Cyclotomic q–Schur algebras

Richard Dipper Mathematische Institut B Universität Stuttgart Postfach 80 11 40 70550 Stuttgart Deutschland

Gordon James Department of Mathematics Imperial College Queen’s Gate London SW7 2BZ England

Andrew Mathas Fakultät für Mathematik Universität Bielefeld P.O. Box 100 131 33501 Bielefeld Deutschland

15th August, 1997

1 Introduction

Recently, we [9] and, independently, Du and Scott [11], defined an analogue of the q–Schur algebra [6, 7] for an Iwahori–Hecke algebra of type B. In this paper we study an analogue of the q–Schur algebra for an arbitrary Ariki–Koike algebra.

The Ariki–Koike algebra ℋ is a cyclotomic algebra of type G(r, 1, n) [2], and it becomes the Iwahori–Hecke algebra of type A or B when r = 1 or 2 respectively. By working over a ring R which contains a primitive rth root of unity, and by specializing the parameters appropriately, the Ariki–Koike algebra turns into the group algebra R(C_{r} ≀𝔖_{n}) of the wreath product of the cyclic group C_{r} of order r with the symmetric group 𝔖_{n} of degree n.

For each multicomposition λ of n, we construct a right ideal M^{λ} of ℋ (see Definition 3.8). The cyclotomic q–Schur algebra is then defined to be 𝒮 (Λ_{r}) = End_{ℋ} ( ⊕_{λ} M^{λ}). (Under the specialization above where ℋ ≅ R(C_{r} ≀ 𝔖_{n}), the module M^{λ} becomes a module induced from a subgroup of the form (C_{r} × ⋯ × C_{r}) ⋊ 𝔖_{λ}.)

A.M.S. subject classification (1991): 16G99, 20C20, 20G05 This paper is a contribution to the DFG project on “Algorithmic number theory and algebra”. The authors acknowledge support from DFG and the Isaac Newton Institute; the third author was also supported in part by SERC grant GR/J37690 and by the Sonderforschungsbereich 343 in Bielefeld.

In this paper we construct a cellular basis for the cyclotomic q–Schur algebra. As a consequence we obtain a Weyl module W^{λ} for each multipartition λ of n. We show that W^{λ} has simple head F^{λ} and that the set {F^{λ}}, as λ ranges over the multipartitions of n, is a complete set of non–isomorphic irreducible 𝒮 (Λ_{r})–modules. Using the cellular structure of 𝒮 (Λ_{r}), it is now easy to see that the cyclotomic q–Schur algebra is quasi–hereditary.

In order to prove these results about the cyclotomic q–Schur algebra, we need to examine the ideals M^{λ} in some detail. Using the cellular structure of the Ariki–Koike algebra ℋ (cf. [10, 13]), we obtain a basis of M^{λ} and a special series of submodules of M^{λ}, known as a Specht series. From the Specht series of M^{λ} we construct the cellular basis of 𝒮 (Λ_{r}).

2 The Ariki–Koike algebra

Throughout this paper, r and n will be fixed positive integers with r ≥ 1 and n ≥ 1.

Let R be a commutative ring with 1 and let q, Q_{1}, …, Q_{r} be elements of R with q invertible. The Ariki–Koike algebra ℋ is the associative unital R–algebra with generators T_{0}, T_{1}, …, T_{n−1} subject to the following relations

(T_{0} − Q_{1}) ⋯ (T_{0} − Q_{r}) = 0
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0}
(T_{i} + 1)(T_{i} − q) = 0  for 1 ≤ i ≤ n − 1
T_{i+1}T_{i}T_{i+1} = T_{i}T_{i+1}T_{i}  for 1 ≤ i ≤ n − 2
T_{i}T_{j} = T_{j}T_{i}  for 0 ≤ i < j − 1 ≤ n − 2.

Suppose that π ∈ R is a primitive rth root of 1 and that q = 1 and Q_{k} = π^{k} for k = 1, 2, …, r. Then it follows from the definition that ℋ is isomorphic to R(C_{r} ≀ 𝔖_{n}).

For each finite set X, let 𝔖(X) be the group of permutations of X; thus, the symmetric group 𝔖_{n} = 𝔖({1, 2, …, n}). Let s_{i} = (i, i + 1) for 1 ≤ i < n; then s_{1}, s_{2}, …, s_{n−1} are the standard Coxeter generators of 𝔖_{n}. If w ∈ 𝔖_{n} then a word w = s_{i_{1}} … s_{i_{k}} for w is a reduced expression for w if k is minimal; in this case we say that w has length k and write ℓ(w) = k. Given a reduced expression s_{i_{1}} … s_{i_{k}} for w ∈ 𝔖_{n}, let T_{w} = T_{i_{1}} … T_{i_{k}}; the relations in ℋ ensure that T_{w} is independent of the choice of reduced expression. We denote by ℋ (𝔖_{n}) the subalgebra of ℋ generated by T_{1}, T_{2}, …, T_{n−1}. Then ℋ (𝔖_{n}) has basis { T_{w} | w ∈ 𝔖_{n} }, and it is isomorphic to an Iwahori–Hecke algebra of type A.

Define elements L_{m} = q^{1−m}T_{m−1} … T_{1}T_{0}T_{1} … T_{m−1} for m = 1, 2, …, n; these are analogues of the q–Murphy operators of the Iwahori–Hecke algebras of type A [5, 15]. Easy calculations using the relations in ℋ (cf. [5, (2.1), (2.2)]) show that we have the following results.

(2.1) Suppose that 1 ≤ i ≤ n − 1 and 1 ≤ m ≤ n. Then 1. L_{i} and L_{m} commute.

2. T_{i} and L_{m} commute if i ≠ m − 1, m.

3. T_{i} commutes with L_{i}L_{i+1} and L_{i} + L_{i+1}.

4. If a ∈ R and i ≠ m then T_{i} commutes with (L_{1} − a)(L_{2} − a) … (L_{m} − a).

Using the elements T_{w} and L_{m} defined above, Ariki and Koike gave a basis for ℋ as follows.

(2.2) Theorem (Ariki–Koike [1, (3.10)]) The algebra ℋ is a free R–module with basis

{ L_{1}^{c_{1}}L_{2}^{c_{2}} … L_{n}^{c_{n}}T_{w} | w ∈ 𝔖_{n} and 0 ≤ c_{m} ≤ r − 1 for m = 1, 2, …, n }.

In particular, ℋ is free of rank r^{n}n!

(2.3) Let ∗ be the R–linear antiautomorphism of ℋ determined by T_{i}^{∗} = T_{i} for all i with 0 ≤ i ≤ n − 1. Then T_{w}^{∗} = T_{w^{−1}} and L_{m}^{∗} = L_{m} for all w ∈ 𝔖_{n} and m = 1, 2, …, n.

We therefore have the following result.

(2.4) { T_{w}L_{1}^{c_{1}}L_{2}^{c_{2}} … L_{n}^{c_{n}} | w ∈ 𝔖_{n} and 0 ≤ c_{m} ≤ r − 1 for m = 1, 2, …, n } is a basis of ℋ .

3 A cellular basis of ℋ

In their paper [12], which introduced the concept of cellular algebras, Graham and Lehrer gave a cellular basis of ℋ , using the Kazhdan–Lusztig basis of ℋ (𝔖_{n}). We require a different cellular basis, namely one similar to the basis of ℋ (𝔖_{n}) introduced by Murphy [16]. Although the construction of the cellular basis of ℋ in this section is similar to that in [10, 13], we are obliged to keep track of new information concerned with the cellular basis (see Corollary 3.24 below).

Consider the R–submodule of ℋ which is spanned by

{ L_{1}^{c_{1}}L_{2}^{c_{2}} … L_{n}^{c_{n}} | 0 ≤ c_{i} ≤ r − 1 for 1 ≤ i ≤ n }.

When q = 1, this is a subalgebra of ℋ , but one of the main difficulties of working with the Ariki–Koike algebra is that it is not a subalgebra in general. (To see this, consider L_{2}^{2} when r = 2.) We shall need certain elements u_{a}^{+} of this R–submodule of ℋ , and we introduce these now.

(3.1) Definition Suppose that a = (a_{1}, …, a_{r}) is an r–tuple of integers a_{i} such that 0 ≤ a_{i} ≤ n for all i. Let u_{a}^{+} = u_{a,1}u_{a,2} … u_{a,r} where

u_{a,k} = ∏_{m=1}^{a_{k}}(L_{m} − Q_{k}) for 1 ≤ k ≤ r.

(3.2) Remarks 1. Suppose that every a_{k} is non–zero. Then (L_{1} −Q_{k}) is a factor of each u_{a,k}; so u_{a}^{+} has a factor

∏_{k=1}^{r}(L_{1} − Q_{k}) = ∏_{k=1}^{r}(T_{0} − Q_{k}) = 0.

Therefore, u_{a}^{+} is zero in this case.

2. Rearranging the order of a_{1}, a_{2}, …, a_{r} amounts just to reordering the parameters Q_{1}, Q_{2}, …, Q_{r}. For example, if we define u_{a}^{−} = u_{(a_{r},a_{r−1},...,a_{1})}^{+} then u_{a}^{−} is obtained from u_{a}^{+} by replacing Q_{k} by Q_{r−k+1} for 1 ≤ k ≤ r.

3. In practice, we shall use u_{a}^{+} only for r–tuples a = (a_{1}, a_{2}, …, a_{r}) such that 0 = a_{1} ≤ a_{2} ≤ … ≤ a_{r} ≤ n. Our last two remarks show that there is no loss in doing this, and that we could equally well work with the elements u_{a}^{−} defined in Remark (ii).

(3.3) Example Suppose that r = 4, n ≥ 5 and a = (0, 2, 4, 5). Then

u_{a}^{+} = (L_{1} − Q_{2})(L_{2} − Q_{2})
× (L_{1} − Q_{3})(L_{2} − Q_{3})(L_{3} − Q_{3})(L_{4} − Q_{3})
× (L_{1} − Q_{4})(L_{2} − Q_{4})(L_{3} − Q_{4})(L_{4} − Q_{4})(L_{5} − Q_{4}).

Our first lemma relates u_{a}^{+} and u_{b}^{+} when b is obtained from a by increasing a single part by one.

(3.4) Lemma Let a = (a_{1}, a_{2}, …, a_{r}) and assume that 1 ≤ k ≤ r and a_{k} + 1 ≤ n. Let b = (a_{1}, …, a_{k−1}, a_{k} + 1, a_{k+1}, …, a_{r}). Then, for some h_{1}, h_{2} ∈ ℋ (𝔖_{n}), we have u_{a}^{+}T_{a_{k}}T_{a_{k}−1} … T_{1}T_{0} = u_{a}^{+}h_{1} + u_{b}^{+}h_{2}.

Proof: The definition of u_{b}^{+} gives u_{b}^{+} = u_{a}^{+}(L_{a_{k}+1} − Q_{k}). Hence,

u_{a}^{+}T_{a_{k}}T_{a_{k}−1} … T_{1}T_{0}T_{1} … T_{a_{k−1}}T_{a_{k}} = q^{a_{k}}u_{a}^{+}L_{a_{k}+1} = q^{a_{k}}Q_{k}u_{a}^{+} + q^{a_{k}}u_{b}^{+}.

The desired result follows by postmultiplying by T_{a_{k}}^{−1} … T_{1}^{−1}.

We next turn our attention to the subalgebra ℋ (𝔖_{n}) of ℋ . Here we shall need the notation and combinatorics of multipartitions.

A composition α = (α_{1}, α_{2}, …) is a finite sequence of non–negative integers; we denote by |α| the sum of this sequence. A multicomposition of n is an ordered r–tuple λ = (λ^{(1)}, …, λ^{(r)}) of compositions λ^{(k)} such that ∑_{k=1}^{r} |λ^{(k)}| = n. We call λ^{(k)} the kth component of λ. A partition is a composition whose parts are non–increasing; a multicomposition is a multipartition if all its components are partitions.

For each composition α = (α_{1}, α_{2}, …) with |α| = m we have a Young subgroup 𝔖_{α} = 𝔖_{α_{1}} × 𝔖_{α_{2}} × ⋯ of 𝔖_{m}. Similarly, to each multicomposition λ = (λ^{(1)}, …, λ^{(r)}) of n we associate the Young subgroup 𝔖_{λ} = 𝔖_{λ^{(1)}} × 𝔖_{λ^{(2)}} × ⋯ × 𝔖_{λ^{(r)}} of 𝔖_{n}.

We are now in a position to define certain key elements m_{λ} of ℋ . The element m_{λ} depends upon the Young subgroup 𝔖_{λ} and also involves one of the elements u_{a}^{+} defined above.

(3.5) Definition Suppose that λ is a multicomposition of n and define a = (a_{1}, a_{2}, …, a_{r}) by a_{k} = ∑_{i=1}^{k−1} |λ^{(i)}|. Let x_{λ} = ∑_{w∈𝔖_{λ}} T_{w} and set m_{λ} = u_{a}^{+}x_{λ}.

(3.6) Example Suppose that λ = ((2, 1), (1^{2}), (2)). Then 𝔖_{λ} = 𝔖_{2} × 𝔖_{1} × 𝔖_{1} × 𝔖_{1} × 𝔖_{2}, a = (0, 3, 5), and

m_{λ} = (L_{1} − Q_{2})(L_{2} − Q_{2})(L_{3} − Q_{2})
× (L_{1} − Q_{3})(L_{2} − Q_{3})(L_{3} − Q_{3})(L_{4} − Q_{3})(L_{5} − Q_{3})
× (1 + T_{1})(1 + T_{6}).

(3.7) Remark If α = (|λ^{(1)}|, |λ^{(2)}|, …, |λ^{(r)}|) then all of the elements in ℋ (𝔖_{α}) commute with u_{a}^{+} by (2.1)(iv). In particular, m_{λ} = u_{a}^{+}x_{λ} = x_{λ}u_{a}^{+}. Hence, m_{λ}^{∗} = m_{λ}, where ∗ is the antiautomorphism of (2.3).

The ℋ –modules which will be our main concern in this paper are the right ideals generated by the m_{λ}, as λ varies over the multicompositions of n. The cyclotomic q–Schur algebra will be built from endomorphisms between such right ideals.

(3.8) Definition Suppose that λ is a multicomposition of n. Let M^{λ} = m_{λ}ℋ .

We leave the proof of the following remarks to the reader.

(3.9) Remarks 1. If the multicomposition μ is obtained from λ by reordering the parts

in each component then M^{μ} ≅ M^{λ}.

2. Suppose that q = 1 and that Q_{k} = π^{k}, for k = 1, 2, …, r, where π is a primitive rth root of unity in R. Then ℋ ≅ R(C_{r} ≀ 𝔖_{n}). Let α_{k} = |λ^{(k)}| for 1 ≤ k ≤ r. Then M^{λ} is induced from a module U for the subgroup (C_{r}^{α_{1}} × C_{r}^{α_{2}} × ⋯ × C_{r}^{α_{r}}) ⋊ 𝔖_{λ}, where C_{r}^{α_{i}} = C_{r} × … × C_{r} (α_{i} factors). The restriction of U to the subgroup C_{r}^{α_{1}} × ⋯ × C_{r}^{α_{r}} has the form U_{1}^{⊗α_{1}} ⊗ ⋯ ⊗ U_{r}^{⊗α_{r}} where U_{k} has rank k for 1 ≤ k ≤ r. The restriction of U to 𝔖_{λ} is the trivial module.

We shall construct a basis of M^{λ}, and study ℋ –homomorphisms between the various modules M^{λ}. To this end, we introduce λ–tableaux.

The diagram of a composition α = (α_{1}, α_{2}, …) is { (i, j) | 1 ≤ i and 1 ≤ j ≤ α_{i} }, which we regard as an array of nodes, or boxes, in the plane. The diagram of a multicomposition is the ordered r–tuple of the diagrams of its components.

Let λ be a multicomposition of n. A λ–tableau 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}) is obtained from the diagram of λ by replacing each node by one of the integers 1, 2, …, n, allowing no repeats. We call the tableaux 𝔱^{(k)} the components of 𝔱.

(3.10) Definition 1. A λ–tableau is row standard if the entries in each row of each com-

ponent increase from left to right.

2. A λ–tableau 𝔱 is standard if λ is a multipartition of n, 𝔱 is row standard and the entries in each column of each component of 𝔱 increase from top to bottom.

3. If λ is a multipartition of n, then let Std(λ) be the set of standard λ–tableaux.

Note, particularly, that we define standard λ–tableaux only when λ is a multipartition whereas row standard tableaux are defined for all multicompositions.

We require partial orders on the set of multicompositions and on the set of row standard tableaux.

If 𝔱 is a row standard λ–tableau and 1 ≤ m ≤ n, then the entries 1, 2, …, m in 𝔱 occupy the diagram of a multicomposition; let 𝔱 ↓ m denote this multicomposition. For example, 𝔱 ↓ n = λ. We use this notation in our next definition.

(3.11) Definition Suppose that λ = (λ^{(1)}, …, λ^{(r)}) and μ = (μ^{(1)}, …, μ^{(r)}) are multicompositions of n.

1. We say that λ dominates μ, and write λ ⊵ μ, if

∑_{i=1}^{k−1} |λ^{(i)}| + ∑_{i=1}^{j} λ_{i}^{(k)} ≥ ∑_{i=1}^{k−1} |μ^{(i)}| + ∑_{i=1}^{j} μ_{i}^{(k)}

for all k and j with 1 ≤ k ≤ r and j ≥ 0. If λ ⊵ μ and λ ≠ μ then we write λ ▷ μ.

2. Suppose that 𝔰 is a row standard λ–tableau and that 𝔱 is a row standard μ–tableau. We say that 𝔰 dominates 𝔱, and write 𝔰 ⊵ 𝔱 if 𝔰 ↓ m ⊵ 𝔱 ↓ m for all m with 1 ≤ m ≤ n. If 𝔰 ⊵ 𝔱 and 𝔰 ≠ 𝔱 then we write 𝔰 ▷ 𝔱.

For example, if n = r = 2, then the multipartitions of 2 are ordered by ((2), (0)) ▷ ((1^{2}), (0)) ▷ ((1), (1)) ▷ ((0), (2)) ▷ ((0), (1^{2})).

Note that if 𝔰 is a row standard λ–tableau and 𝔱 is a row standard μ tableau such that 𝔰 ⊵ 𝔱 then λ ⊵ μ.

Our next definition gives another relation between tableaux.

(3.12) Definition Suppose that 𝔰 is a tableau and that 1 ≤ j ≤ n. We write comp_{𝔰}(j) = k if j appears in the kth component 𝔰^{(k)} of 𝔰.

Suppose that 𝔱 is another tableau. Then we write comp_{𝔰} = comp_{𝔱} if comp_{𝔰} (j) = comp_{𝔱}(j) for all j with 1 ≤ j ≤ n. We also write comp_{𝔰} ≥ comp_{𝔱} if comp_{𝔰}(j) ≥ comp_{𝔱}(j) for all 1 ≤ j ≤ n; and comp_{𝔰} > comp_{𝔱} if comp_{𝔰} ≥ comp_{𝔱} and comp_{𝔰} ≠ comp_{𝔱}.

(3.13) Example Let

𝔰 = ( \begin{matrix}1 & 4 \\ 5 & 6 \end{matrix} , \begin{matrix}3 & 8 & 9 \\ 2 & 7 \end{matrix} ), 𝔱 = ( \begin{matrix}2 & 3 \\ 4 & 7 \end{matrix} , \begin{matrix}1 & 6 & 8 \\ 5 & 9 \end{matrix} ) and 𝔲 = ( \begin{matrix}1 & 2 & 3 \\ 4 & 6 & 7 \end{matrix} , 5 9 8 ).

Then 𝔰 is row standard, and 𝔱 and 𝔲 are standard. We have 𝔲 ▷ 𝔰 and 𝔲 ▷ 𝔱 but 𝔰 and 𝔱 are incomparable in the dominance order. Also, comp_{𝔱} > comp_{𝔲}, but there are no other equations or inequalities between comp_{𝔰}, comp_{𝔱} and comp_{𝔲}.

Let λ be a multicomposition of n. The symmetric group 𝔖_{n} acts from the right on the set of λ–tableaux by permuting the entries in each tableau. Let 𝔱^{λ} be the λ–tableau where 1, 2, …, n appear in order along the rows of the first component, and then along the rows of the second component, and so on. The row stabilizer of 𝔱^{λ} is the Young subgroup 𝔖_{λ} of 𝔖_{n}. For example, if λ = ((3, 2), (1^{2}), (3)) then

𝔱^{λ} = ( \begin{matrix}1 & 2 & 3 \\ 4 & 5 \end{matrix} , 6 7 , 8 9 10 ).

For a row standard λ–tableau 𝔰, let d(𝔰) be the element of 𝔖_{n} such that 𝔰 = 𝔱^{λ}d(𝔰). Then d(𝔰) is a distinguished right coset representative of 𝔖_{λ} in 𝔖_{n} and we obtain, in this way, a correspondence between the set of row standard λ–tableaux and the set of right coset representatives of 𝔖_{λ} in 𝔖_{n}.

Recall the antiautomorphism ∗ of ℋ from (2.3).

(3.14) Definition Suppose that λ is a multicomposition of n and that 𝔰 and 𝔱 are row standard λ–tableaux. Let m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}.

Note that m_{λ} = m_{𝔱^{λ}}_{𝔱^{λ}}. Also, m_{𝔰𝔱}^{∗} = m_{𝔱𝔰} (cf. Remark 3.7).

One of the aims of this section is to show that the elements m_{𝔰𝔱}, as (𝔰, 𝔱) varies over the ordered pairs of standard tableaux of the same shape, give a cellular basis of ℋ . We shall also establish useful properties of the right ideals M^{λ}. Initially, though, we concentrate upon the two–sided ideal generated by m_{λ}.

(3.15) Lemma Suppose that λ is a multicomposition of n and that 𝔰 and 𝔱 are row standard λ–tableaux. Let h ∈ ℋ (𝔖_{n}). Then m_{𝔰𝔱}h is a linear combination of terms of the form m_{𝔰𝔳} where each 𝔳 is a row standard λ–tableau.

Proof: Suppose that w ∈ 𝔖_{n}. Then there exist y ∈ 𝔖_{λ} and a distinguished right coset representative d of 𝔖_{λ} in 𝔖_{n} such that w = yd and ℓ(w) = ℓ(y) + ℓ(d). Hence, x_{λ}T_{w} = x_{λ}T_{y}T_{d} = q^{ℓ(y)}x_{λ}T_{d}. If m_{λ} = u_{a}^{+}x_{λ} then

m_{𝔰𝔱^{λ}}T_{w} = T_{d(𝔰)}^{∗}u_{a}^{+}x_{λ}T_{w} = q^{ℓ(y)}T_{d(𝔰)}^{∗}u_{a}^{+}x_{λ}T_{d} = q^{ℓ(y)}m_{𝔰𝔳}

where the tableau 𝔳 = 𝔱^{λ}d is row standard.

Now, m_{𝔰𝔱}h = m_{𝔰𝔱^{λ}}T_{d(𝔱)}h, and this is a linear combination of terms of the form m_{𝔰𝔱^{λ}}T_{w} with w ∈ 𝔖_{n}. Therefore, the required result follows.

In order to apply a result of Murphy in Proposition 3.18 below, we require a combinatorial lemma which concerns the dominance order on row standard tableaux. A preliminary definition sets the scene.

(3.16) Definition We say that a tableau 𝔰 is of the initial kind (for λ) if comp_{𝔰} = comp_{𝔱^{λ}}.

Note that a μ–tableau 𝔰 can be of the initial kind for λ even though μ ≠ λ.

(3.17) Lemma Suppose that λ = (λ^{(1)}, …, λ^{(r)}) is a multicomposition of n, and let α = (|λ^{(1)}|, …, |λ^{(r)}|). Suppose that w is a distinguished right coset representative of 𝔖_{α} in 𝔖_{n} and that 𝔰 is a row standard λ–tableau of the initial kind. Then the following hold.

1. The tableau 𝔰w is row standard.

2. If 𝔰 is standard then 𝔰w is standard.

3. If 𝔱 is a row standard λ–tableau of the initial kind with 𝔰 ▷ 𝔱 then 𝔰w ▷ 𝔱w.

Proof: The fact that w is a distinguished right coset representative of 𝔖_{α} in 𝔖_{n} implies that whenever x and y are integers such that x and y are in the same component of 𝔰 and x < y, then xw < yw. Hence, (i) and (ii) are true.

Now assume that 𝔱 is a row standard λ–tableau of the initial kind, and that 𝔰 ▷ 𝔱. Then d(𝔰) ▷ d(𝔱) in the Bruhat–Chevalley order ▷ on 𝔖_{n}, by [4, (1.3)]. (Our notation for the order is such that 1 ⊵ v for all v ∈ 𝔖_{n}.) Since w is a distinguished right coset representative of 𝔖_{α} in 𝔖_{n}, we also have ℓ(d(𝔱)w) = ℓ(d(𝔱)) + ℓ(w). The well–known “cancellation” property of the Bruhat–Chevalley order now lets us conclude that d(𝔰)w ▷ d(𝔱)w. But 𝔰w and 𝔱w are row standard, so d(𝔰)w = d(𝔰w) and d(𝔱)w = d(𝔱w). By applying [4, (1.3)] again, we deduce that 𝔰w ▷ 𝔱w.

(3.18) Proposition Suppose that λ is a multicomposition of n and that 𝔰 and 𝔱 are row standard λ–tableaux. Then m_{𝔰𝔱} is a linear combination of terms of the form m_{𝔲𝔳} where 𝔲 and 𝔳 are standard μ–tableaux for some multipartition μ of n, and

1. 𝔲 ⊵ 𝔰 and comp_{𝔲} = comp_{𝔰}; and,

2. 𝔳 ⊵ 𝔱 and comp_{𝔳} = comp_{𝔱}.

Proof: When r = 1, this is a Theorem of Murphy [16, (4.18)]. We deduce the general case from this.

Let α = (|λ^{(1)}|, …, |λ^{(r)}|). We may write 𝔰 = 𝔰′w_{1} and 𝔱 = 𝔱′w_{2} where 𝔰′ and 𝔱′ are row standard λ–tableau of the initial kind, and w_{1} and w_{2} are distinguished right coset representatives for 𝔖_{α} in 𝔖_{n}.

Let a = (a_{1}, …, a_{r}) where a_{k} = ∑_{i=1}^{k−1} |λ^{(i)}|, as in the definition of m_{λ}. We have m_{𝔰𝔱} = T_{w_{1}}^{∗}m_{𝔰′𝔱′}T_{w_{2}} and m_{𝔰′𝔱′} = T_{d(𝔰′)}^{∗}u_{a}^{+}x_{λ}T_{d(𝔱′)} = u_{a}^{+}T_{d(𝔰′)}^{∗}x_{λ}T_{d(𝔱′)} since d(𝔰′) ∈ 𝔖_{α}.

We may write T_{d(𝔰′)}^{∗}x_{λ}T_{d(𝔱′)} as a product of r commuting terms, one for each component of λ; say, T_{d(𝔰′)}^{∗}x_{λ}T_{d(𝔱′)} = x_{1}x_{2} … x_{r}, where the kth term x_{k} involves only elements T_{w} with w ∈ 𝔖({a_{k} + 1, a_{k} + 2 …, a_{k+1}}). For example, x_{1} = T_{d(𝔰′_{1})}^{∗}x_{λ^{(1)}}T_{d(𝔱′_{1})} where 𝔰′_{1} is the first component of 𝔰′ and 𝔱′_{1} is the first component of 𝔱′. By applying Murphy’s result [16, (4.18)] to the Hecke algebra ℋ (𝔖_{α_{1}}) we may write x_{1} as a linear combination of terms T_{d(𝔲′_{1})}^{∗}x_{μ^{(1)}}T_{d(𝔳′_{1})} where 𝔲′_{1} and 𝔳′_{1} are standard μ^{(1)}–tableaux for some partition μ^{(1)} of a_{1}, and 𝔲′_{1} ⊵ 𝔰′_{1} and 𝔳′_{1} ⊵ 𝔱′_{1}.

We can apply the same technique for the other factors x_{2}, …, x_{r}, to conclude that T_{d(𝔰′)}^{∗}x_{λ}T_{d(𝔱′)} is a linear combination of terms of the form T_{d(𝔲′)}^{∗}x_{μ}T_{d(𝔳′)} where 𝔲′ and 𝔳′ are standard μ–tableaux for some multipartition μ of n, and 𝔲′ ⊵ 𝔰′ and 𝔳′ ⊵ 𝔱′. Also, 𝔲′ and 𝔳′ are of the initial kind, and |μ^{(k)}| = |λ^{(k)}| for 1 ≤ k ≤ r. Therefore, m_{𝔰′𝔱′} is a linear combination of terms m_{𝔲′𝔳′} where the sum runs over the same set of pairs (𝔲′, 𝔳′). Consequently, m_{𝔰𝔱} is a linear combination of terms T_{w_{1}}^{∗}m_{𝔲′𝔳′}T_{w_{2}}.

Now, T_{d(𝔳′)}T_{w_{2}} = T_{d(𝔳′)w_{2}} = T_{d(𝔳′w_{2})}, so T_{w_{1}}^{∗}m_{𝔲′𝔳′}T_{w_{2}} = T_{w_{1}}^{∗}m_{𝔲′𝔳} where 𝔳 = 𝔳′w_{2}. By Lemma 3.17, since 𝔳′ is of the initial kind, 𝔳 = 𝔳′w_{2} is standard, and 𝔳′w_{2} ⊵ 𝔱′w_{2}; that is, 𝔳 ⊵ 𝔱. Moreover, comp_{𝔳′w_{2}} = comp_{𝔱′w_{2}} since 𝔳′ and 𝔱′ are of the initial kind.

Similar remarks applied to T_{w_{1}}^{∗}T_{d(𝔲′)}^{∗} now complete the proof of the Proposition.

(3.19) Corollary Suppose that λ is a multicomposition of n and that 𝔰 and 𝔱 are row standard λ–tableaux. If h ∈ ℋ (𝔖_{n}) then m_{𝔰𝔱}h is a linear combination of terms of the form m_{𝔲𝔳} where 𝔲 and 𝔳 are standard μ–tableaux for some multipartition μ of n, and μ ⊵ λ, 𝔲 ⊵ 𝔰 and comp_{𝔲} = comp_{𝔰}.

Proof: Combine Lemma 3.15 and Proposition 3.18, and recall that 𝔲 ⊵ 𝔰 implies that μ ⊵ λ.

Corollary 3.19 provides the kind of information we need when we multiply m_{𝔰𝔱} by elements of ℋ (𝔖_{n}). More complicated is our next proposition, which shows what happens when we multiply m_{𝔰𝔱} by T_{0}. It is vital to the proposition that λ is a multipartition, not merely a multicomposition.

(3.20) Proposition Suppose that λ is a multipartition of n, and that 𝔰 and 𝔱 are standard λ–tableaux. Then m_{𝔰𝔱}T_{0} = x_{1} + x_{2} where

1. x_{1} is a linear combination of terms of the form m_{𝔲𝔳} where 𝔲 and 𝔳 are standard λ– tableaux, with 𝔲 ⊵ 𝔰 and comp_{𝔲} = comp_{𝔰}, and

2. x_{2} is a linear combination of terms of the form m_{𝔲𝔳} where 𝔲 and 𝔳 are standard μ– tableaux for some multipartition μ of n, with μ ▷ λ and comp_{𝔰} > comp_{𝔲}.

Proof: Let α = (|λ^{(1)}|, …, |λ^{(r)}|) and let a = (a_{1}, a_{2}, …, a_{r}) where a_{k} = ∑_{i=1}^{k−1} |λ^{(i)}| for 1 ≤ k ≤ r. Write d(𝔱) = yc with y ∈ 𝔖_{α} and c a distinguished right coset representative for 𝔖_{α} in 𝔖_{n}. Then the α–tableau 𝔱^{α}c is row standard. Assume that 1 is in row k of 𝔱^{α}c; thus, 1 ≤ k ≤ r.

Now, c = (a_{k}, a_{k} + 1)(a_{k} − 1, a_{k}) … (1, 2)c′, where ℓ(c) = a_{k} + ℓ(c′) and c′ fixes 1. Thus, T_{c} = T_{a_{k}}T_{a_{k−1}} … T_{1}T_{c′} and T_{c′}T_{0} = T_{0}T_{c′}. Let b = (a_{1}, …, a_{k} + 1, …, a_{r}). Then

u_{a}^{+}T_{c}T_{0} = u_{a}^{+}T_{a_{k}}T_{a_{k−1}} … T_{1}T_{0}T_{c′} = u_{a}^{+}h_{1} + u_{b}^{+}h_{2}

for some h_{1}, h_{2} ∈ 𝔖_{n} by Lemma 3.4. Since y ∈ 𝔖_{α} and 𝔱^{λ}y is standard, y fixes a_{k} + 1; therefore, T_{y} commutes with u_{b}^{+}. Hence, using Remark 3.7, we have

\begin{aligned}m_{𝔰𝔱}T_{0} & = T_{d(𝔰)}^{∗}u_{a}^{+}x_{λ}T_{d(𝔱)}T_{0} = T_{d(𝔰)}^{∗}x_{λ}u_{a}^{+}T_{y}T_{c}T_{0} = T_{d(𝔰)}^{∗}x_{λ}T_{y}u_{a}^{+}T_{c}T_{0} \\ & = T_{d(𝔰)}^{∗}x_{λ}T_{y}(u_{a}^{+}h_{1} + u_{b}^{+}h_{2}) = T_{d(𝔰)}^{∗}x_{λ}(u_{a}^{+}T_{y}h_{1} + u_{b}^{+}T_{y}h_{2}). \end{aligned}

The first term, T_{d(𝔰)}^{∗}x_{λ}u_{a}^{+}T_{y}h_{1}, is a linear combination of terms of the required form by Corollary 3.19. If k = 1 then u_{b}^{+} = 0 by Remark 3.2(i), and the proof is complete in this case. Assume therefore that k ≥ 2. We turn out attention to the term T_{d(𝔰)}^{∗}x_{λ}u_{b}^{+}T_{y}h_{2}. We now need to digress in order to prove that T_{d(𝔰)}^{∗}x_{λ}u_{b}^{+}T_{y}h_{2} has the required form.

Define the multicomposition ν = (ν^{(1)}, …, ν^{(r)}) of n by ν^{(i)} = λ^{(i)} for i ≠ k − 1, k, ν^{(k−1)} = (λ_{1}^{(k−1)}, λ_{2}^{(k−1)}, …, 1) and ν^{(k)} = (λ_{1}^{(k)} − 1, λ_{2}^{(k)}, …). (Thus we reduce the first part of the kth component of λ by 1, and adjoin a new part of size 1 to the end of the (k − 1)th component of λ.) Note that ν ▷ λ.

Let l = λ_{1}^{(k)} be the first part of λ^{(k)}. The entries in the first row of the kth component of 𝔱^{λ} are then a_{k} + 1, a_{k} + 2, …, a_{k} + l. Let G_{1} be the symmetric group on these numbers, and let G_{2} be the symmetric group on a_{k} + 2, …, a_{k} + l. Let c_{1}, c_{2}, …, c_{l} be the distinguished right coset representatives for G_{2} in G_{1}, ordered in terms of increasing length. Then the tableaux 𝔱^{ν}c_{i} are row standard, and they agree with 𝔱^{ν} except on the last row of the (k −1)th component and the first row of the kth component. The last row of the (k − 1)th component of 𝔱^{ν}c_{i} contains the single entry a_{k} + i.

We are given that 𝔰 = 𝔱^{λ}d(𝔰) is a row standard λ–tableau. Let 𝔲_{i} = 𝔱^{ν}c_{i}d(𝔰) for 1 ≤ i ≤ l. Each 𝔲_{i} agrees with 𝔰 except on the last row of the (k − 1)th component of 𝔲_{i} and on the first row of the kth component of 𝔲_{i}. Furthermore, since (a_{k} + 1)d(𝔰) < ⋯ < (a_{k} + l)d(𝔰), it follows that each 𝔲_{i} is row standard.

We also have comp_{𝔲_{i}}(j) = comp_{𝔰}(j) if j ∉ {(a_{k} + 1)d(𝔰), …, (a_{k} + l)d(𝔰)} and, for j ∈ {(a_{k} + 1)d(𝔰), …, (a_{k} + l)d(𝔰)}, we have comp_{𝔲_{i}}(j) = k − 1 or k and comp_{𝔰}(j) = k. Therefore, comp_{𝔰} > comp_{𝔲_{i}}.

Now x_{λ} has a factor

∑_{w∈G_{1}} T_{w} = ∑_{i=1}^{l} T_{c_{i}}^{∗} ∑_{w∈G_{2}} T_{w}.

Hence, x_{λ} = ∑_{i=1}^{l} T_{c_{i}}^{∗}x_{ν}. Note that m_{ν} = x_{ν}u_{b}^{+}. Thus,

T_{d(𝔰)}^{∗}x_{λ}u_{b}^{+} = T_{d(𝔰)}^{∗} ∑_{i=1}^{l} T_{c_{i}}^{∗}x_{ν}u_{b}^{+} = ∑_{i=1}^{l} T_{c_{i}d(𝔰)}^{∗}x_{ν}u_{b}^{+} = ∑_{i=1}^{l} m_{𝔲_{i}𝔱^{ν}}.

Hence, by Corollary 3.19, T_{d(𝔰)}^{∗}x_{λ}u_{b}^{+}T_{y}h_{2} is a linear combination of the required form. This completes the proof of the Proposition.

(3.21) Definition Suppose that λ is a multicomposition of n.

1. Let N^{λ} be the R–module spanned by

{ m_{𝔰𝔱} | 𝔰 and 𝔱 are standard μ–tableaux for some multipartition μ of n with μ ⊵ λ }.

2. Let \overline{N^{λ}} be the R–module spanned by

{ m_{𝔰𝔱} | 𝔰 and 𝔱 are standard μ–tableaux for some multipartition μ of n with μ ▷ λ }.

We now apply Propositions 3.18 and 3.20 to obtain a sequence of useful results.

(3.22) Proposition Suppose that λ is a multicomposition of n. Then N^{λ} and \overline{N^{λ}} are two– sided ideals of ℋ .

Proof: Corollary 3.19 shows that N^{λ} is closed under postmultiplication by T_{1}, T_{2}, …, T_{n−1} and Proposition 3.20 shows that N^{λ} is also closed under postmultiplication by T_{0}. Because ℋ is generated by T_{0}, T_{1}, …, T_{n−1}, we deduce that N^{λ} is a right ideal of ℋ . Since m_{𝔰𝔱}^{∗} = m_{𝔱𝔰}, by applying the antiautomorphism ∗ we see that N^{λ} is also a left ideal of ℋ .

Finally, \overline{N^{λ}} is a two sided ideal of ℋ since \overline{N^{λ}} = ∑_{μ▷λ} N^{μ}.

A proof very similar to that of Proposition 3.22 gives our next result.

(3.23) Proposition Suppose that λ is a multicomposition of n and that 𝔰 is a row standard λ–tableau. Let I_{𝔰} be the R–module spanned by

{ m_{𝔲𝔳} | 𝔲 and 𝔳 are standard μ–tableaux for some multipartition μ of n with μ ⊵ λ and comp_{𝔰} ≥ comp_{𝔲} }.

Then I_{𝔰} is a right ideal of ℋ .

Recall that M^{λ} = m_{λ}ℋ .

(3.24) Corollary Suppose that λ is a multicomposition of n. Then every element of M^{λ} is a linear combination of terms of the form m_{𝔲𝔳} where 𝔲 and 𝔳 are standard μ–tableaux for some multipartition μ of n with μ ⊵ λ and comp_{𝔱^{λ}} ≥ comp_{𝔲}.

Proof: Since m_{λ} = m_{𝔱^{λ}}_{𝔱^{λ}}, Proposition 3.18 shows that m_{λ} ∈ I_{𝔱^{λ}} (note that λ may not be a multipartition). Therefore, M^{λ} ⊆ I_{𝔱^{λ}} and the desired result follows by Proposition 3.23.

(3.25) Proposition Suppose that λ is a multipartition of n and that 𝔱 is a standard λ– tableau. Let h ∈ ℋ . Then for every standard λ–tableau 𝔳 there exists r_{𝔳} ∈ R such that, for all standard λ–tableau 𝔰, we have

m_{𝔰𝔱}h ≡ ∑_{𝔳∈Std(λ)}r_{𝔳}m_{𝔰𝔳} mod \overline{N^{λ}}.

Proof: Let U be the R–module spanned by { m_{𝔱^{λ}𝔱′} | 𝔱′ is a standard λ–tableau }. Note that 𝔱^{λ} is the unique λ–tableau 𝔲 such that 𝔲 ⊵ 𝔱^{λ}. Therefore, U + \overline{N^{λ}} is closed under postmultiplication by T_{1}, T_{2}, …, T_{n−1} by Proposition 3.18, and it is closed under postmultiplication by T_{0} by Proposition 3.20. Hence U + \overline{N^{λ}} is a right ideal of ℋ . Thus, for each 𝔳 ∈ Std(λ) there exists r_{𝔳} ∈ R such that

m_{𝔱^{λ}𝔱}h ≡ ∑_{𝔳∈Std(λ)}r_{𝔳}m_{𝔱^{λ}𝔳} mod \overline{N^{λ}}.

Multiply this congruence on the left by T_{d(𝔰)}^{∗}, and use the fact that \overline{N^{λ}} is an ideal, to obtain the congruence of the proposition.

(3.26) Theorem ( [13, (1.7)]) The algebra ℋ is a free R–module with basis

ℳ = { m_{𝔰𝔱} | 𝔰 and 𝔱 are standard λ–tableaux for some multipartition λ of n }.

Moreover, ℳ is a cellular basis of ℋ .

Proof: Since m_{λ} = 1 when λ = ((0), …, (0), (1^{n})), Proposition 3.25 shows that ℳ spans ℋ . By Theorem 2.2, ℋ is free of rank r^{n}n! Since this is also the cardinality of ℳ, we deduce that ℳ is a basis of ℋ . Finally, the properties that ℳ must satisfy in order to be a cellular basis of ℋ (as given in [12, (1.1)]), are covered by Proposition 3.25 and the fact that m_{𝔰𝔱}^{∗} = m_{𝔱𝔰}.

(3.27) Definition We call ℳ the standard basis of ℋ .

We can now apply Graham and Lehrer’s theory of cellular algebras [12] to describe the representation theory of ℋ .

(3.28) Definition Suppose that λ is a multipartition of n. Let z_{λ} = (\overline{N^{λ}} + m_{λ})/\overline{N^{λ}}. The Specht module S^{λ} is the submodule of ℋ /\overline{N^{λ}} given by S^{λ} = z_{λ}ℋ .

Directly from Proposition 3.25 and Theorem 3.26, S^{λ} is a free R–module with basis { z_{λ}T_{d(𝔱)} | 𝔱 a standard λ–tableau }.

Define a bilinear form ⟨ , ⟩ on the Specht module S^{λ} by

m_{λ}T_{d(𝔰)}T_{d(𝔱)}^{∗}m_{λ} ≡ ⟨z_{λ}T_{d(𝔰)}, z_{λ}T_{d(𝔱)}⟩m_{λ} mod \overline{N^{λ}}.

By Proposition 3.25 (and the version of Proposition 3.25 obtained by applying the antiautomorphism ∗), the bilinear form is well defined (cf. [12]). Moreover, the bilinear form is symmetric and ⟨uh, v⟩ = ⟨u, vh^{∗}⟩ for all u, v ∈ S^{λ} and all h ∈ ℋ [12, (2.4)]. Consequently, rad S^{λ}, the radical of the bilinear form, is an ℋ –module.

(3.29) Definition Suppose that λ is a multipartition of n. Let D^{λ} = S^{λ}/ rad S^{λ}.

(3.30) Theorem Suppose that R is a field. Then the non–zero ℋ modules in

{ D^{λ} | λ a multipartition of n }

form a complete set of non–isomorphic irreducible ℋ –modules. Moreover, each irreducible module D^{λ} is absolutely irreducible.

Proof: Since ℳ is a cellular basis of ℋ the Theorem follows from the general theory of cellular algebras [12, (3.4)].

The theory in [12, (3.6)] also shows that if D^{μ} ≠ 0 and D^{μ} is a composition factor of S^{λ} then λ ⊵ μ.

When r = 1 the partitions μ for which D^{μ} ≠ 0 have been classified in [4, (7.6)]. When r > 1, if q = 1, or if the parameters Q_{k} are powers of q, the multipartitions μ for which D^{μ} ≠ 0 are classified by the results of [12, (5.9)], [13, (1.17)] and [14, (3.7)]; in type B, see also [8, (5.3)] and [10, (8.19)].

4 A Specht series for M^{μ}

In the last section, we used various elements m_{μ} of ℋ to construct a cellular basis of ℋ . We saw that the right ideal M^{μ} generated by m_{μ} is an analogue of an induced module; indeed, if r = 1 and q = 1 then M^{μ} is the permutation module of 𝔖_{n} on the Young subgroup 𝔖_{μ}. We shall use the right ideals M^{μ} of ℋ to construct the cyclotomic q–Schur algebra, but, before that, we study the individual modules M^{μ} in more detail. In particular, we shall give a semistandard basis of M^{μ} and construct a Specht series for M^{μ}.

We are going to define a new kind of λ–tableau. This will have entries which are ordered pairs (i, k) where i is a positive integer and 1 ≤ k ≤ r. We denote such tableaux by capital letters.

(4.1) Definition Suppose that λ is a multipartition of n and that μ is a multicomposition of n. A λ–tableau S has type μ if its entries are ordered pairs (i, k), as above, and for all m and k the number of times (i, k) is an entry in S is μ_{i}^{(k)}.

Next, we introduce a function which converts standard λ–tableaux (or, indeed any tableau whose entries are 1, 2, …, n) into a λ–tableau of type μ.

(4.2) Definition Suppose that λ is a multipartition of n and that μ is a multicomposition of n. Let 𝔰 be a standard λ–tableau. Define μ(𝔰) to be the λ–tableau obtained from 𝔰 by replacing each entry m in 𝔰 by (i, k) if m is in row i of the kth component of 𝔱^{μ}.

In our examples, we shall always write i_{k} for the ordered pair (i, k).

(4.3) Example Suppose that μ = ((3, 1), (2), (2, 1^{2})). Then

𝔱^{μ} = ( 1 2 3 4 , 5 6 , 7 8 9 10 ) and μ(𝔱^{μ}) = ( 1_{1} 1_{1} 1_{1} 2_{1} , 1_{2} 1_{2} , 1_{3} 1_{3} 2_{3} 3_{3} ).

Suppose that

𝔰_{1} = ( 1 2 3 5 4 , 6 10 , 7 8 9 ), 𝔰_{2} = ( 1 2 3 6 4 , 5 10 , 7 8 9 )

and

𝔰_{3} = ( 1 2 5 9 7 , 3 10 , 4 6 8 ).

Then

μ(𝔰_{1}) = μ(𝔰_{2}) = ( 1_{1} 1_{1} 1_{1} 1_{2} 2_{1} , 1_{2} 3_{3} , 1_{3} 1_{3} 2_{3} )

and

μ(𝔰_{3}) = ( 1_{1} 1_{1} 1_{2} 2_{3} 1_{3} , 1_{1} 3_{3} , 2_{1} 1_{2} 1_{3} ).

Given two pairs (i_{1}, k_{1}) and (i_{2}, k_{2}), we write (i_{1}, k_{1}) < (i_{2}, k_{2}) if k_{1} < k_{2}, or k_{1} = k_{2} and i_{1} < i_{2}. Note that if 𝔰 is a standard λ–tableau, then μ(𝔰) is a λ–tableau of type μ whose entries are weakly increasing along rows and down columns. We next single out some of these tableaux μ(𝔰).

(4.4) Definition Suppose that λ is a multipartition of n and that μ is a multicomposition of n. Let S = (S^{(1)}, …, S^{(r)}) be a λ–tableau of type μ. Then S is semistandard if

1. the entries in each row of each component S^{(k)} of S are non–decreasing; and

2. the entries in each column of each component S^{(k)} of S are strictly increasing; and 3. for each k with 1 ≤ k ≤ r, no entry in S^{(k)} has the form (i, l) with l < k. Let T_{0}(λ, μ) denote the set of semistandard λ–tableaux of type μ.

For example, μ(𝔰_{1}) in Example 4.3 is semistandard, but μ(𝔰_{3}) is not, since the third part of Definition 4.4 is not satisfied.

The point of the third condition in Definition 4.4 is the following result (cf. Corollary 3.24).

(4.5) Suppose that 𝔰 is a standard λ–tableau. Then μ(𝔰) satisfies Definition 4.4(iii) if and only if comp_{𝔱^{μ}} ≥ comp_{𝔰}

If λ is a multipartition and S ∈ T_{0}(λ, μ), then it is clear that there exists a standard λ–tableau 𝔰 with S = μ(𝔰). In order to say more about this, we introduce another definition.

(4.6) Definition If 𝔰 is a standard λ–tableau and 1 ≤ m ≤ n then let row_{𝔰}(m) = (i, k) if m belongs to row i of the kth component of 𝔰.

The definition of row_{𝔰} allows us to recover 𝔰 from the function row_{𝔰}

Suppose that S ∈ T_{0}(λ, μ) and let 𝔰 be a standard λ–tableau such that μ(𝔰) = S. If there exists an integer i such that row_{𝔰}(i) ≠ row_{𝔰}(i+1) and i and i+1 are in the same row of 𝔱^{μ} then the tableau 𝔰_{1} = 𝔰(i, i + 1) is standard and μ(𝔰_{1}) = S. Also, 𝔰 ▷ 𝔰_{1} if row_{𝔰}(i) < row_{𝔰}(i + 1), and 𝔰_{1} ▷ 𝔰 if row_{𝔰}(i) > row_{𝔰}(i + 1). Hence we have shown the following.

(4.7) Suppose that S ∈ T_{0}(λ, μ). Then there exist standard λ–tableaux first(S) and last(S) such that

1. μ(first(S)) = μ(last(S)) = S; and,

2. if 𝔰 is any standard λ–tableau such that μ(𝔰) = S then first(S) ⊵ 𝔰 ⊵ last(S).

(4.8) Definition Suppose that λ is a multipartition of n and that μ is a multicomposition of n. Let S ∈ T_{0}(λ, μ) and 𝔱 ∈ Std(λ). Then m_{S𝔱} is the element of ℋ given by

m_{S𝔱} = ∑_{μ(𝔰)=S^{𝔰∈Std(λ)}}m_{𝔰𝔱}.

Note that m_{S𝔱} is a sum of standard basis elements m_{𝔰𝔱} where first(S) ⊵ 𝔰 ⊵ last(S).

(4.9) Example Suppose that λ = ((4, 3), (2, 1), (2, 1)), μ = ((3^{2}, 1), (1^{2}), (2, 1^{2})) and

S = ( \begin{matrix}1_{1} & 1_{1} & 1_{1} & 2_{1} \\ 2_{1} & 2_{1} & 3_{1} \end{matrix} , 1_{2} 3_{3} 2_{2} , 1_{3} 1_{3} 2_{3} ).

Then S ∈ T_{0}(λ, μ) and for any 𝔱 ∈ Std(λ) we have m_{S𝔱} = m_{𝔰_{1}𝔱} + m_{𝔰_{2}𝔱} + m_{𝔰_{3}𝔱} where

𝔰_{1} = ( \begin{matrix}1 & 2 & 3 & 4 \\ 5 & 6 & 7 \end{matrix} , 8 13 9 , 10 11 12 ), 𝔰_{2} = ( \begin{matrix}1 & 2 & 3 & 5 \\ 4 & 6 & 7 \end{matrix} , 8 13 9 , 10 11 12 )

and

𝔰_{3} = ( \begin{matrix}1 & 2 & 3 & 6 \\ 4 & 5 & 7 \end{matrix} , 8 13 9 , 10 11 12 ).

Here 𝔰_{1} = first(S) and 𝔰_{3} = last(S).

(4.10) Lemma Suppose that λ is a multipartition of n and that μ is a multicomposition of n. Let S ∈ T_{0}(λ, μ) and 𝔱 ∈ Std(λ). Then m_{S𝔱} ∈ M^{μ}.

Proof: Let a = (a_{1}, …, a_{r}) where a_{k} = ∑_{i=1}^{k−1} |μ^{(i)}|, and b = (b_{1}, …, b_{r}) where b_{k} = ∑_{i=1}^{k−1} |λ^{(i)}|.

Let 𝔰_{1} = first(S) and let d = d(𝔰_{1}). Then, as in [4],

∑_{μ(𝔰)=S^{𝔰∈Std(λ)}}x_{λ}T_{d(𝔰)} = ∑_{w∈𝔖_{λ}d𝔖_{μ}} T_{w} = hT_{d}x_{μ},

where h = ∑ T_{v}, the sum running over certain elements v ∈ 𝔖_{λ}. Since h ∈ ℋ (𝔖_{λ}), we have h^{∗}u_{b}^{+} = u_{b}^{+}h^{∗}; so we obtain

m_{S𝔱} = ∑_{μ(𝔰)=S^{𝔰∈Std(λ)}}m_{𝔰𝔱} = ∑_{μ(𝔰)=S^{𝔰∈Std(λ)}}T_{d(𝔰)}^{∗}x_{λ}u_{b}^{+}T_{d(𝔱)} = x_{μ}T_{d}^{∗}u_{b}^{+}h^{∗}T_{d(𝔱)}.

Therefore, since m_{μ} = x_{μ}u_{a}^{+}, it is sufficient to prove that T_{d}^{∗}u_{b}^{+} ∈ u_{a}^{+}ℋ .

Assume that 1 ≤ k ≤ r. Recall that 𝔰_{1} = first(S), a_{k} = ∑_{i=1}^{k−1} |μ^{(i)}| and b_{k} = ∑_{i=1}^{k−1} |λ^{(i)}| and note that a_{k} ≤ b_{k} since S is semistandard. Define 𝔱_{k} to be the standard λ–tableau such that 1, 2, …, a_{k} occupy the same positions in 𝔰_{1} and 𝔱_{k}; b_{k} + 1, …, n occupy the same positions in 𝔱^{λ} and 𝔱_{k}; and that for a_{k} + 1 ≤ i < j ≤ b_{k} we have row_{𝔱_{k}}(i) ≤ row_{𝔱_{k}}(j) — thus the numbers a_{k}+1, …, b_{k} are inserted into 𝔱_{k} in “row order”. The fact that S is semistandard ensures that 𝔱_{k} is well defined. Note that 𝔱_{1} = 𝔱^{λ} and 𝔱_{r} = 𝔰_{1}.

Now define w_{k} ∈ 𝔖_{n} inductively by requiring that 𝔱_{k} = 𝔱^{λ}w_{1}w_{2}...w_{k}. Then w_{1} = 1 and d = w_{1}w_{2}...w_{r}; moreover, ℓ(d) = ℓ(w_{1}) + ... + ℓ(w_{r}). By construction, the element w_{k} fixes each of the numbers 1, 2, …, a_{k−1} and it also fixes each of b_{k} + 1, b_{k} + 2, …, n.

For 1 ≤ k ≤ r, let u_{a,k} and u_{b,k} be as defined in Definition 3.1, and let v_{k} = ∏_{m=a_{k}+1}^{b_{k}}(L_{m}− Q_{k}). Then u_{b,k} = u_{a,k}v_{k}.

Now, T_{w_{k}} commutes with u_{a,l} for l < k (since w_{k} fixes 1, 2, …, a_{k−1}), and T_{w_{k}} commutes with u_{b,l} for l ≥ k (since w_{k} fixes b_{k} + 1, b_{k} + 2, …, n); see (2.1)(iv). Therefore,

T_{d}^{∗}u_{b}^{+} =  T_{w_{r}}^{∗} … T_{w_{1}}^{∗}u_{b,1}u_{b,2} … u_{b,r}
=  T_{w_{r}}^{∗} … T_{w_{2}}^{∗}u_{b,1}u_{b,2} … u_{b,r}T_{w_{1}}^{∗}
=  u_{a,1}T_{w_{r}}^{∗} … T_{w_{2}}^{∗}u_{b,2} … u_{b,r}v_{1}T_{w_{1}}^{∗}
=  u_{a,1}u_{a,2}T_{w_{r}}^{∗} … T_{w_{3}}^{∗}u_{b,3} … u_{b,r}v_{2}T_{w_{2}}^{∗}v_{1}T_{w_{1}}^{∗}
⋮  ⋮
= u_{a,1} … u_{a,r}v_{r}T_{w_{r}}^{∗}v_{r−1}T_{w_{r−1}}^{∗} … v_{2}T_{w_{2}}^{∗}v_{1}T_{w_{1}}^{∗}.

Thus, T_{d}^{∗}u_{b}^{+} ∈ u_{a}^{+}ℋ , as required.

Having shown that m_{S𝔱} ∈ M^{μ}, our next aim is to prove that the set of elements of the form m_{S𝔱} give a basis of M^{μ}.

Let a = (a_{1}, …, a_{r}) where a_{k} = ∑_{i=1}^{k−1} |μ^{(i)}| as in the previous proof; then m_{μ} = u_{a}^{+}x_{μ}. Since u_{a}^{+} and x_{μ} commute, we have M^{μ} = m_{μ}ℋ ⊆ u_{a}^{+}ℋ ∩ x_{μ}ℋ . We shall show that we have equality here.

We have seen that ℋ has a standard basis which consists of elements m_{𝔰𝔱} with 𝔰 and 𝔱 standard λ–tableaux for some multipartition λ of n. Suppose that h ∈ ℋ and let h = ∑_{𝔰,𝔱} r_{𝔰𝔱}m_{𝔰𝔱}, with r_{𝔰𝔱} ∈ R, be the unique expression for h in terms of the standard basis. We say that m_{𝔰𝔱} is involved in h if r_{𝔰𝔱} ≠ 0.

(4.11) Lemma Suppose that μ is a multipartition of n and that m_{μ} = u_{a}^{+}x_{μ}. Let h ∈ x_{μ}ℋ ∩ u_{a}^{+}ℋ and suppose that (𝔰, 𝔱) is a pair of standard tableaux of the same shape such that

1. m_{𝔰𝔱} is involved in h; and,

2. if (𝔲, 𝔳) is a pair of standard tableaux of the same shape such that 𝔰 ⊵ 𝔲, 𝔱 ⊵ 𝔳 and

(𝔰, 𝔱) ≠ (𝔲, 𝔳) then m_{𝔲𝔳} is not involved in h. Let S = μ(𝔰). Then S is semistandard and 𝔰 = last(S).

Proof: Since 𝔰 is standard, the entries in S are non–decreasing down rows and columns.

Suppose that i and i + 1 belong to the same row of 𝔱^{μ}. Then T_{i}x_{μ} = qx_{μ}, so T_{i}h = qh. Therefore, by [16, (4.19)], i and i + 1 do not belong to the same column of 𝔰. Hence the entries in S are strictly increasing down columns.

Since h ∈ u_{a}^{+}ℋ we have comp_{𝔱^{μ}} ≥ comp_{𝔰} by Corollary 3.24 (applied to the multicomposition ((1^{α_{1}}), (1^{α_{2}}), …, (1^{α_{r}})), where α = (|μ^{(1)}|, …, |μ^{(r)}|)). Hence, by (4.5), S is semistandard.

Finally, suppose that 𝔰 ≠ last(S). Then there exist integers i and i + 1 in the same row of 𝔱^{μ} such that the tableau 𝔰′ = 𝔰(i, i + 1) is standard and 𝔰 ▷ 𝔰′. As in [16, (4.19)], m_{𝔰′𝔱} is involved in T_{i}h = qh, contradicting part (ii) of our hypothesis. Therefore, 𝔰 = last(S).

(4.12) Corollary Suppose that μ is a multicomposition of n and m_{μ} = u_{a}^{+}x_{μ}. Then u_{a}^{+}ℋ ∩ x_{μ}ℋ is spanned by

{ m_{S𝔱} | S ∈ T_{0}(λ, μ) and 𝔱 ∈ Std(λ) for some multipartition λ of n }.

Proof: Note that m_{S𝔱} ∈ M^{μ} by Lemma 4.10.

For each multipartition λ of n let

T_{l}(λ, μ) = { 𝔰 ∈ Std(λ) | μ(𝔰) = last(S) for some S ∈ T_{0}(λ, μ) }.

By Lemma 4.11 every non–zero element of u_{a}^{+}ℋ ∩ x_{μ}ℋ involves a standard basis element m_{𝔰𝔱} where 𝔰 ∈ T_{l}(λ, μ) and 𝔱 ∈ Std(λ) for some multipartition λ.

Now suppose that h ∈ u_{a}^{+}ℋ ∩ x_{μ}ℋ and write h in terms of the standard basis; say, h = ∑ r_{𝔲𝔳}m_{𝔲𝔳} with r_{𝔲𝔳} ∈ R. Let

h′ = h − ∑_{λ} ∑_{𝔰∈T_{l}(λ,μ)} ∑_{𝔱∈Std(λ)}r_{𝔰𝔱}m_{μ(𝔰)𝔱}.

Then h′ ∈ u_{a}^{+}ℋ ∩x_{μ}ℋ , but h′ does not involve any term m_{𝔰𝔱} for any 𝔰 ∈ T_{l}(λ, μ). Therefore, h′ = 0, and we have obtained an expression for h as a linear combination of the required form.

(4.13) Corollary Suppose that μ is a multicomposition of n and m_{μ} = u_{a}^{+}x_{μ}. Then M^{μ} = u_{a}^{+}ℋ ∩ x_{μ}ℋ .

Proof: We have that M^{μ} ⊆ u_{a}^{+}ℋ ∩ x_{μ}ℋ . The inclusion u_{a}^{+}ℋ ∩ x_{μ}ℋ ⊆ M^{μ} follows from Corollary 4.12 and Lemma 4.10

The next theorem for Iwahori–Hecke algebras of type A (that is, the case r = 1), is due to Murphy [16]; for Iwahori–Hecke algebras of type B (that is, the case r = 2), the result is due to Du and Scott [11]

(4.14) Theorem Suppose that μ is a multicomposition of n. Then M^{μ} is free as an R– module with basis

{ m_{S𝔱} | S ∈ T_{0}(λ, μ) and 𝔱 ∈ Std(λ) for some multipartition λ of n }.

Proof: By Lemma 4.10 each element m_{S𝔱} belongs to M^{μ}. Since distinct elements m_{S𝔱} involve distinct standard basis elements m_{𝔰𝔱}, the elements m_{S𝔱} are linearly independent. Finally, Corollaries 4.12 and 4.13 show that the elements m_{S𝔱} span M^{μ}.

(4.15) Corollary Suppose that μ is a multicomposition of n. Then there exists a filtration of M^{μ},

M^{μ} = M_{1} > M_{2} > ⋯ > M_{k+1} = 0

such that for each i with 1 ≤ i ≤ k there exists a multipartition λ_{i} of n with M_{i}/M_{i+1} ≅ S^{λ_{i}}. Moreover, if λ is a multipartition of n, then the number of i such that λ = λ_{i} is equal to the number of semistandard λ–tableaux of type μ.

Proof: Choose an ordering S_{1} > S_{2} > ⋯ > S_{k} on the set of semistandard tableaux of type μ such that j > i if λ_{i} ▷ λ_{j} where S_{i} ∈ T_{0}(λ_{i}, μ) and S_{j} ∈ T_{0}(λ_{j}, μ). Let M_{i} be the R–submodule of M^{μ} with basis { m_{S_{j}𝔱} | j ≥ i and 𝔱 ∈ Std(λ_{j}) }. Then

M^{μ} = M_{1} > M_{2} > ⋯ > M_{k+1} = 0,

and, by Proposition 3.25 and Theorem 4.14, each M_{i} is a right ideal of ℋ .

Suppose that 1 ≤ i ≤ k. Then M_{i} ∩ \overline{N^{λ_{i}}} ⊆ M_{i+1}. Hence, there is a well defined ℋ – homomorphism θ from S^{λ_{i}} onto M_{i}/M_{i+1} such that θ(z_{λ_{i}}) = m_{S_{i}𝔱^{λ_{i}}} + M_{i+1}. Since both S^{λ_{i}} and M_{i}/M_{i+1} have rank equal to the number of standard λ_{i}–tableaux, θ is an isomorphism. The Corollary now follows.

5 The double annihilator of m_{μ}

The purpose of this section is to compute the double annihilator of the element m_{μ} of ℋ for any multicomposition μ of n. This will enable us to calculate a basis for Hom_{ℋ} (M^{ν}, M^{μ}) in the next section.

(5.1) Definition Suppose S is a subset of ℋ and define r(S) = { h ∈ ℋ | Sh = 0 } and l(S) = { h ∈ ℋ | hS = 0 }. The double annihilator of S is lr(S) = l(r(S)).

It is elementary that lr(S) contains the left ideal of ℋ generated by S. If ℋ is a quasi– Frobenius algebra, then lr(S) is equal to the left ideal of ℋ generated by S [3, (61.2)]. Although ℋ is quasi–Frobenius for r ≤ 2 (see, for example, [4, section 2]), we do not know whether it is quasi–Frobenius for r > 2. If ℋ is quasi–Frobenius for all r, then the main result of this section, namely Theorem 5.16, is immediate.

A connection between double annihilators and homomorphisms is provided by the following easy lemma.

(5.2) Lemma Suppose that m ∈ ℋ and that lr(m) = ℋ m. Let I be a right ideal of ℋ . 1. For all φ ∈ Hom_{ℋ} (mℋ , I) there exists h_{φ} ∈ ℋ such that φ(m) = h_{φ}m.

2. Hom_{ℋ} (mℋ , I) ≅ ℋ m ∩ I.

Proof: For all x ∈ r(m) we have φ(m)x = φ(mx) = 0. Therefore, φ(m) ∈ lr(m) = ℋ m; so φ(m) = h_{φ}m for some h_{φ} ∈ ℋ . The proof of part (ii) is now straightforward.

The main result of this section is that lr(m_{μ}) = ℋ m_{μ}. Because the Iwahori–Hecke algebra ℋ (𝔖_{n}) is quasi–Frobenius, our first step is easy.

(5.3) Lemma Suppose that μ is a multicomposition of n. Then lr(x_{μ}) = ℋ x_{μ}.

Proof: Since ℋ (𝔖_{n}) is a quasi–Frobenius algebra, we have lr(x_{μ})∩ℋ (𝔖_{n}) = ℋ (𝔖_{n})x_{μ}. Now suppose that z ∈ lr(x_{μ}). By Theorem 2.2, we may write

z = ∑_{c∈𝒞}L_{1}^{c_{1}}, … L_{n}^{c_{n}}h_{c}

where 𝒞 = { c = (c_{1}, …, c_{r}) | 0 ≤ c_{j} < r for 1 ≤ j ≤ n } and h_{c} ∈ ℋ (𝔖_{n}). Then, for all y ∈ r(x_{μ}) ∩ ℋ (𝔖_{n}), we have

0 = zy = ∑_{c∈𝒞}L_{1}^{c_{1}}, … L_{n}^{c_{n}}h_{c}y.

Therefore, by Theorem 2.2, we have h_{c}y = 0 for all c ∈ 𝒞. Thus, h_{c} ∈ lr(x_{μ}) ∩ ℋ (𝔖_{n}) = ℋ (𝔖_{n})x_{μ}, and so z ∈ ℋ x_{μ}.

For the remainder of this section we fix a multicomposition μ and define a = (a_{1}, …, a_{r}) by a_{k} = ∑_{i=1}^{k−1} |μ^{(i)}|; so 0 = a_{1} ≤ a_{2} ≤ … ≤ a_{r} ≤ n. Our main task in this section is to prove that lr(u_{a}^{+}) = ℋ u_{a}^{+}. To achieve this goal, we use the next result.

(5.4) Lemma Assume that I is a left ideal of ℋ and suppose that S is a subset of r(I) such that l(S) ⊆ I. Then lr(I) = I.

Proof: By assumption S ⊆ r(I) and l(S) ⊆ I. Also, I ⊆ lr(I). Therefore, lr(I) ⊆ l(S) ⊆ I ⊆ lr(I); so lr(I) = I as required.

Our definition of u_{a}^{+} expresses u_{a}^{+} as a double product

u_{a}^{+} = ∏_{k=1}^{r} ∏_{m=1}^{a_{k}}(L_{m} − Q_{k}).

We now introduce notation which allows us to reverse the order of the multiplication here.

(5.5) Definition Suppose that 1 ≤ i ≤ n.

1. Let γ_{i} = k if k is maximal such that a_{k} < i. Let γ = (γ_{1}, γ_{2}, …, γ_{n}).

2. Let v_{i} = ∏_{k=γ_{i}+1}^{r}(L_{i} − Q_{k}).

Then u_{a}^{+} = v_{1}v_{2} … v_{n} and all of these factors commute.

(5.6) Definition Suppose that 1 ≤ i ≤ n. Let y_{i} = T_{i−1} … T_{2}T_{1} ∏_{k=1}^{γ_{i}}(L_{1} − Q_{k}).

(5.7) Example Suppose that r = 4, n = 5 and a = (0, 2, 4, 5). Then

u_{a}^{+} = (L_{1} − Q_{2})(L_{2} − Q_{2})
× (L_{1} − Q_{3})(L_{2} − Q_{3})(L_{3} − Q_{3})(L_{4} − Q_{3})
× (L_{1} − Q_{4})(L_{2} − Q_{4})(L_{3} − Q_{4})(L_{4} − Q_{4})(L_{5} − Q_{4}).

Also γ = (1, 1, 2, 2, 3) and, for 1 ≤ i ≤ n, the product of the factors in the ith column of the above array is v_{i}. We have

y_{1} =  (L_{1} − Q_{1})
y_{2} =  T_{1}(L_{1} − Q_{1})
y_{3} =  T_{2}T_{1}(L_{1} − Q_{1})(L_{1} − Q_{2})
y_{4} = T_{3}T_{2}T_{1}(L_{1} − Q_{1})(L_{1} − Q_{2})
y_{5} = T_{4}T_{3}T_{2}T_{1}(L_{1} − Q_{1})(L_{1} − Q_{2})(L_{1} − Q_{3}).

(5.8) Lemma Suppose that 1 ≤ i ≤ n. Then v_{1}v_{2} … v_{i}y_{i} = 0.

Proof: Assume that k satisfies γ_{i} + 1 ≤ k ≤ r. Then v_{1}v_{2} … v_{i} has a factor (L_{1} − Q_{k})(L_{2} − Q_{k}) … (L_{i} − Q_{k}). By (2.1)(iv), this factor commutes with T_{i−1} … T_{2}T_{1}. Hence, v_{1}v_{2} … v_{i}T_{i−1} … T_{2}T_{1} has a right factor ∏_{k=γ_{i}+1}^{r}(L_{1} − Q_{k}). However,

∏_{k=1}^{r}(L_{1} − Q_{k}) = ∏_{k=1}^{r}(T_{0} − Q_{k}) = 0;

so it follows that v_{1}v_{2} … v_{i}y_{i} = 0 as claimed.

Since u_{a}^{+} = v_{1}v_{2} … v_{n} we have the following Corollary.

(5.9) Corollary Suppose that 1 ≤ i ≤ n. Then u_{a}^{+}y_{i} = 0.

It turns out that y_{1}, y_{2}, …, y_{n} generate r(u_{a}^{+}). More importantly, together with Lemma 5.4 the elements y_{i} will allow us to prove that lr(u_{a}^{+}) = ℋ u_{a}^{+}. For our proof, we require the technical Lemma 5.13 below; for this result we need some preparation.

(5.10) Lemma Suppose that 1 ≤ i < n and that a, b ∈ {0, 1, …, r − 1}.

1. If a ≤ b then L_{i}^{a}L_{i+1}^{b}T_{i} = T_{i}L_{i}^{b}L_{i+1}^{a} + (q − 1) ∑_{c=1}^{b−a} L_{i}^{b−c}L_{i+1}^{a+c}.

2. If a > b then L_{i}^{a}L_{i+1}^{b}T_{i} = qT_{i}^{−1}L_{i}^{b}L_{i+1}^{a} − (q − 1) ∑_{c=1}^{a−b−1} L_{i}^{b+c}L_{i+1}^{a−c}.

Proof: (i) Assume that a ≤ b. If b = a then T_{i} commutes with L_{i}^{a}L_{i+1}^{a} by (2.1)(iii), so the result is correct in this case. Now suppose that b > a. Since L_{i+1}T_{i} = T_{i}L_{i} + (q − 1)L_{i+1} we have

L_{i}^{a}L_{i+1}^{b}T_{i} = L_{i}^{a}L_{i+1}^{b−1}(T_{i}L_{i} + (q − 1)L_{i+1}),

which gives the required result by induction on b − a.

(ii) Either argue similarly, or apply the antiautomorphism ∗ to the result of part (i) and rearrange, interchanging a and b.

We next generalize Lemma 5.10 as follows.

(5.11) Lemma Suppose that 1 ≤ i ≤ n and b_{j} ∈ {0, 1, …, r − 1} for j = 1, 2, …, n. Then there exists an integer b and ϵ_{1}, …, ϵ_{i−1} ∈ {±1} such that

L_{1}^{b_{1}}L_{2}^{b_{2}} … L_{i}^{b_{i}}T_{i−1} … T_{2}T_{1} = x_{1} + x_{2},

where x_{1} = q^{b}T_{i−1}^{ϵ_{i−1}} … T_{2}^{ϵ_{2}}T_{1}^{ϵ_{1}}L_{1}^{b_{i}}L_{2}^{b_{1}}L_{3}^{b_{2}} … L_{i}^{b_{i−1}} and x_{2} is a linear combination of terms of the form T_{w}L_{1}^{c_{1}}L_{2}^{c_{2}} … L_{i}^{c_{i}} where

1. w ∈ 𝔖_{n}; and 2. c_{1}, …, c_{i} ∈ {0, 1, …, r − 1} with c_{1} + c_{2} + ⋯ + c_{i} = b_{1} + b_{2} + ⋯ + b_{i}; and

3. either ∏_{j=1}^{i}(c_{j} + 1) < ∏_{j=1}^{i}(b_{j} + 1), or c_{1}, c_{2}, …, c_{i} is a permutation of b_{1}, b_{2}, …, b_{i} and

c_{1} < b_{i}.

Proof: The result is true when i = 1. (In this case, x_{1} = L_{1}^{b_{1}} and x_{2} = 0.) Assume, inductively, that i < n and that the result as stated is true. By Lemma 5.10, there exists b′ ∈ {0, 1} and ϵ_{i} = ±1 such that

L_{i}^{b_{i}}L_{i+1}^{b_{i+1}}T_{i} = q^{b′}T_{i}^{ϵ_{i}}L_{i}^{b_{i+1}}L_{i+1}^{b_{i}} ± (q − 1) ∑ L_{i}^{d_{i}}L_{i+1}^{d_{i+1}}

where d_{i}, d_{i+1} ∈ {0, 1, …, r − 1} and d_{i} + d_{i+1} = b_{i} + b_{i+1} and either (d_{i} + 1)(d_{i+1} + 1) < (b_{i} + 1)(b_{i+1} + 1) or d_{i} = b_{i} < b_{i+1}. Using this result, and (2.1)(ii), we obtain

\begin{aligned}L_{1}^{b_{1}}L_{2}^{b_{2}} … L_{i}^{b_{i}}T_{i−1} … T_{2}T_{1} = & q^{b′}T_{i}^{ϵ_{i}}L_{1}^{b_{1}} … L_{i−1}^{b_{i−1}}L_{i}^{b_{i+1}}T_{i−1} … T_{2}T_{1}L_{i+1}^{b_{i}} \\ & ± (q − 1) ∑ L_{1}^{b_{1}} … L_{i−1}^{b_{i−1}}L_{i}^{d_{i}}T_{i−1} … T_{2}T_{1}L_{i+1}^{d_{i+1}}, \end{aligned}

where b′, d_{i} and d_{i+1} are as above. The result now follows from our inductive hypothesis.

For the statement and proof of our next Lemma, we need some more notation.

(5.12) Notation Suppose that 0 ≤ i ≤ n. Let V_{i} be the R–module with basis

{ T_{w}L_{1}^{b_{1}}L_{2}^{b_{2}} … L_{n}^{b_{n}}v_{1}v_{2} … v_{i} | w ∈ 𝔖_{n}, 0 ≤ b_{j} < γ_{j}, for 1 ≤ j ≤ i, and 0 ≤ b_{j} < r, for i + 1 ≤ j ≤ n }.

Result (2.4) shows the elements in these sets are indeed linearly independent. Note also that V_{0} = ℋ and V_{n} = ℋ u_{a}^{+}.

We shall use (2.4) extensively in the proof of Lemma 5.13.

(5.13) Lemma Suppose that 1 ≤ i ≤ n and that z ∈ V_{i−1} and zy_{i} = 0. Then z ∈ V_{i}.

Proof: Since z ∈ V_{i−1} we may write z as a linear combination of linearly independent terms

T_{w}L_{1}^{b_{1}} … L_{i−1}^{b_{i−1}}f(L_{i})L_{i+1}^{b_{i+1}} … L_{n}^{b_{n}}v_{1}v_{2} … v_{i−1}

where w ∈ 𝔖_{n} and 0 ≤ b_{j} < γ_{j} for 1 ≤ j ≤ i − 1 and 0 ≤ b_{j} < r for i + 1 ≤ j ≤ n, and f(X) is a polynomial in R[X] of degree less than r. Write

f(X) = g(X) ∏_{k=γ_{i}+1}^{r}(X − Q_{k}) + h(X),

where g(X) and h(X) are polynomials in R[X] such that deg g(X) < γ_{i} and deg h(X) < r − γ_{i}. We may apply the “Euclidean algorithm” in this way, even though R[X] need not be a Euclidean domain, because the divisor is a monic polynomial.

Note that ∏_{k=γ_{i}+1}^{r}(L_{i} − Q_{k}) = v_{i}. As a consequence, we have written z as z_{1} + z_{2} where z_{1} ∈ V_{i} and z_{2} is a linear combination of terms

T_{w}L_{1}^{b_{1}} … L_{i−1}^{b_{i−1}}h(L_{i})L_{i+1}^{b_{i+1}} … L_{n}^{b_{n}}v_{1}v_{2} … v_{i−1}.

We shall prove that z_{2} = 0.

Note that z_{1}y_{i} = 0, by Lemma 5.8, since z_{1} ∈ V_{i}. Therefore, because zy_{i} = 0 by assumption, it follows that z_{2}y_{i} = 0.

For each n–tuple c = (c_{1}, …, c_{n}) with 0 ≤ c_{i} < r, for all i with 1 ≤ i ≤ n, let U_{c} = ℋ (𝔖_{n})L_{1}^{c_{1}}, … L_{n}^{c_{n}}. Then we may write

z_{2} = ∑_{c∈𝒞}z_{c} where z_{c} ∈ U_{c}

and 𝒞 = { c = (c_{1}, …, c_{n}) | 0 ≤ c_{j} < r for 1 ≤ j ≤ n and c_{i} < r − γ_{i} }.

Assume that z_{2} ≠ 0. Among the c such that z_{c} ≠ 0 choose d = (d_{1}, …, d_{n}) such that

(5.14) 1. d_{1} + d_{2} + ⋯ + d_{n} is maximal; and, 2. (d_{1} + 1)(d_{2} + 1) … (d_{n} + 1) is maximal subject to (i); and,

3. d_{i} is maximal subject to (i) and (ii).

Then z_{d} = h_{d}L_{1}^{d_{1}}, … L_{n}^{d_{n}} for some non–zero element h_{d} of ℋ (𝔖_{n}).

Now, T_{i−1} … T_{2}T_{1} commutes with each of the elements L_{i+1}, L_{i+2}, …, L_{n}. Therefore, by Lemma 5.11 there exists an invertible element h ∈ ℋ (𝔖_{n}) (namely, h = q^{b}T_{i−1}^{ϵ_{i−1}} … T_{2}^{ϵ_{2}}T_{1}^{ϵ_{1}}) such that

z_{2}T_{i−1} … T_{2}T_{1} = h_{d}hL_{1}^{d_{i}}L_{2}^{d_{1}}L_{3}^{d_{2}} … L_{i}^{d_{i−1}}L_{i+1}^{d_{i+1}} … L_{n}^{d_{n}} + terms in the sets U_{e},

where e = (e_{1}, …, e_{n}) and e_{1} + ⋯ + e_{n} ≤ d_{1} + ⋯ + d_{n}, and either 1. e_{1} + ⋯ + e_{n} < d_{1} + ⋯ + d_{n} (by Lemma 5.11(ii) and (5.14)(i)); or,

2. (e_{1} + 1) … (e_{n} + 1) < (d_{1} + 1) … (d_{n} + 1) (using Lemma 5.11(iii) and (5.14)(ii)); or,

3. e_{1}, …, e_{n} is a permutation of d_{1}, …, d_{n} and e_{1} < d_{i} (using Lemma 5.11(iii) and (5.14)(iii)); or,

4. e_{1}, …, e_{n} is a permutation of d_{1}, …, d_{n} but e ≠ (d_{i}, d_{1}, d_{2}, …, d_{i−1}, d_{i+1}, …, d_{n}) (con-

sider the term x_{1} in Lemma 5.11). In particular, no e on the right hand side is equal to (d_{i}, d_{1}, d_{2}, …, d_{i−1}, d_{i+1}, …, d_{n}).

Next we postmultiply our expression for z_{2}T_{i−1} … T_{2}T_{1} by ∏_{m=1}^{γ_{i}}(L_{1} − Q_{m}) to obtain z_{2}y_{i}. Note that if v_{e} ∈ U_{e} and k is any positive integer then v_{e}L_{1}^{k} is a linear combination of terms v_{f} ∈ U_{f} where f = (f_{1}, e_{2}, …, e_{n}) and f_{1} < r. Thus, if e_{1} + ⋯ + e_{n} ≤ d_{1} + ⋯ + d_{n} and e_{1}+k ≥ r then v_{e}L_{1}^{k} is a linear combination of terms v_{f} with f_{1}+e_{2}+⋯+e_{n} < d_{1}+⋯+d_{n}+k. Therefore,

z_{2}y_{i} = h_{d}hL_{1}^{d_{i}+γ_{i}}L_{2}^{d_{1}}L_{3}^{d_{2}} … L_{i}^{d_{i−1}}L_{i+1}^{d_{i+1}} … L_{n}^{d_{n}} + terms in the sets U_{f}

where f ≠ (d_{i} + γ_{i}, d_{1}, d_{2}, …, d_{i−1}, d_{i+1}, …, d_{n}).

Note that d_{i} + γ_{i} < r since d ∈ 𝒞. Therefore, since h_{d} ≠ 0 and h is invertible, z_{2}y_{i} ≠ 0 giving a contradiction. Hence, our assumption that z_{2} ≠ 0 is false. Thus, z = z_{1} ∈ V_{i} as required.

(5.15) Corollary We have lr(u_{a}^{+}) = ℋ u_{a}^{+}.

Proof: Let I = ℋ u_{a}^{+} and S = {y_{1}, y_{1}, …, y_{n}}. Then Corollary 5.9 shows that S ⊆ r(I).

Suppose that z ∈ ℋ and zy_{i} = 0 for all i with 1 ≤ i ≤ n. Since V_{0} = ℋ and V_{n} = ℋ u_{a}^{+} = I, it follows from Lemma 5.13 that z ∈ I. Thus, l(S) ⊆ I. Lemma 5.4 now implies that lr(I) = I; that is, lr(u_{a}^{+}) = ℋ u_{a}^{+}.

Finally, we can prove the main result of this section.

(5.16) Theorem Suppose that μ is a multicomposition of n. Then lr(m_{μ}) = ℋ m_{μ}.

Proof: Write m_{μ} = u_{a}^{+}x_{μ} as in Definition 3.5. By applying the definitions, Corollary 5.15 and Lemma 5.3, we obtain

ℋ m_{μ} ⊆ lr(m_{μ}) ⊆ lr(u_{a}^{+}) ∩ lr(x_{μ}) = ℋ u_{a}^{+} ∩ ℋ x_{μ}.

However, (u_{a}^{+}ℋ ∩x_{μ}ℋ )^{∗} = (m_{μ}ℋ )^{∗} by Corollary 4.13. Hence, we have equality throughout and lr(m_{μ}) = ℋ m_{μ} as claimed.

(5.17) Corollary Suppose that μ and ν are multicompositions of n. Then

1. For every element φ in Hom_{ℋ} (M^{ν}, M^{μ}) there exists h_{φ} ∈ ℋ such that φ(m_{ν}) = h_{φ}m_{ν}; in particular, φ(m_{ν}) ∈ M^{ν}^{∗} ∩ M^{μ}.

2. Hom_{ℋ} (M^{ν}, M^{μ}) ≅ M^{ν}^{∗} ∩ M^{μ}.

Proof: By Theorem 5.16 we may apply Lemma 5.2(i) with m = m_{ν} to obtain that φ(m_{ν}) = h_{φ}m_{ν} for some h_{φ} ∈ ℋ . Thus, φ(m_{ν}) ∈ M^{ν}^{∗}. It is clear that φ(m_{ν}) ∈ M^{μ}. Part (ii) follows from Lemma 5.2(ii), where an explicit isomorphism is given by φ⟼φ(m_{ν}).

6 The cyclotomic q–Schur algebra

In order to make our results as general as possible, let Λ_{r} be a poset ideal in the set of all multicompositions of n in the sense that if μ ∈ Λ_{r} and ν is a multipartition such that ν ▷ μ then ν ∈ Λ_{r}. We let Λ_{r}^{+} be the set of multipartitions in Λ_{r}.

(6.1) Definition The cyclotomic q–Schur algebra is the endomorphism algebra

𝒮 (Λ_{r}) = End_{ℋ} ( ⨁_{μ∈Λ_{r}} M^{μ}).

Thus, 𝒮 (Λ_{r}) ≅ ⨁_{μ,ν∈Λ_{r}} Hom_{ℋ} (M^{ν}, M^{μ}).

Suppose that φ ∈ Hom_{ℋ} (M^{ν}, M^{μ}). Then φ(m_{ν}h) = φ(m_{ν})h for all h ∈ ℋ ; thus φ is completely determined by φ(m_{ν}). Moreover, φ(m_{ν}) ∈ M^{ν}^{∗} ∩ M^{μ} by Corollary 5.17. These remarks motivate us to construct a basis of M^{ν}^{∗} ∩ M^{μ}.

(6.2) Definition Suppose that μ and ν are multicompositions of n and that λ is a multipartition of n. Assume that S ∈ T_{0}(λ, μ) and that T ∈ T_{0}(λ, ν). Let

m_{ST} = ∑_{𝔰,𝔱}m_{𝔰𝔱}

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

Note that m_{ST}^{∗} = m_{TS}.

(6.3) Proposition Suppose that μ and ν are multicompositions of n. Then

{ m_{ST} | S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν) for some multipartition λ of n}

is a basis of M^{ν}^{∗} ∩ M^{μ}.

Proof: Since

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

Lemma 4.10 shows that m_{ST} ∈ M^{ν}^{∗} ∩ M^{μ}. Moreover, the elements m_{ST} are linearly independent since they involve distinct elements m_{𝔰𝔱} of the standard basis of ℋ .

Now suppose that h ∈ M^{ν}^{∗} ∩ M^{μ}. Since h ∈ ℋ , we may express h in terms of the standard basis ℳ; say

h = ∑_{m_{𝔰𝔱}∈ℳ} r_{𝔰𝔱}m_{𝔰𝔱}

where r_{𝔰𝔱} ∈ R. Since h ∈ M^{μ}, we have r_{𝔰𝔱} = r_{𝔰′𝔱} if μ(𝔰) = μ(𝔰′), by Theorem 4.14. Similarly, since h ∈ M^{ν}^{∗}, we have r_{𝔰𝔱} = r_{𝔰𝔱′} if ν(𝔱) = ν(𝔱′). Thus, if μ(𝔰) = μ(𝔰′) and ν(𝔱) = ν(𝔱′) then r_{𝔰𝔱} = r_{𝔰′𝔱} = r_{𝔰′𝔱′}. Furthermore, r_{𝔰𝔱} = 0 unless both μ(𝔰) and ν(𝔱) are semistandard. Therefore, h is a linear combination of elements m_{ST}. This completes the proof of the Proposition.

Proposition 6.3 shows that, in our next Definition, φ_{ST} is a well defined ℋ –homomorphism from M^{ν} into M^{μ}.

(6.4) Definition Suppose that μ, ν ∈ Λ_{r} and λ ∈ Λ_{r}^{+}. Assume that S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν). Define φ_{ST} ∈ Hom_{ℋ} (M^{ν}, M^{μ}) by

φ_{ST}(m_{ν}h) = m_{ST}h

for all h ∈ ℋ . Extend φ_{ST} to an element of the cyclotomic q–Schur algebra 𝒮 (Λ_{r}) by defining φ_{ST} to be zero on M^{κ} when ν ≠ κ ∈ Λ_{r}.

(6.5) Definition Suppose that λ ∈ Λ_{r}^{+}. Let \overline{𝒮 (Λ_{r})_{λ}} be the R–submodule of 𝒮 (Λ_{r}) spanned by

{ φ_{UV} |  U ∈ T_{0}(α, μ), V ∈ T_{0}(α, ν) for some μ, ν ∈ Λ_{r} and α ∈ Λ_{r}^{+} with α ▷ λ  }.

This brings us to one of the main results of our paper. The work in Sections 4 and 5 was aimed at proving part (i) of the Theorem.

(6.6) Theorem (The Semistandard Basis Theorem)

1. The cyclotomic q–Schur algebra 𝒮 (Λ_{r}) is free as an R–module with basis

{ φ_{ST} | S ∈ T_{0}(λ, μ), T ∈ T_{0}(λ, ν) for some μ, ν ∈ Λ_{r} and λ ∈ Λ_{r}^{+} }.

2. Suppose that μ, ν ∈ Λ_{r} and λ ∈ Λ_{r}^{+} and let φ ∈ 𝒮 (Λ_{r}). Then, for every κ ∈ Λ_{r} and every T′ ∈ T_{0}(λ, κ) there exists r_{T′} ∈ R such that for all S ∈ T_{0}(λ, μ), we have

φ_{ST}φ ≡ ∑_{κ∈Λ_{r}} ∑_{T′∈T_{0}(λ,κ)} r_{T′}φ_{ST′} mod \overline{𝒮 (Λ_{r})_{λ}} .

Proof: (i) By Corollary 5.17(ii), Hom_{ℋ} (M^{ν}, M^{μ}) ≅ M^{ν}^{∗} ∩ M^{μ}, where an isomorphism is given by φ⟼φ(m_{ν}). Hence, by Proposition 6.3 and Definition 6.4, the set {φ_{ST}} is a basis of 𝒮 (Λ_{r}).

(ii) It is sufficient to consider the case where φ ∈ Hom_{ℋ} (M^{κ}, M^{ν}) for some κ ∈ Λ_{r}. Suppose that φ(m_{κ}) = m_{ν}h where h ∈ ℋ . Then, for all S ∈ T_{0}(λ, μ) and T ∈ T_{0}(λ, ν), we have

φ_{ST}φ(m_{κ}) = m_{ST}h ∈ M^{κ}^{∗} ∩ M^{μ}.

By Proposition 6.3, m_{ST}h = ∑ r_{UV}m_{UV}, where r_{UV} ∈ R and the sum is over U ∈ T_{0}(α, μ) and V ∈ T_{0}(α, κ) for some α ∈ Λ_{r}^{+}. By applying Proposition 3.25 we deduce that

m_{ST}h = ∑_{T′∈T_{0}(λ,κ)} r_{T′}m_{ST′} + ∑_{U′,V′} r_{U′V′}m_{U′V′}

where r_{T′}, r_{U′V′} ∈ R and the second sum is over U′ ∈ T_{0}(α, μ) and V′ ∈ T_{0}(α, ν) for some α ∈ Λ_{r}^{+} with α ▷ λ. Therefore,

φ_{ST}φ ≡ ∑_{T′∈T_{0}(λ,κ)} r_{T′}φ_{ST′} mod \overline{𝒮 (Λ_{r})_{λ}} .

This completes the proof.

(6.7) Definition We call the basis {φ_{ST}} the semistandard basis of 𝒮 (Λ_{r}).

(6.8) Remark It is straightforward to prove that, up to Morita equivalence, 𝒮 (Λ_{r}) depends only upon the set Λ_{r}^{+} (and not on Λ_{r}).

In order to show that the semistandard basis of 𝒮 (Λ_{r}) is a cellular basis we need an appropriate antiautomorphism ∗ for 𝒮 (Λ_{r}).

(6.9) Proposition Let ∗: 𝒮 (Λ_{r}) ⟶ 𝒮 (Λ_{r}) be the unique R–linear map such that φ_{ST}^{∗} = φ_{TS} for all elements φ_{ST} in the semistandard basis. Then ∗ is an antiautomorphism of 𝒮 (Λ_{r}).

Proof: Assume that S ∈ T_{0}(λ, μ) and that T ∈ T_{0}(λ, ν) for some μ, ν ∈ Λ_{r} and λ ∈ Λ_{r}^{+}. Then

φ_{ST}(m_{ν}) = m_{ST} = (m_{TS})^{∗} = (φ_{TS}(m_{μ}))^{∗}.

The proposition now follows from the next general lemma.

(6.10) Lemma Suppose that ℋ is an algebra with an antiautomorphism ∗. Let { m_{μ} | μ ∈ Λ } be a set of elements of ℋ such that m_{μ}^{∗} = m_{μ} for all μ ∈ Λ and let

S = End_{ℋ} ( ⨁_{μ∈Λ} m_{μ}ℋ ).

Assume that, for all μ, ν ∈ Λ, every ℋ –homomorphism from m_{ν}ℋ to m_{μ}ℋ is given by left multiplication by an element of ℋ . Then the following hold.

1. For each φ ∈ Hom_{ℋ} (m_{ν}ℋ , m_{μ}ℋ ) there is a unique φ^{∗} ∈ Hom_{ℋ} (m_{μ}ℋ , m_{ν}ℋ ) such that φ^{∗}(m_{μ}) = (φ(m_{ν}))^{∗}.

2. The map ∗ is an antiautomorphism of 𝒮 (Λ_{r}).

Proof: (i) Suppose that φ ∈ Hom_{ℋ} (m_{ν}ℋ , m_{μ}ℋ ). Then φ(m_{ν}) = x_{1}m_{ν} = m_{μ}y for some x_{1}, y ∈ ℋ . Define φ^{∗} ∈ Hom_{ℋ} (m_{μ}ℋ , m_{ν}ℋ ) by φ^{∗}(m_{μ}h) = y^{∗}m_{μ}h for all h ∈ ℋ . Then φ^{∗} is a well–defined ℋ –homomorphism; also, it maps into m_{ν}ℋ , since y^{∗}m_{μ} = m_{ν}x_{1}^{∗}. Since φ^{∗}(m_{μ}) = (φ(m_{ν}))^{∗}, the proof of (i) is complete.

(ii) Suppose that ψ ∈ Hom_{ℋ} (m_{μ}ℋ , m_{λ}ℋ ) for some λ ∈ Λ. Then ψ(m_{μ}) = x_{2}m_{μ} for some x_{2} ∈ ℋ . We have,

\begin{aligned}(ψφ)^{∗}(m_{λ}) & = (ψφ(m_{ν}))^{∗} & = (x_{2}x_{1}m_{ν})^{∗} & = m_{ν}x_{1}^{∗}x_{2}^{∗} \\ & = φ^{∗}(m_{μ})x_{2}^{∗} & = φ^{∗}(m_{μ}x_{2}^{∗}) & = φ^{∗}ψ^{∗}(m_{λ}). \end{aligned}

Therefore, (ψφ)^{∗} = φ^{∗}ψ^{∗}, and it follows that ∗ is an antiautomorphism of 𝒮 (Λ_{r}).

(6.11) Corollary The R–module \overline{𝒮 (Λ_{r})_{λ}} is a two–sided ideal of 𝒮 (Λ_{r}).

Proof: Theorem 6.6(ii) shows that \overline{𝒮 (Λ_{r})_{λ}} is a right ideal. By applying the antiautomorphism ∗, we deduce that it is also a left ideal.

(6.12) Theorem The semistandard basis of 𝒮 (Λ_{r}) is a cellular basis.

Proof: This follows at once from Theorem 6.6 and Proposition 6.9.

We now apply the theory of cellular algebras to the representation theory of 𝒮 (Λ_{r}), just as we treated ℋ in Section 3.

Suppose that λ is a multipartition of n. Let T^{λ} = λ(𝔱^{λ}) (see Definition 4.2). Then T^{λ} is the unique semistandard λ–tableau of type λ (cf. Example 4.3). Define φ_{λ} = φ_{T^{λ}}_{T^{λ}}; then φ_{λ} is the identity map on M^{λ}.

(6.13) Definition Suppose that λ ∈ Λ_{r}^{+}, The Weyl module W^{λ} is the submodule of 𝒮 (Λ_{r})/\overline{𝒮 (Λ_{r})_{λ}} given by W^{λ} = 𝒮 (Λ_{r})(φ_{λ} + \overline{𝒮 (Λ_{r})_{λ}} ).

If S is a semistandard λ–tableau let φ_{S} = φ_{ST^{λ}}(φ_{λ} + \overline{𝒮 (Λ_{r})_{λ}} ) = φ_{ST^{λ}} + \overline{𝒮 (Λ_{r})_{λ}} . Then, from Theorem 6.12, we obtain the following result.

(6.14) Corollary The Weyl module W^{λ} is a free R–module with basis

{ φ_{S} | S ∈ T_{0}(λ, μ) for some μ ∈ Λ_{r} }.

Define a bilinear form ⟨ , ⟩ on W^{λ} by requiring that

φ_{T^{λ}S}φ_{TT^{λ}} ≡ ⟨φ_{S}, φ_{T}⟩φ_{λ} mod \overline{𝒮 (Λ_{r})_{λ}}

for all semistandard λ–tableaux S and T. This bilinear form is well–defined and symmetric, and also satisfies ⟨su, v⟩ = ⟨u, s^{∗}v⟩ for all u, v ∈ W^{λ} and all s ∈ 𝒮 (Λ_{r}) [12, (2.4)]. Consequently, rad W^{λ} = { u ∈ W^{λ} | ⟨u, v⟩ = 0 for all v ∈ W^{λ} } is a submodule of W^{λ}.

(6.15) Definition Suppose that λ ∈ Λ_{r}^{+}. Let F^{λ} = W^{λ}/ rad W^{λ}.

(6.16) Theorem Suppose that R is a field. Then

{ F^{λ} | λ ∈ Λ_{r}^{+} }

is a complete set of non–isomorphic irreducible 𝒮 (Λ_{r})–modules. Moreover, each F^{λ} is absolutely irreducible.

Proof: Let λ ∈ Λ_{r}^{+}. From the definition of the bilinear form on W^{λ}, we have

φ_{T^{λ}}_{T^{λ}}φ_{T^{λ}}_{T^{λ}} ≡ ⟨φ_{T^{λ}}, φ_{T^{λ}}⟩φ_{λ} mod \overline{𝒮 (Λ_{r})_{λ}} .

However, φ_{T^{λ}}_{T^{λ}}φ_{T^{λ}}_{T^{λ}} = φ_{λ} is the identity on M^{λ}; so ⟨φ_{T^{λ}}, φ_{T^{λ}}⟩ = 1. Consequently, F^{λ} is non–zero. The Theorem now follows from [12, (3.4)].

If λ, μ ∈ Λ_{r}^{+}, let d_{λμ} denote the composition multiplicity of F^{μ} as a composition factor of W^{λ}. Then

(d_{λμ})_{λ,μ∈Λ_{r}^{+}}

is the decomposition matrix of 𝒮 (Λ_{r}). The theory of cellular algebras [12, (3.6)] yields the following.

(6.17) Corollary The decomposition matrix of 𝒮 (Λ_{r}) is unitriangular. That is, for λ, μ ∈ Λ_{r}^{+}, we have d_{μμ} = 1 and d_{λμ} ≠ 0 only if λ ⊵ μ.

Finally, Theorem 6.16 combined with [12, (3.10)], gives us our last result.

(6.18) Corollary The cyclotomic q–Schur algebra is quasi–hereditary.

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 E–mail addresses: rdipper@mathematik.uni-stuttgart.de

                 g.james@ic.ac.uk
                 sfb32@Mathematik.Uni-Bielefeld.de