplicity of D_{♮}^{μ} in S_{♮}^{λ}. Similarly we define d_{λμ} = [S^{λ} : D^{μ}] for H-modules S^{λ} and D^{μ}. By a general theory of cellular algebras, it is known, in the case where R is a field, that d_{μμ} = 1, and d_{λμ} = 0 unless λ ⊵ μ, where λ, μ ∈ P_{n,r} such that D^{μ} ≠ 0, and similar facts hold also for H̃. Note that the H-modules S^{λ}, D^{μ} can be regarded as H̃-modules via the homomorphism ρ_{0} : H̃ → H.

Next result gives an estimate of the decomposition numbers for the Ariki-Koike algebra H̃ in terms of the decomposition numbers for the modified Ariki-Koike algebra H.

Theorem 4.13. Assume that R is a field.

(i) For H̃-modules D^{μ} and D_{♮}^{ν}, we have [D^{μ} : D_{♮}^{ν}] = 0 unless ν ⊵ μ.

(ii) If D_{♮}^{μ} ≠ 0, then D^{μ} ≠ 0. Moreover in this case, we have [D^{μ} : D_{♮}^{μ}] = 1.

(iii) For any λ, μ ∈ P_{n,r} such that D_{♮}^{μ} ≠ 0, we have [S^{λ} : D^{μ}] ≤ [S_{♮}^{λ} : D_{♮}^{μ}].

(iv) Assume that H ≃ H̃. Then D^{μ} ≃ D_{♮}^{μ} for any μ, and the parametrization of irreducible modules in terms of the cellular basis C and of C^{♮} coincides with each other.

Proof. For a Specht module S^{λ}, we consider a composition series whose successive quotients are of the form D^{μ}. This gives a filtration of S^{λ} as H̃-modules, and by an isomorphism S^{λ} ≃ S_{♮}^{λ} it gives a filtration of S_{♮}^{λ} whose successive quotients are isomorphic to D^{μ} as H̃-modules. Now (i) follows from this by induction on the dominance order, by making use of the property of decomposition numbers d_{λμ} and d_{λμ}^{♮}.

We show (ii). Assume that D_{♮}^{μ} ≠ 0. If D^{μ} = 0, the Specht module S^{μ} has a composition series whose quotients consist of D^{ν} such that ν ▷ μ. Then by (i), S_{♮}^{μ} does not contain D_{♮}^{μ}, which contradicts d_{μμ}^{♮} = 1. Hence we have D^{μ} ≠ 0. The equality [D^{μ} : D_{♮}^{μ}] = 1 follows from (i) noticing that d_{μμ} = 1, d_{μμ}^{♮} = 1.

Now assume that D_{♮}^{μ} ≠ 0. Then by (ii), D^{μ} ≠ 0, and the H̃-module D^{μ} contains the H̃-module D_{♮}^{μ} with multiplicity one, as a composition factor. This proves (iii).

Finally (iv) follows easily from (ii). □

5. Cyclotomic q-Schur algebras

5.1. Let μ, ν ∈ P̃_{n,r} be such that α(μ) = α(ν) = α. Put α = (n_{1}, …, n_{r}). Let us take d ∈ D_{μν} ∩ 𝔖_{α}. We have d = (d_{1}, …, d_{r}) with d_{k} ∈ D_{μ^{(k)}ν^{(k)}} with respect to 𝔖_{n_{k}}. Then, by a similar argument as in the proof of Lemma 4.7, one can define a map φ_{μν}^{d} : M^{ν} → M^{μ} by

φ_{μν}^{d}(m_{ν}h) = ∑_{w∈𝔖_{μ}d𝔖_{ν}} q^{l(w)}F_{α}T_{w}h

for all h ∈ H. In fact, by (4.2.1), (see also (4.3.2)), we have

(5.1.1)  ∑_{\begin{matrix}y∈D_{ν}∩𝔖_{α} \\ μ(𝔱^{ν}y)=T \end{matrix}} q^{l(y)}F_{α}T_{y}^{∗}m_{\{ν\}} = ∑_{w∈𝔖_{μ}d𝔖_{ν}} q^{l(w)}F_{α}T_{w} = ∑_{\begin{matrix}x∈D_{μ}∩𝔖_{α} \\ ν(𝔱^{μ}x)=S \end{matrix}} q^{l(x)}m_{μ}T_{x},

where S = μ(𝔱^{ν}d), T = ν(𝔱^{μ}d^{−1}) are row semistandard Tableaux. Since y ∈ 𝔖_{α}, we have F_{α}T_{y}^{∗}m_{{ν}} = T_{y}^{∗}F_{α}m_{{ν}} = T_{y}^{∗}m_{ν}, and φ_{μν}^{d} is well-defined.

As an analogy to the case of H_{n} ([M1, Theorem 4.7]), we have the following result.

Proposition 5.2. Let μ, ν ∈ P̃_{n,r}. Then (i) Assume that α(μ) ≠ α(ν). Then Hom_{H}(M^{ν}, M^{μ}) = 0. (ii) Assume that α(μ) = α(ν). Then Hom_{H}(M^{ν}, M^{μ}) is a free R-module with basis

{φ_{μν}^{d} | d ∈ D_{μν} ∩ 𝔖_{α}}.

Proof. Suppose that φ ∈ Hom_{H}(M^{ν}, M^{μ}). Then for h ∈ H, we have φ(m_{ν}h) = φ(m_{ν})h, and φ is determined completely by φ(m_{ν}). Since φ(m_{ν}) ∈ M^{μ} one can write, by Lemma 2.5, that φ(m_{ν}) = ∑_{x∈D_{μ}} c_{x}m_{μ}T_{x}, with c_{x} ∈ R. Then by Lemma 2.6 (iii), we have

(5.2.1)  φ(m_{ν}ξ_{k}) = φ(Q_{ε(k)}m_{ν}) = ∑_{x∈D_{μ}} Q_{ε(k)}c_{x}m_{μ}T_{x}

for each k, where ε(k) is determined by the condition v_{α} = v_{j_{1}}^{(ε(1))} ⊗ ⋯ ⊗ v_{j_{n}}^{(ε(n))}, with α = α(ν). Now assume that c_{y} ≠ 0 for some y ∈ D_{μ}. Then we have

(5.2.2)  (m_{μ}T_{y})ξ_{k} = Q_{ε′(k)}m_{μ}T_{y},

where ε′(k) is determined by the condition that v_{α′}T_{y} = v_{j_{1}}^{(ε′(1))} ⊗ ⋯ ⊗ v_{j_{n}}^{(ε′(n))}, with α′ = α(μ). By comparing (5.2.1) and (5.2.2), we have ε(k) = ε′(k) for k = 1, …, r. This implies that α = α′ since T_{y} preserves the weight spaces. So (i) is proved.

Now assume that α(μ) = α(ν). Since T_{y} acts as a permutation of factors on v_{α}, the condition ε(k) = ε′(k) implies that y ∈ 𝔖_{α}. Let d be the unique minimal length element in 𝔖_{μ}y𝔖_{ν}. Then d ∈ D_{μν} ∩𝔖_{α}, and a similar argument as in the proof of [M1, Theorem 4.7] implies that c_{d} ≠ 0. Put ψ = φ−c_{d}φ_{μν}^{d}. Then ψ ∈ Hom_{H}(M^{ν}, M^{μ}), and ψ(m_{ν}) can be written as ψ(m_{ν}) = ∑_{x∈D_{μ}} a_{x}m_{μ}T_{x}, where a_{x} = c_{x} if 𝔖_{μ}x𝔖_{ν} ≠ 𝔖_{μ}d𝔖_{ν}, and a_{x} = 0 for x ∈ 𝔖_{μ}d𝔖_{ν} by the argument there. Thus by induction, φ can be written as a linear combination of φ_{μν}^{d}.

We have only to show that {φ_{μν}^{d} | d ∈ D_{μν} ∩ 𝔖_{α}} is linearly independent. But {φ_{μν}^{d}(m_{ν})} is a linearly independent subset of M^{μ} since a set {F_{α}T_{w}} is linearly independent by (1.2.7). □

We write M^{ν∗} = (M^{ν})^{∗} = m_{ν}H. As a corollary to Proposition 5.2, we have the following.

Corollary 5.3. Let μ, ν ∈ P̃_{n,r}.

(i) Hom_{H}(M^{ν}, M^{μ}) and M^{ν∗} ∩ M^{μ} are canonically isomorphic as R-modules. The isomorphism Hom_{H}(M^{ν}, M^{μ}) → M^{ν∗} ∩ M^{μ} is given by φ ↦ φ(m_{ν}).

(ii) Assume that α = α(μ) = α(ν). Then we have

M^{ν∗} ∩ M^{μ} = F_{α}(H_{α}m_{{ν}} ∩ m_{{μ}}H_{α}),

where m_{{ν}} and m_{{μ}} are regarded as elements in H_{α} = H(𝔖_{α}) in a natural way.

Proof. The assertion (i) is proved in a similar way as [M1, Corollary 4.8], thanks to Proposition 5.2. We show (ii). By (i) together with Proposition 5.2, we see that

{F_{α} ∑_{w∈𝔖_{μ}d𝔖_{ν}} q^{l(w)}T_{w} | d ∈ D_{μν} ∩ 𝔖_{α}}

forms an R-basis of M^{ν∗} ∩ M^{μ}. But it is already known from the results in the case of H_{α} (cf [M1, 4.6, 4.8]) that {∑_{w∈𝔖_{μ}d𝔖_{ν}} q^{l(w)}T_{w} | d ∈ D_{μν} ∩ 𝔖_{α}} forms a basis of H_{α}m_{{ν}} ∩ m_{{μ}}H_{α}. Thus the assertion (ii) holds. □

5.4. We define a cyclotomic q-Schur algebra S(m, n) by S(m, n) = End_{H} V^{⊗n}. By (3.3.1), we have V^{⊗n} = ⨁_{μ∈P̃_{n,r}} V_{μ}^{⊗n} ≃ ⨁_{μ∈P̃_{n,r}} M^{μ′}. Thus one can write

S(m, n) = End_{H}( ⨁_{μ∈P̃_{n,r}} M^{μ}) ≃ ⨁_{μ,ν∈P̃_{n,r}} Hom_{H}(M^{ν}, M^{μ}).

For an r-tuples α ∈ P̃_{n,1}, let M^{α} = ⨁_{μ; α(μ)=α} M^{μ}. Then by Proposition 5.2 (i), we have

(5.4.1)  S(m, n) ≃ ⨁_{α∈P̃_{n,1}} End_{H} M^{α}

as R-algebras.

Recall that U_{q} is the quantum algebra over R. Since U_{q} acts on V^{⊗n} commuting with H, we have an algebra homomorphism τ : U_{q} → End_{H} V^{⊗n} = S(m, n). The following result is in analogy to the result of Ariki [A, Theorem 3.2], where he proved a similar fact for the cyclotomic q-Schur algebra EndH̃ V^{⊗n} (associated to the Ariki-Koike algebra H̃) under the separation condition (5.4.2) q^{2k}Q_{i} − Q_{j} ∈ R are inverstible in R for |k| < n, i ≠ j,

but without any restriction on m = (m_{1}, …, m_{r}). Note that the condition (5.4.2) is stronger than our condition (1.2.1).

Theorem 5.5. Let S(m, n) be the cyclotomic q-Schur algebra associated to the modified Ariki-Koike algebra H. Then we have

(i) Let S(m_{i}, n_{i}) be the q-Schur algebra assocaited to the Iwahori-Hecke algebra H_{n_{i}} as in [M1, Chap. 4]. Then there exists an isomorphism of R-algebras

(5.5.1)  S(m, n) ≃ ⨁_{\begin{matrix}(n_{1},...,n_{r}) \\ n=n_{1}+···+n_{r} \end{matrix}} S(m_{1}, n_{1}) ⊗ ⋯ ⊗ S(m_{r}, n_{r}).

(ii) The map τ : U_{q} → S(m, n) is surjective.

Proof. First we show (i). For each μ ∈ P̃_{n,r} we denote by \overline{M}^{μ} the right ideal m_{\{μ\}}H_{α} of H_{α}. For α ∈ P̃_{n,1}, let \overline{M}^{α} = ⨁_{μ; α(μ)=α} \overline{M}^{μ}. It is easy to see that

End_{H_{α}} \overline{M}^{α} ≃ S(m_{1}, n_{1}) ⊗ ⋯ ⊗ S(m_{r}, n_{r})

as R-algebras, with α = (n_{1}, …, n_{r}). Thus, in view of (5.4.1), it is enough to show that

(5.5.2)  End_{H} M^{α} ≃ End_{H_{α}} \overline{M}^{α}

as R-algebras. We show (5.5.2). By Corollary 5.3 (ii), we have an isomorphism of R-modules \overline{M}^{ν∗} ∩ \overline{M}^{μ} → M^{ν∗} ∩ M^{μ} via h ↦ F_{α}h. By applying Corollary 5.3 (i) for both sides, one can define an R-linear map g : End_{H_{α}} \overline{M}^{α} → End_{H} M^{α}, which is clearly bijective. It remains to show that the map g is a homomorphism of R-algebras.

Let τ, ν, μ ∈ P̃_{n,r} be such that α(τ) = α(ν) = α(μ). Take x ∈ M^{τ∗} ∩ M^{ν}, y ∈ M^{ν∗} ∩ M^{μ}, and let φ_{x} ∈ Hom_{H}(M^{τ}, M^{ν}), φ_{y} ∈ Hom_{H}(M^{ν}, M^{μ}) be the elements corresponding to x, y under the isomorphism in Corollary 5.3 (i), respectively. By using Corollary 5.3 (ii), one can write x = m_{ν}h = F_{α}m_{{ν}}h for some h ∈ H_{α}. By definition, we have φ_{x}(m_{τ}) = x, φ_{y}(m_{ν}) = y. Hence φ_{y}φ_{x}(m_{τ}) = φ_{y}(m_{ν}h) = φ_{y}(m_{ν})h = yh. Thus we see that yh ∈ M^{τ∗} ∩ M^{μ} corresponds to φ_{y}φ_{x} ∈ Hom_{H}(M^{τ}, M^{μ}). If we write x = F_{α}x′, y = F_{α}y′ with x′ ∈ \overline{M}^{τ∗} ∩ \overline{M}^{ν}, y′ ∈ \overline{M}^{ν∗} ∩ \overline{M}^{μ}, a completely similar rule for the composition holds between φ_{x′} ∈ Hom_{H_{α}}(\overline{M}^{τ}, \overline{M}^{ν}) and φ_{y′} ∈ Hom_{H_{α}}(\overline{M}^{ν}, \overline{M}^{μ}), i.e., y′h corresponds to φ_{y′}φ_{x′} ∈ Hom_{H_{α}}(\overline{M}^{τ}, \overline{M}^{μ}). It follows that g : End_{H_{α}} \overline{M}^{α} → End_{H} M^{α} is a homomorphism. Hence (5.5.2) holds, and (i) follows.

Next we show (ii). Take α = (n_{1}, …, n_{r}) ∈ P̃_{n,1} and put α′ = (n_{r}, …, n_{1}). Put V_{α}^{⊗n} = ⨁_{μ;α(μ)=α} V_{μ}^{⊗n}. Then V^{⊗n} = ⨁_{α} V_{α}^{⊗n}, and V_{α}^{⊗n} is isomorphic to M^{α′} as H-modules. Moreover, V_{α}^{⊗n} can be expressed as

V_{α}^{⊗n} = ⨁_{w∈𝔖_{α′}\𝔖_{n}}(V_{r}^{⊗n_{r}} ⊗ V_{r−1}^{⊗n_{r−1}} ⊗ ⋯ ⊗ V_{1}^{⊗n_{1}})w.

Here V_{r}^{⊗n_{r}} ⊗ ⋯ ⊗ V_{1}^{⊗n_{1}} is isomorphic to \overline{M}^{α′} as H_{α′}-modules. We note that, if we choose w ∈ D_{α′},

(V_{r}^{⊗n_{r}} ⊗ ⋯ ⊗ V_{1}^{⊗n_{1}})w = (V_{r}^{⊗n_{r}} ⊗ ⋯ ⊗ V_{1}^{⊗n_{1}})T_{w}.

Hence the map φ_{r} ⊗ ⋯ ⊗ φ_{1} ∈ End_{H_{α′}}(V_{r}^{⊗n_{r}} ⊗ ⋯ ⊗ V_{1}^{⊗n_{1}}) with φ_{i} ∈ End_{H_{n_{i}}} V_{i}^{⊗n_{i}} determines the map φ ∈ End V_{α}^{⊗n}. It follows from (5.5.2) that the correspondence φ_{r}⊗⋯⊗φ_{1} ↦ φ gives an R-isomorphism between (End_{H_{nr}} V_{r}^{⊗n_{r}})⊗⋯⊗(End_{H_{n_{1}}} V_{1}^{⊗n_{1}}) and End_{H} V_{α}^{⊗n}. On the other hand, it is known by J. Du [D, Theorem 3.4, Corollary 3.5] that the natural map

U_{q,R}(𝔤𝔩_{m}) → End_{H_{n}} V^{⊗n}

(i.e., the map τ in the case where r = 1) is surjective. It follows that the map

τ_{1} : U_{q}(𝔤𝔩_{m_{r}}) ⊗ ⋯ ⊗ U_{q}(𝔤𝔩_{m_{1}}) → (End_{H_{nr}} V_{r}^{⊗n_{r}}) ⊗ ⋯ ⊗ (End_{H_{n_{1}}} V_{1}^{⊗n_{1}})

is surjective. But the map τ_{1} is compatible with τ : U_{q} → End_{H} V^{⊗n} under the identifications

(End_{H_{nr}} V_{r}^{⊗n_{r}}) ⊗ ⋯ ⊗ (End_{H_{n_{1}}} V_{1}^{⊗n_{1}}) ≃ End_{H} V_{α}^{⊗n} ↪ End_{H} V^{⊗n}.

and U_{q} ≃ U_{q,R}(𝔤𝔩_{m_{r}}) ⊗ ⋯ ⊗ U_{q,R}(𝔤𝔩_{m_{1}}). Since End_{H} V^{⊗n} ≃ ⨁_{α} End_{H} V_{α}^{⊗n} by (5.4.1), this proves the surjectivity of τ. □

5.6 Let μ, ν ∈ P̃_{n,r} and λ ∈ P_{n,r}. We assume that α(μ) = α(ν) = α(λ). For S ∈ T_{0}^{+}(λ, μ) and T ∈ T_{0}^{+}(λ, ν), put

(5.6.1)  m_{ST} = ∑_{𝔰,𝔱}m_{𝔰𝔱},

where the sum is taken over all 𝔰, 𝔱 ∈ Std(λ) such that μ(𝔰) = S and ν(𝔱) = T. Since

m_{ST} = ∑_{\begin{matrix}𝔰∈Std(λ) \\ μ(𝔰)=S \end{matrix}}(m_{T 𝔰})^{∗} = ∑_{\begin{matrix}𝔱∈Std(λ) \\ ν(𝔱)=T \end{matrix}}m_{S𝔱},

we see that m_{ST} ∈ M^{μ} ∩ M^{ν∗} by Lemma 4.7. The following result can be proved in a similar way as in [DJM, Prop. 6.3], [M1, Cor. 4.11], by using Proposition 4.8.

Proposition 5.7. Let μ, ν ∈ P̃_{n,r} be such that α(μ) = α(ν). Then the set

{m_{ST} | S ∈ T_{0}^{+}(λ, μ), T ∈ T_{0}^{+}(λ, ν) for some λ ∈ P_{n,r}}

is a basis of M^{ν∗} ∩ M^{μ}.

5.8. Let S ∈ T_{0}^{+}(λ, μ) and T ∈ T_{0}^{+}(λ, ν). In view of Proposition 5.7, one can define φ_{ST} ∈ Hom_{H}(M^{ν}, M^{μ}) by

φ_{ST} (m_{ν}h) = m_{ST} h.

We extend φ_{ST} to an element of S(m, n) by defining φ_{ST} is zero on M^{κ} for ν ≠ κ ∈ P̃_{_{n},r}. For any partition λ ∈ P_{n,r}, let T_{0}^{+}(λ) = ⋃_{μ∈P̃_{n,r}} T_{0}^{+}(λ, μ). We denote by Š(m, n)^{λ} the R-submodule of S(m, n) spanned by φ_{ST} such that S, T ∈ T_{0}^{+}(α) with α ▷ λ. Similarly, we define S(m, n)^{λ} as the R-submodule spanned by φ_{ST} with α ⊵ λ. We have the following theorem.

Theorem 5.9. The cyclotomic q-Schur algebra S(m, n) has a free basis

C(m, n) = {φ_{ST} | λ ∈ P_{n,r}, S, T ∈ T_{0}^{+}(λ)}.

satisfying the following properties.

(i) The R-linear map ∗ : S(m, n) → S(m, n) determined by φ_{ST}^{∗} = φ_{T S}, for all S, T ∈ T_{0}^{+}(λ) and all λ ∈ P_{n,r}, is an anti-automorphism of S(m, n).

(ii) Let T ∈ T_{0}^{+}(λ). Then for all φ ∈ S(m, n), and every V ∈ T_{0}^{+}(λ), there exists r_{V} = r_{V,T,φ} ∈ R such that for any S ∈ T_{0}^{+}(λ),

(5.9.1)  φ_{ST} φ ≡ ∑_{V ∈T_{0}^{+}(λ)} r_{V} φ_{SV} mod Š(m, n)^{λ}.

In particular, C(m, n) is a cellular basis of S(m, n).

Proof. The proof is similar to [M1, Theorem 4.13] and [DJM, Theorem 6.6]. By Proposition 5.7 and Corollary 5.3, C(m, n) gives an R-basis of S(m, n). Take S ∈ T_{0}^{+}(λ, μ), T ∈ T_{0}^{+}(λ, ν). Then φ_{ST}^{∗} (m_{μ}) = m_{T S} = m_{ST}^{∗} = φ_{ST} (m_{ν})^{∗}, and by the Rlinearity, we have φ^{∗}(m_{μ}) = φ(m_{ν})^{∗} for any φ ∈ Hom_{H}(M^{ν}, M^{μ}). Then the assertion (i) follows from the standard functorial argument (see the proof of [M1, Theorem 4.13]).

Next we show (ii). Take S ∈ T_{0}^{+}(λ, μ), T ∈ T_{0}^{+}(λ, ν). We may assume that φ ∈ Hom_{H}(M^{τ}, M^{ν}) for some τ ∈ P̃_{n,r} such that α(λ) = α(τ). We have φ(m_{τ}) = m_{ν}h for some h ∈ H. Then φ_{ST} φ(m_{τ}) = m_{ST} h. By Corollary 5.3, we see that m_{ST} h ∈ M^{τ∗} ∩ M^{μ}. Hence by Proposition 5.7, one can write m_{ST} h = ∑_{U,V} r_{UV} m_{UV} with r_{UV} ∈ R, where U, V are elements in T_{0}^{+}(α) for some α ∈ P_{n,r}. But in view of Theorem 2.19 (ii), m_{ST} h can be rewritten as

m_{ST} h = ∑_{V ∈T_{0}^{+}(λ,τ)} r_{V} m_{SV} + ∑_{\begin{matrix}α∈P_{n,r} \\ α▷λ \end{matrix}} ∑_{\begin{matrix}U′∈T_{0}^{+}(α,μ) \\ V′∈T_{0}^{+}(α,τ) \end{matrix}} r_{U′V′}m_{U′V′}

with r_{V} , r_{U′V′} ∈ R. This implies the formula (5.9.1) evaluated at m_{τ}, and so (5.9.1) follows. □

5.10. For an r-partition λ ∈ P_{n,r}, let T^{λ} = λ(𝔱^{λ}). Then T^{λ} ∈ T_{0}^{+}(λ, λ), and in fact T^{λ} is the unique semistandard λ-Tableau of type λ. Moreover 𝔱 = 𝔱^{λ} is the unique element in Std(λ) such that λ(𝔱) = T^{λ}. Thus, m_{T^{λ}T^{λ}} = m_{𝔱^{λ}𝔱^{λ}} = m_{λ}, and φ_{λ} = φ_{T^{λ}T^{λ}} is the identity map on M^{λ}. We define the Weyl module W^{λ} as the right S(m, n)- submodule of S(m, n)/Š(m, n)^{λ} spanned by the image of φ_{λ}. For each S ∈ T_{0}^{+}(λ, μ), we denote by φ_{T} the image of φ_{T^{λ}T} in S(m, n)/Š(m, n)^{λ}. Then by Theorem 5.9, we see that

(5.10.1) The Weyl module W^{λ} is R-free with basis {φ_{T} | T ∈ T_{0}^{+}(λ)}.

The Weyl module W^{λ} posses an associative symmetric bilinear form, defined by the equation

φ_{T^{λ}S}φ_{T T^{λ}} ≡ ⟨φ_{S}, φ_{T} ⟩ φ_{λ} mod Š(m, n)^{λ}.

for all S, T ∈ T_{0}^{+}(λ). Note that ⟨φ_{S}, φ_{T} ⟩ = 0 unless S and T are semistandard Tableau of the same type. Let L^{λ} = W^{λ}/ rad W^{λ}, where rad W^{λ} = {x ∈ W^{λ} | ⟨x, y⟩ = 0 for all y ∈ W^{λ}}.

As in the case of q-Schur algebras ([M1, Theorem 4.15]), we have

Proposition 5.11. Suppose that R is a field.

(i) For an r-partition λ ∈ P_{n,r}, L^{λ} is an absolutely irreducible S(m, n)-module. (ii) {L^{λ} | λ ∈ P_{n,r}} is a complete set of non-isomorphic irreducible S(m, n)-

modules.

Proof. We have

φ_{T^{λ}T^{λ}}φ_{T^{λ}T^{λ}} ≡ ⟨φ_{T^{λ}}, φ_{T^{λ}}⟩ φ_{λ} mod Š(m, n)^{λ}.

But since φ_{T^{λ}T^{λ}}φ_{T^{λ}T^{λ}} = φ_{λ} is the identity map on M^{λ}, we see that ⟨φ_{T^{λ}}, φ_{T^{λ}}⟩ = 1, and so L^{λ} is non-zero. Then the assertions (i) and (ii) follow from the general properties of cellular algebras [GL]. □

5.12. We give here a more explicit description of the Weyl module W^{λ}. We consider Hom_{H}(⨁_{μ} M^{μ}, S^{λ}) = Hom_{H}(V^{⊗n}, S^{λ}), which has a structure of right S(m, n)- module by the composition of maps. For each 𝔱 ∈ Std(λ), let m_{𝔱} ∈ S^{λ} be the image of m_{𝔱^{λ}𝔱} under the map H → H/Ň^{λ}. Then \{m_{𝔱}\} = \{\overline{m}_{λ}T_{d(𝔱)}\} gives a basis of S^{λ} by 2.19. For T ∈ T_{0}^{+}(λ, μ), put m_{T} = ∑_{𝔱} m_{𝔱} ∈ S^{λ}, where the sum is taken over all 𝔱 such that μ(𝔱) = T. Since m_{T} is the image of m_{T^{λ}T}, one obtains a welldefined map φ_{T} ∈ Hom_{H}(M^{μ}, S^{λ}) by φ_{T} (m_{μ}) = m_{T}, which we regard as an element of Hom_{H}(⨁ M^{μ}, S^{λ}) by extending by 0 outside. In a similar way as in [M1, 4.14], we see that

(5.12.1) W^{λ} is isomorphic to the S(m, n)-submodule of Hom_{H}(⨁_{μ} M^{μ}, S^{λ}) with basis {φ_{T} | T ∈ S_{0}^{+}(λ)}.

5.13. We shall describe the structures of W^{λ} and L^{μ} in connection with the decomposition of S(m, n) in (5.5.1). Note that in the case where r = 1, the notation for S(m, n) coincides with the standard notation for q-Schur algebras discussed as in [M1, Chap.4]. So, we use freely such a notation. For λ, μ ∈ P_{n,r}, we denote by [W^{λ} : L^{μ}] the composition multiplicity of L^{μ} in W^{λ}. Similarly, [W^{λ^{(i)}} : L^{μ^{(i)}}] (1 ≤ i ≤ r) is defined as the composition multiplicity of L^{μ^{(i)}} in W^{λ^{(i)}} for λ = (λ^{(1)}, …, λ^{(r)}) and μ = (μ^{(1)}, …, μ^{(r)}). Then we have the following.

Proposition 5.14. Suppose that R is a field. Let λ, μ ∈ P_{n,r}. Then under the isomorphism (5.5.1), we have the following.

(i) W^{λ} ≃ W^{λ^{(1)}} ⊗ ⋯ ⊗ W^{λ^{(r)}}.

(ii) L^{μ} ≃ L^{μ^{(1)}} ⊗ ⋯ ⊗ L^{μ^{(r)}}.

(iii) [W^{λ} : L^{μ}] = \begin{cases}∏_{i=1}^{r}[W^{λ^{(i)}} : L^{μ^{(i)}}] & if α(λ) = α(μ), \\ 0 & otherwise. \end{cases}

(In (i) and (ii), the right hand sides are regarded as S(m, n)-modules by the projection from S(m, n) to S(m_{1}, n_{1}) ⊗ ⋯ ⊗ S(m_{r}, n_{r}) via (5.5.1).)

Proof. We shall see how the cellular basis of S(m, n) is mapped by the isomorphism (5.5.1). Let μ, ν ∈ P̃_{n,r} be such that α(μ) = α(ν) = (n_{1}, …, n_{r}). Take S ∈ T_{0}^{+}(λ, μ) and T ∈ T_{0}^{+}(λ, ν) with S = (S^{(1)}, …, S^{(r)}), T = (T^{(1)}, …, T^{(r)}). Then under the bijection in (4.3.1), S^{(i)} ∈ T_{0}(λ^{(i)}, μ^{(i)}) and T^{(i)} ∈ T_{0}(λ^{(i)}, ν^{(i)}) are semistandard λ^{(i)}- Tableaux of type μ^{(i)} or ν^{(i)} in the case of type A. We define \overline{m}_{ST} ∈ \overline{M}^{ν∗} ∩ \overline{M}^{μ}, and \overline{φ}_{ST} ∈ Hom_{H_{α}}(\overline{M}^{ν}, \overline{M}^{μ}) by

\overline{m}_{ST} = m_{S^{1)}T^{(1)}} ⊗ ⋯ ⊗ m_{S^{r)}T^{(r)}},
\overline{φ}_{ST} = φ_{S^{(1)}T^{(1)}} ⊗ ⋯ ⊗ φ_{S^{(r)}T^{(r)}},

where m_{S^{(i)}T^{(i)}} (resp. φ_{S^{(i)}T^{(i)}}) are corresponding elements in H_{n_{i}}m_{ν^{(i)}} ∩m_{μ^{(i)}}H_{n_{i}} (resp. Hom_{H_{ν_{i}}}(m_{ν^{(i)}}H_{n_{i}}, m_{μ^{(i)}}H_{n_{i}}). Then under the R-isomorphism M^{ν∗} ∩ M^{μ} ≃ \overline{M}^{ν∗} ∩ \overline{M}^{μ}, the basis elements m_{ST} are mapped to \overline{m}_{ST}. (Note that any element m_{𝔰𝔱} occurring in m_{ST} satisfies the property that d(𝔰), d(𝔱) ∈ 𝔖_{α} by 4.3). It follows that φ_{ST} are mapped to \overline{φ}_{ST} under the R-isomorphism Hom_{H}(M^{ν}, M^{μ}) ≃ Hom_{H_{α}}(\overline{M}^{μ}, \overline{M}^{μ}). This shows that, under the isomorphism (5.5.1), the basis element φ_{ST} of S(m, n) is mapped to the basis element \overline{φ}_{ST} in S(m_{1}, n_{1}) ⊗ ⋯ ⊗ S(m_{r}, n_{r}). Note that for λ, κ ∈ P_{n,r} such that α(λ) = α(κ), the condition κ ▷ λ is equivalent to the condition κ^{(i)} ▷ λ^{(i)} for i = 1, …, r, where λ = (λ^{(1)}, …, λ^{(r)}) an d κ = (κ^{(1)}, …, κ^{(r)}). This implies that, via (5.5.1), Š(m, n)^{λ} is mapped to Š(m_{1}, n_{1})^{λ^{(1)}} ⊗ ⋯ ⊗ Š(m_{r}, n_{r})^{λ^{(r)}}. Since φ_{λ} is mapped to \overline{φ}_{λ} = φ_{λ^{(1)}} ⊗ ⋯ ⊗ φ_{λ^{(r)}}, and φ_{T} is mapped to \overline{φ}_{T} = φ_{T^{(}1)} ⊗ ⋯ ⊗ φ_{T^{(r)}}, we see that the Weyl module W^{λ} is mapped onto W^{λ^{(1)}} ⊗ ⋯ ⊗ W^{λ^{(r)}}. This proves (i).

Next we show (ii) and (iii). Let L_{1}^{μ} = L^{μ^{(1)}} ⊗ ⋯ ⊗ L^{μ^{(r)}} be the irreducible module of S(m_{1}, n_{1}) ⊗ ⋯ ⊗ S(m_{r}, n_{r}). Then under the isomorphism (5.5.1), L_{1}^{μ} gives an irreducible S(m, n)-module, and all the irreducible modules of S(m, n) are obtained in this way. We have only to show that L^{μ} ≃ L_{1}^{μ}. Now the decomposition matrix for L^{μ} with respect to W^{λ} is unitriangular with respect to the dominance order of P_{n,r} by [GL]. On the other hand, by (i), and by the results in the case of type A, the decomposition matrix for L_{1}^{μ} is also unitriangular. Hence by induction on the dominance order, we can conclude that L^{μ} ≃ L_{1}^{μ}. This argument also shows that the decomposition number [W^{λ} : L^{μ}] is equal to zero unless α(λ) = α(μ), and in that case it coincides with the corresponding decomposition number in S(m_{1}, n_{1})⊗⋯⊗S(m_{r}, n_{r}). Hence (ii) and (iii) holds. □

6. A comparison of cyclotomic q-Schur algebras

6.1. We shall compare the cyclotomic q-Schur algebras S(m, n) with the original cyclotomic q-Schur algebras defined by Dipper-James-Mathas [DJM], associated to H̃.

Recall that the cyclotomic q-Schur algebra S̃(m, n) defined in [DJM] is of the form

S̃(m, n) = EndH̃( ⨁_{μ∈P̃_{n,r}} M_{♮}^{μ}) = ⨁_{μ,ν∈P̃_{n,r}} HomH̃(M_{♮}^{ν}, M_{♮}^{μ}).

(Note that the algebra S̃(m, n) given above is the cyclotomic q-Schur algebra S(Λ) with Λ = P̃_{n,r}, under the notation of [DJM].) We review some properties of S̃(m, n) following [DJM, 6]. For each μ, ν ∈ P̃_{n,r} and λ ∈ P_{n,r}, take S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν). Let

m_{ST}^{♮} = ∑_{𝔰,𝔱}m_{𝔰𝔱}^{♮}

where the sum is taken over all 𝔰, 𝔱 ∈ Std(λ) such that μ(𝔰) = S, ν(𝔱) = T. Then

(6.1.1) The set {m_{ST}^{♮} | S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν)} gives a basis of M_{♮}^{ν∗} ∩ M_{♮}^{μ}.

Let λ, μ, ν be as above. For each S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν), we define φ_{ST}^{♮} ∈ HomH̃(M_{♮}^{ν}, M_{♮}^{μ}) by

φ_{ST}^{♮} (m_{ν}^{♮}h) = m_{ST}^{♮} h.

for any h ∈ H̃. Then φ_{ST}^{♮} is well-defined. We extend φ_{T S}^{♮} to an element in S̃(m, n) by defining φ_{ST}^{♮} = 0 on M_{♮}^{κ} for ν ≠ κ ∈ P̃_{n,r}. Let T_{0}(λ) = ⋃_{μ∈P̃_{n,r}} T_{0}(λ, μ). By [DJM, Theorem 6.6] we have

(6.1.2) S̃(m, n) is a cellular algebra with cellular basis

C^{♮}(m, n) = {φ_{ST}^{♮} | λ ∈ P_{n,r}, S, T ∈ T_{0}(λ)}.

For each λ ∈ P_{n,r}, let S̃^{∨}(m, n)^{λ} be the R-span of φ_{ST}^{♮} such that S, T ∈ T_{0}(α) with α ▷ λ, which is a two-sided ideal of S̃(m, n). We define the Weyl module W_{♮}^{λ} by the right S̃(m, n)-submodule of S̃(m, n)/S̃^{∨}(m, n)^{λ} generated by the image φ_{λ}^{♮} of φ_{T^{λ}T^{λ}}^{♮} ∈ S̃(m, n). For each T ∈ T_{0}(λ), let φ_{T}^{♮} be the image of φ_{T^{λ}T}^{♮} in W_{♮}^{λ}. Then

(6.1.3) The Weyl module W_{♮}^{λ} is R-free with basis {φ_{T}^{♮} | T ∈ T_{0}(λ)}.

A bilinear form ⟨, ⟩_{♮} on W_{♮}^{λ} is defined in a similar way as in 5.10, and the S̃(m, n)- module L_{♮}^{λ} is defined as the quotient of W_{♮}^{λ} by rad W_{♮}^{λ}, where rad W_{♮}^{λ} is the radical of the bilinear form ⟨ , ⟩_{♮}. Then in the case where R is a field, {L_{♮}^{λ} | λ ∈ P_{n,r}} gives a complete set of non-isomorphic irreducible S̃(m, n)-modules.

6.2. Let M̂_{♮} = ⨁_{μ∈P̃_{n,r}} M̂_{♮}^{μ}, where M̂_{♮}^{μ} is as in 3.3. If φ ∈ S̃(m, n) satisfies the property that φ(M̂_{♮}) ⊂ M̂_{♮}, then φ induces a map \overline{φ} ∈ End M, where M = ⨁_{μ∈P̃_{n,r}} M^{μ} ≃ M_{♮}/M̂_{♮}. Here \overline{φ} is H̃-equivariant, but not necessarily H-equivariant. We define an R-subalgebra S^{0}(m, n) of S̃(m, n) by

S^{0}(m, n) = \{φ ∈ S̃(m, n) | φ(M̂_{♮}) ⊂ M̂_{♮}, \overline{φ} ∈ End_{H} M\}.

Clearly, we have

S^{0}(m, n) = ⨁_{μ,ν∈P̃_{n,r}} Hom0 ; H̃(M_{♮}^{ν}, M_{♮}^{μ})

where

Hom0 ; H̃(M_{♮}^{ν}, M_{♮}^{μ}) =
\{φ ∈ HomH̃(M_{♮}^{ν}, M_{♮}^{μ}) | φ(M̂_{♮}^{ν}) ⊂ M̂_{♮}^{μ}, \overline{φ} ∈ Hom_{H}(M^{ν}, M^{μ})\}.

We have the following lemma.

Lemma 6.3. Suppose that μ, ν ∈ P̃_{n,r}. Then

Hom0 ; H̃(M_{♮}^{ν}, M_{♮}^{μ}) = \begin{cases}HomH̃(M_{♮}^{ν}, M_{♮}^{μ}) & if α(μ) = α(ν), \\ \{φ ∈ HomH̃(M_{♮}^{ν}, M_{♮}^{μ}) | φ(m_{ν}^{♮}) ∈ M̂_{♮}^{μ}\} & if α(μ) ≠ α(ν). \end{cases}

Proof. First assume that α(μ) = α(ν). Take φ ∈ HomH̃(M_{♮}^{ν}, M_{♮}^{μ}). We show that φ maps M̂_{♮}^{ν} into M̂_{♮}^{μ}. By [DJM, 5.16], there exists h_{φ} ∈ H such that φ(m_{ν}^{♮}h) = h_{φ}m_{ν}^{♮}h for any h ∈ H. Note that any element in M̂_{♮}^{ν} is a linear combination of m_{S𝔱}^{♮} with S ∈ T_{0}(λ, ν) and 𝔱 ∈ Std(λ) such that a(λ) > a(ν). Assume that m_{ν}^{♮}h = m_{S𝔱}^{♮} for such m_{S𝔱}^{♮}. Then by the property of cellular basis, φ(m_{ν}^{♮}h) = h_{φ}m_{S𝔱}^{♮} is a linear combination of m_{𝔰′𝔱′}^{♮}, where 𝔰′, 𝔱′ ∈ Std(λ′) with a(λ′) ≥ a(λ) > a(ν). Since a(ν) = a(μ) by our assumption, we see that φ(m_{♮}^{ν}h) ∈ M̂_{♮}^{μ}. Hence φ induces an H̃-equivariant map \overline{φ} : M^{ν} → M^{μ}. In particular \overline{φ} is H_{n}-equivariant, and \overline{φ} is determined uniquely by \overline{φ}(m_{ν}). Thus, in order to see \overline{φ} is H-equivariant, we have only to show that \overline{φ}(m_{ν}) ∈ M^{ν∗} ∩ M^{μ}. But by our construction, \overline{φ}(m_{ν}) coincides with f(φ(m_{ν}^{♮})) under the map f : M_{♮}^{μ} → M^{μ}. Our assertion then follows from the following fact

(6.3.1) Let f : M_{♮}^{μ} → M^{μ} be the natural map as before. Then we have f(m_{ST}^{♮}) = m_{ST} for any S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν). In particular, f(M_{♮}^{ν∗} ∩ M_{♮}^{μ}) = M^{ν∗} ∩ M^{μ}.

In fact, by (6.1.1), M_{♮}^{ν∗} ∩ M_{♮}^{μ} has a basis {m_{ST}^{♮} | S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν))}. By Proposition 5.7, M^{ν∗} ∩M^{μ} has a basis {m_{ST} | S ∈ T_{0}^{+}(λ, μ), T ∈ T_{0}^{+}(λ, ν)}. Moreover by Theorem 4.9, f maps M_{♮}^{ν∗}∩M_{♮}^{μ} onto M^{ν∗}∩M^{μ}. This shows (6.3.1), and (i) follows.

Next assume that α(μ) ≠ α(ν), and take φ ∈ Hom0 ; H̃(M_{♮}^{ν}, M_{♮}^{μ}). Then φ induces a map \overline{φ} ∈ Hom_{H}(M^{ν}, M^{μ}), which is a zero map by Proposition 5.2 (i). It follows that φ(M_{♮}^{ν}) ⊆ M̂_{♮}^{μ}, which is equivalent to φ(m_{ν}^{♮}) ∈ M̂_{♮}^{μ}. This proves the second assertion. □

The following result gives a relationship between Dipper-James-Mathas’ cyclotomic q-Schur algebra S̃(m, n) and our algebra S(m, n) by means of S^{0}(m, n).

Proposition 6.4. (i) S^{0}(m, n) is free as an R-module with basis

\begin{matrix}C^{0}(m, n) = ⋃_{μ,ν∈P̃_{n,r},λ∈P_{n,r}}\{φ_{ST}^{♮} ∈ C^{♮}(m, n) & | S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν), \\ a(λ) > a(μ) if α(μ) ≠ α(ν)\}. \end{matrix}

(ii) There exists a surjective algebra homomorphism f̂: S^{0}(m, n) → S(m, n) satisfying the following. For φ_{ST}^{♮} ∈ C^{0}(m, n) with S, T ∈ T_{0}(λ),

(6.4.1)  f̂(φ_{ST}^{♮}) = \begin{cases}φ_{ST} & if S, T ∈ T_{0}^{+}(λ), \\ 0 & otherwise. \end{cases}

Proof. By (6.1.1), M_{♮}^{ν∗} ∩ M̂_{♮}^{μ} is R-free with basis {m_{ST}^{♮} } such that S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν) for some λ ∈ P_{n,r} with a(λ) > a(μ). Then (i) follows from Lemma 6.5 and the fact that HomH̃(M_{♮}^{ν}, M_{♮}^{μ}) ≃ M_{♮}^{ν∗} ∩ M_{♮}^{μ} by φ ↦ φ(m_{ν}^{♮}) (cf. [DJM, 5.2]).

Next we show (ii). Take φ ∈ Ŝ(m, n). By definition, it induces an endomorphism \overline{φ} ∈ End_{H} M = S(m, n). The map φ ↦ \overline{φ} gives an algebra homomorphism f̂ : Ŝ(m, n) → S(m, n). The equation (6.4.1) is immediate from (6.3.1). The surjectivity of f̂ then follows from (6.4.1). □

Remark 6.5. The basis C^{0}(m, n) satisfies the following property. Let μ, ν ∈ P̃_{n,r}, λ ∈ P_{n,r} and φ ∈ S^{0}(m, n). Then for any κ ∈ P̃_{n,r} and T′ ∈ T_{0}(λ, κ), there exists r_{T′} ∈ R such that for all S ∈ T_{0}(λ, μ), we have

(6.5.1)  φ_{ST}^{♮} φ ≡ ∑_{κ∈P̃_{n,r}} ∑_{T′∈T_{0}(λ,κ)} r_{T′}φ_{ST′}^{♮} mod Š^{0}(m, n)^{λ}

where φ_{ST}^{♮} , φ_{ST′}^{♮} ∈ C^{0}(m, n) and Š^{0}(m, n)^{λ} = S^{0}(m, n)∩S̃^{∨}(m, n)^{λ}. However, C^{0}(m, n) is not a cellular basis since ∗ : φ_{ST}^{♮} ↦ φ_{T S}^{♮} is not defined.

6.6. For λ ∈ P_{n,r}, φ_{λ}^{♮} = φ_{T^{λ}T^{λ}}^{♮} is an element in S^{0}(m, n). Hence one can define a Weyl module Z^{λ} of S^{0}(m, n) as the right S^{0}(m, n)-submodule of S^{0}(m, n)/Š^{0}(m, n)^{λ} spanned by the image of φ_{λ}^{♮}. We denote by φ_{T}^{0} the image of φ_{T^{λ}T}^{♮} in S^{0}(m, n)/Š^{0}(m, n)^{λ}. It follows by (6.5.1) that

(6.6.1) The Weyl module Z^{λ} is R-free with basis {φ_{T}^{0} | T ∈ T_{0}^{+}(λ)}.

The following lemma describes the relationship of various Weyl modules.

Lemma 6.7. Let W_{♮}^{λ}, Z^{λ}, W^{λ} be the Weyl modules of S̃(m, n), S^{0}(m, n), S(m, n), respectively.

(i) There exists an injective R-linear map f_{1} : Z^{λ} → W_{♮}^{λ}, compatible with the action of S^{0}(m, n) such that f_{1}(φ_{T}^{0} ) = φ_{T}^{♮} for T ∈ T_{0}^{+}(λ).

(ii) There exists an S^{0}(m, n)-module isomorphism f_{2} : Z^{λ} → W^{λ} such that f_{2}(φ_{T}^{0} ) = φ_{T} for any T ∈ T_{0}^{+}(λ), where W^{λ} is regarded as an S^{0}(m, n)-module via the map f̂ in Proposition 6.4 (ii).

Proof. The inclusion S^{0}(m, n) ↪ S̃(m, n) induces an injective S^{0}(m, n)-equivariant map S^{0}(m, n)/Š^{0}(m, n)^{λ} → S̃(m, n)/S̃^{∨}(m, n)^{λ}, which maps φ_{T}^{0} to φ_{T}^{♮} for any T ∈ T_{0}(λ). Hence (i) holds.

The surjection f̂ : S^{0}(m, n) → S(m, n) induces a map S^{0}(m, n)/Š^{0}(m, n)^{λ} → S(m, n)/Š^{λ}(m, n), which is comaptible with the action of S^{0}(m, n). This map sends the basis {φ_{T}^{0} } of Z^{λ} to the basis {φ_{T} } of W^{λ}. Thus (ii) holds. □

Concerning the irreducible modules, we have the following.

Lemma 6.8. Suppose that R is a field. Let L_{♮}^{μ} (resp. L^{μ}) be the irreducible S̃(m, n)- module (resp. S(m, n)-module). We regard them as S^{0}(m, n)-modules, via the restriction and via the map f̂. Then S^{0}(m, n)-module L_{♮}^{μ} contains L^{μ} as an irreducible submodule.

Proof. Let ⟨, ⟩_{♮} (resp. ⟨, ⟩) be the bilinear form on W_{♮}^{λ} (resp. on W^{λ} ). By identifying Z^{λ} with W^{λ} by the map f_{2}, we show that f_{1} : Z^{λ} → W_{♮}^{λ} maps rad W^{λ} into rad W_{♮}^{λ}. In fact, it is easy to see that ⟨φ_{S}^{♮}, φ_{T}^{♮} ⟩_{♮} = ⟨φ_{S}, φ_{T} ⟩ for any S, T ∈ T_{0}^{+}(λ). On the other hand, if S ∈ T_{0}^{+}(λ, μ), T ∈ T_{0}(λ, ν)\T_{0}^{+}(λ, ν), then α(μ) ≠ α(ν), and so μ ≠ ν. It follows that φ_{T^{λ}S}^{♮}φ_{T T^{λ}}^{♮} = 0, and ⟨φ_{S}^{♮}, φ_{T}^{♮} ⟩_{♮} = 0. Hence f_{1}(rad W^{λ}) ⊂ rad W_{♮}^{λ}. Now f_{1} induces an S^{0}(m, n)-equivariant map L^{λ} → L_{♮}^{λ}, which is clearly non-trivial. Since L^{μ} is an irreducible S^{0}(m, n)-module, L_{♮}^{μ} contains an irreducible module isomorphic to L^{μ}. □

7. Schur functors

7.1. Recall the R-submodule U of V^{⊗n} defined in 1.4. Under the identification V^{⊗n} ≃ ⨁_{μ∈P̃_{n,r}} M^{μ} in 5.4, U can be identified with ⨁_{ω∈Ω} M^{ω}, where Ω is the set of r-

compositions ω ∈ P̃_{n,r} of the form ω = (ω^{(1)}, …, ω^{(r)}) such that ω_{j}^{(k)} ∈ {0, 1} and that ∑_{1≤k≤r} ω_{j}^{(k)} = 1 for j = 1, …, n. U = v_{0}H is a regular H-module, and Hom_{H}(U, U) is isomorphic to H as R-algebras via φ_{h} ↔ h, where φ_{h} is a map on U determined by φ_{h}(v_{0}) = v_{0}h. Put φ_{Ω} = ∑_{ω∈Ω} φ_{T^{ω}T^{ω}}. Then φ_{Ω} is the identity map on U and is zero on the complement of U. For each ω ∈ Ω, we consider v_{ω}^{♮} ∈ V^{⊗n} (see (3.3.2)). Then there exists a unique d(ω) ∈ D_{α(ω)} such that v_{ω}^{♮}T_{d(ω)} = v_{J}^{c} with J = (n, n − 1, …, 1) and some c ∈ [1, r]^{n}. It follows that v_{0} can be written as

(7.1.1)  v_{0} = ∑_{ω∈Ω} v_{ω}^{♮}T_{d(ω)},

and so v_{0} corresponds to an element ∑_{ω∈Ω} m_{ω}T_{d(ω)} ∈ ⨁_{ω∈Ω} M^{ω}.

Now φ_{Ω} is an idempotent in S(m, n) and φ_{Ω}S(m, n)φ_{Ω} = Hom_{H}(U, U) ≃ H. Hence one can define a functor, which we call a Schur functor, from the category of right S(m, n)-modules to the category of right H-modules, by associating Mφ_{Ω} to any S(m, n)-module M.

The following result is an analogy of the case of Iwahori-Hecke algebras H_{n} ([M1, 4.18]), and of H̃ ([JM]).

Proposition 7.2. For each λ ∈ P_{n,r}, there exist isomorphisms of right H-modules. (i) W^{λ}φ_{Ω} ≃ S^{λ}, (ii) L^{λ}φ_{Ω} ≃ D^{λ}.

Furthermore, for μ ∈ P_{n,r} such that D^{μ} ≠ 0, we have [W^{λ} : L^{μ}] = [S^{λ} : D^{μ}].

Proof. Let T ∈ T_{0}^{+}(λ, ν) and φ_{T} ∈ Hom(⨁_{μ} M^{μ}, S^{λ}) be the element in the semistandard basis of W^{λ} (cf. (5.12.1)). Then we see that φ_{T} φ_{Ω} = 0 unless ν = ω for some ω ∈ Ω. If ν = ω, we have φ_{T} φ_{Ω} = φ_{T}. Thus W^{λ}φ_{Ω} is a free R-module with basis

(7.2.1)  {φ_{T} | T ∈ ⋃_{ω∈Ω} T_{0}^{+}(λ, ω)}.

Here any T ∈ T_{0}^{+}(λ, ω) can be seen as a standard tableau 𝔱 ∈ Std(λ) in an obvious way. Then T_{0}^{+}(λ, ω) is identified with the set of 𝔱 ∈ Std(λ) such that the entries in 𝔱^{(i)} consists of {j | ω_{j}^{(i)} ≠ 0}. It follows that the set ⋃_{ω∈Ω} T_{0}^{+}(λ, ω) may be identified with Std(λ). For T ∈ T_{0}^{+}(λ, ω), there exists a unique 𝔱_{1} ∈ Std(λ) such that λ(𝔱_{1}) = T. Moreover it is easy to see that 𝔱 = 𝔱_{1}d(ω). We note that 𝔱_{1} is a standard tableau of the type the entries of 𝔱_{1}^{(i)} consist of integers in the set {p | a_{i} < p ≤ a_{i+1}} for a(ω) = (a_{1}, …, a_{r}). We claim that

(7.2.2) For T = 𝔱 ∈ T_{0}^{+}(λ, ω), we have φ_{T} (v_{0}) = m_{𝔱} ∈ S^{λ}.

In fact, by (7.1.1) and the remark after that, we have

φ_{T} (v_{0}) = φ_{T} (m_{ω}T_{d(ω)}) = φ_{T} (m_{ω})T_{d(ω)} = m_{T} T_{d(ω)}.

By definition, m_{T} = m_{𝔱_{1}} = m_{𝔱^{λ}𝔱_{1}} + Š(m, n)^{λ} ∈ S^{λ}. Since

m_{𝔱^{λ}𝔱_{1}}T_{d(ω)} = m_{λ}T_{d(𝔱_{1})}T_{d(ω)} = m_{λ}T_{d(𝔱)} = m_{𝔱^{λ}𝔱},

we obtain (7.2.2).

Now, by the H-equivariant map W^{λ}φ_{Ω} → S^{λ}, φ ↦ φ(v_{0}), the basis of W^{λ}φ_{Ω} in (7.2.1) is mapped to the basis {m_{𝔱} | 𝔱 ∈ Std(λ)} of S^{λ}. This proves (i).

We show (ii). Let S, T ∈ T_{0}^{+}(λ, ω), and let α = α(λ) = α(ω). First we note that

(7.2.3)  m_{T T^{λ}} = F_{α}m_{T T^{λ}} = m_{ω}m_{T T^{λ}}.

In fact, m_{T T^{λ}} = m_{𝔱_{1}𝔱^{λ}} = T_{d(𝔱_{1})}^{∗}m_{λ}. Now, m_{λ} ∈ H corresponds to v_{λ′}^{♮} ∈ V_{λ′}^{⊗n} under the isomorphism m_{λ}H = M^{λ} ≃ V_{λ′}^{⊗n}. Let α = α(λ) = (n_{1}, …, n_{r}). Then v_{λ′}^{♮} is expressed as v_{λ′}^{♮} = v_{I}^{c} for some I with

c = (\underbrace{r, …, r,}_{n_{1}−times} …, \underbrace{1, …, 1}_{n_{r}−times})

by (3.3.2). This implies that m_{λ}F_{α} = m_{λ} by Lemma 2.11. Since d(𝔱_{1}) ∈ 𝔖_{α}, F_{α} commutes with m_{λ} and T_{d(𝔱_{1})}^{∗}. This shows the first equality. Since m_{{ω}} = 1, we have m_{ω} = F_{α}m_{{ω}} = F_{α}. The second equality follows from this.

It follows from (7.2.3) that

(7.2.4)  φ_{T^{λ}S}φ_{T T^{λ}}(m_{λ}) = φ_{T^{λ}S}(m_{T T^{λ}}) = φ_{T^{λ}S}(m_{ω})m_{T T^{λ}} = m_{T^{λ}S}m_{T T^{λ}}.

We now consider the defining equation

⟨φ_{S}, φ_{T} ⟩ φ_{T^{λ}T^{λ}} ≡ φ_{T^{λ}S}φ_{T T^{λ}} mod Š(m, n)^{λ}.

By evaluating both sides by m_{λ}, together with (7.2.4), we have

⟨φ_{S}, φ_{T} ⟩ m_{λ} ≡ m_{T^{λ}S}m_{T T^{λ}} mod Ň^{λ}.

This implies that ⟨φ_{S}, φ_{T} ⟩ = ⟨m_{𝔰}, m_{𝔱}⟩ for 𝔰, 𝔱 ∈ Std(λ) corresponding to S, T respectively. (ii) follows from this.

The last statement is completely similar to [M1, 4.18]. We consider a composition series W^{λ} = W^{1} > ⋯ > W^{k} > 0. Then S^{λ} ≃ W^{1}φ_{Ω} ≥ ⋯ ≥ W^{k}φ_{Ω} ≥ 0, and this implies that W^{i}φ_{Ω}/W^{i+1}φ_{Ω} ≃ L^{μ_{i}}φ_{Ω} ≃ D^{μ_{i}} for some μ_{i} ∈ P_{n,r}. Hence if D^{μ} ≠ 0, the number of D^{μ} occurring in the series of S^{λ} is the same as the number of L^{μ} occurring in the series of W^{λ}. □

Recall that a 1-partition ν = (ν_{1}, …, ν_{m}) ∈ P_{n,1} is called e-restricted if ν_{i}−ν_{i+1} < e for each i, where e is the smallest positive integer such that 1 + q + ⋯ + q^{e−1} = 0. It is known in the case of Iwahori-Hecke algebra H_{n}, D^{ν} ≠ 0 if and only if ν is erestricted. The following result is a consequence of Corollary 4.13, Proposition 5.13 and Proposition 7.2.

Corollary 7.3. Suppose that R is a filed. Let λ = (λ^{(1)}, …, λ^{(r)}), μ = (μ^{(1)}, …, μ^{(r)}) ∈ P_{n,r}. Suppose that D_{♮}^{μ} ≠ 0 and that μ^{(i)} is e-restricted for i = 1, …, r.

(i) Suppose that α(λ) = α(μ) = (n_{1}, …, n_{r}). Then we have

[S_{♮}^{λ} : D_{♮}^{μ}] ≥ ∏_{i=1}^{r}[S^{λ^{(i)}} : D^{μ^{(i)}}],

where [S^{λ^{(i)}} : D^{μ^{(i)}}] is the composition multiplicity in the Iwahori-Hecke algebra H_{n_{i}}.

(ii) Suppose that H ≃ H̃. Then [S_{♮}^{λ} : D_{♮}^{μ}] = 0 unless α(λ) = α(μ). If α(λ) = α(μ), then the equality holds in (i).

Remark 7.4. As the remark in the next section (7.3.2) shows, H ≃ H̃ if the parameters satisfy the separation condition (5.4.2).

By applying the Schur functor on the isomorphism in Theorem 5.5, we have the following result, which was obtained by [HS] in the case where R is a field, in a different method.

Proposition 7.5. For each r-tuple α = (n_{1}, …, n_{r}) such that n_{1} + ⋯ n_{r} = n, put n_{α} = n!/(n_{1}! ⋯ n_{r}!). Then we have an isomorphism of R-algebras

H ≃ ⨁_{\begin{matrix}α=(n_{1},...,n_{r}) \\ n_{1}+···+n_{r}=n \end{matrix}} M_{n_{α}}(H_{n_{1}} ⊗ ⋯ ⊗ H_{n_{r}}),

where M_{n}(A) denotes the matrix algebra of degree n over the algebra A.

Proof. We have

H ≃ φ_{Ω}S(m, n)φ_{Ω} ≃ ⨁_{α} End_{H}(M^{α} ∩ U).

But by (5.5.2),

End_{H}(M^{α} ∩ U) ≃ ⨁_{\begin{matrix}ω,ω′∈Ω \\ α(ω)=α(ω′)=α \end{matrix}} Hom_{H_{α}}(\overline{M}^{ω}, \overline{M}^{ω′}).

Here \overline{M}^{ω} are isomorphic each other as H_{α}-modules for any ω ∈ Ω such that α(ω) = α, and we have

End_{H_{α}} \overline{M}^{ω} ≃ H_{n_{1}} ⊗ ⋯ ⊗ H_{n_{r}}.

The proposition follows from this by noticing that ♯{ω ∈ Ω | α(ω) = α} = n_{α}. □

Remark 7.6. Proposition 7.5 implies that D^{μ} ≠ 0 if and only if μ = (μ^{(1)}, …, μ^{(r)}) with all μ^{(i)} e-restricted. By combining this with Theorem 4.13 (ii), we recover the result in [DJMu, Theorem 6.9]; If D_{♮}^{μ} ≠ 0, then all the partitions μ^{(i)} are e-restricted.

8. Schur-Weyl reciprocity

8.1. The tensor space V^{⊗n} has a structure of U_{q}-H-bimodule (the left action of U_{q} and the right action of H), which is the same as the S(m, n)-H-bimodule structure on V^{⊗n} by Theorem 5.5 (ii). It is known in [SS] that the Schur-Weyl reciprocity (i.e., the double centralizer property) between U_{q} and H holds in the case where R = K_{0}. In this section, we shall extend this to the case of R.

Theorem 8.2. Let ρ : H → End V^{⊗n}, τ : U_{q} → End V^{⊗n} be the representations of H, U_{q} on V^{⊗n}, respectively. Then we have

τ(U_{q}) = End_{H} V^{⊗n}, ρ(H) = End_{U_{q}} V^{⊗n}.

Proof. The first equality is equivalent to Theorem 5.5, (ii). We show the second equality. Instead of discussing with U_{q}, we will work on S(m, n). It is enough to show that

(8.2.1)  ρ(H) = End_{S(m,n)} V^{⊗n}.

The proof is similar to [M1, Theorem 4.19]. For each μ ∈ P̃_{n,r}, let φ_{μ} = φ_{T^{μ}T^{μ}} be the identity map on M^{μ}. As before, we regard φ_{μ} as an element of S(m, n). Then ∑_{μ} φ_{μ} is the identity map of S(m, n). First we note that φ_{μ}S(m, n)φ_{Ω} ≃ M^{μ} as H-modules. Since φ_{μ}S(m, n)φ_{Ω} = ⨁_{ω∈Ω} Hom_{H}(M^{ω}, M^{μ}), we see that φ_{μ}S(m, n)φ_{Ω} is a free Rmodule with basis {φ_{ST} }, where S ∈ T_{0}^{+}(λ, μ), T ∈ T_{0}^{+}(λ, ω) for some λ ∈ P_{n,r} and ω ∈ Ω. As in the proof of (8.2.2) (under the notation there), we have

φ_{ST} (v_{0}) = φ_{ST} (m_{ω}T_{d(ω)}) = m_{ST} T_{d(ω)} = m_{S𝔱},

where 𝔱 ∈ Std(λ) is the underlying standard tableau of T. Since {m_{S𝔱}}, where S ∈ T_{0}^{+}(λ, μ), 𝔱 ∈ Std(λ) is a basis of M^{μ}, we see that φ ↦ φ(v_{0}) gives an H-module isomorphism between φ_{μ}S(m, n)φ_{Ω} and M^{μ}.

It follows that

V^{⊗n} ≃ ⨁_{μ∈P̃_{n,r}} M^{μ} ≃ ⨁_{μ∈P̃_{n,r}} Hom_{H}(U, M^{μ}) ≃ ⨁_{μ∈P̃_{n,r}} φ_{μ}S(m, n)φ_{Ω}.

Since ∑_{μ} φ_{μ} is the identity map on ⨁_{μ∈P̃_{n,r}} M^{μ}, we see that ⨁_{μ} M^{μ} ≃ S(m, n)φ_{Ω} as left S(m, n)-modules. This implies that

End_{S(m,n)} V^{⊗n} ≃ End_{S(m,n)}(S(m, n)φ_{Ω})
≃ φ_{Ω}S(m, n)φ_{Ω}
≃ H.

This isomorphism is explicitly given by the map φ ↦ h_{φ} for φ ∈ End_{S(m,n)} V^{⊗n}, where h_{φ} ∈ H is determined by the condition that φ(v_{0}) = v_{0}h_{φ}. Now ρ(H) is contained in End_{S(m,n)} V^{⊗n}, and one can check that h_{φ} = h for any φ = ρ(h). This shows that ρ(H) = End_{S(m,n)} V^{⊗n}. Hence (8.2.1) holds. □

Remarks 8.3. (i) In [HS, Theorem 3.3], Hu and Stoll extended the action of H on the space V^{⊗n} to the general case where no restrictions on m_{i} are assumed, and proved the second equality of Theorem 8.2 for this general case, in a different method, under the assumption that R is a field.

(ii) They also proved in [HS, Theorem 1.10] a similar formula for the Ariki-Koike algebra H̃, subject to the condition that R is a field with the separation condition (5.4.2), i.e.,

(8.3.1)  ρ̃(H̃) = End_{U_{q}} V^{⊗n}.

This implies the following result.

(8.3.2) Suppose that R is a field satisfying the separation condition. Then H̃ ≃ H.

In fact, we consider the case where m_{i} ≥ n for any i. By Theorem 8.2 and (8.3.1), we have ρ(H) = ρ̃(H̃). Thus by Lemma 1.5, we have H ≃ H̃.

It is likely that the separation condition will be the necessary and sufficient condition for the isomorphism H ≃ H̃.

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