The (Q, q)–Schur Algebra

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

11th February 1997

Abstract In this paper we use the Hecke algebra of type B to define a new algebra S which is an analogue of the q–Schur algebra. We construct Weyl modules for S and obtain, as factor modules, a family of irreducible S–modules over any field.

1 Introduction

The ordinary Schur algebra is of key importance in the study of the representation theory of general linear groups in the describing characteristic, and it provides a link between the general linear groups and the symmetric groups. In [8, 9] we introduced the q–Schur algebra, and demonstrated its usefulness in the representation theory of GL_{n}(q) over a field of non– describing characteristic. In [5] it was shown that the q–Schur algebra is given as the dual of a homogeneous part of quantum–GL_{n}, or alternatively as the factor of quantum–GL_{n} or the corresponding quantum enveloping algebra modulo the kernel of its action on quantum tensor space (compare [1]). In particular, the representations of q–Schur algebras are precisely the homogeneous polynomial representations of quantum–GL_{n} in a fixed degree (compare [4]).

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; the third author was also supported in part by SERC grant GR/J37690

The construction of the q–Schur algebra involves the Hecke algebra of type A; in this paper we use the Hecke algebra of type B to build an algebra which we call the (Q, q)–Schur algebra. Others have devised a version of a Schur algebra of type B [13, 15], but ours is a larger algebra. Applications to the representation theory of finite symplectic groups in the non–describing characteristic case have already been provided [16, 6], and we expect that further applications will ensue, using our larger and more complicated algebra.

Hecke algebras of type B have been studied in [10, 11, 18]. We begin by recalling and extending some of the notation and results which we used in those papers. The remainder of the paper is then devoted to introducing the (Q, q)–Schur algebra and investigating its main properties. In particular we construct a generic basis of the (Q, q)–Schur algebra and we define (Q, q)–Weyl modules. The Weyl modules are labelled by bipartitions and they have unique maximal submodules. The corresponding factor modules are pairwise non–isomorphic irreducible representations of the (Q, q)–Schur algebra. We show that the decomposition matrix which describes the composition multiplicities of these irreducible modules in the (Q, q)–Weyl modules is unitriangular, and we construct a “semistandard basis” for each (Q, q)–Weyl module.

In a forthcoming paper we shall construct a cellular basis of the (Q, q)-Schur algebra S. As a consequence, every irreducible representation of S is isomorphic to one of the irreducible representations which we construct here and, in addition, S is quasi–hereditary. Furthermore, we shall generalize some of our results to construct Schur algebras of the Ariki–Koike algebras.

After this manuscript was submitted for publication, we received a preprint by Jie Du and Leonard Scott entitled “The q-Schur^{2} algebra” where they independently construct an algebra which is Morita equivalent to the one considered here; furthermore they show that it is quasi–hereditary.

2 The Hecke algebra of type B

Let W_{r} be the group C_{2} ≀ 𝔖_{r}, where 𝔖_{r} is the symmetric group of degree r. Then W_{r} is generated by elements s_{0}, s_{1}, …, s_{r−1} which satisfy the following relations:

(2.1)  \begin{aligned}s_{i}^{2} & = & 1 & for 0 ≤ i ≤ r − 1 \\ s_{i}s_{j} & = & s_{j}s_{i} & if 1 ≤ i + 1 < j ≤ r − 1 \\ s_{i}s_{i+1}s_{i} & = & s_{i+1}s_{i}s_{i+1} & if 1 ≤ i ≤ r − 2 \\ s_{0}s_{1}s_{0}s_{1} & = & s_{1}s_{0}s_{1}s_{0}. \end{aligned}

Let r^{+} = {1, 2, …, r} and r^{−} = {−1, −2, …, −r}. We identify W_{r} with a subgroup of the symmetric group on r^{±} = r^{+} ∪ r^{−} by letting

s_{0} = (1, −1)
s_{i} = (i, i + 1)(−i, −i − 1) for 1 ≤ i ≤ r − 1.

It is easy to see that W_{r} acts transitively on r^{±}. The subgroup of W_{r} generated by s_{1}, …, s_{r−1} is the symmetric group 𝔖_{r} of degree r.

The concepts of a reduced expression for an element of W_{r} and the length ℓ(w) of w ∈ W_{r} are defined in the usual way. It is useful to note the following well-known method for calculating ℓ(w). Consider the ℚW_{r}–module V whose basis is {e_{i} | i ∈ r^{+}} and where, for i ∈ r^{+} and w ∈ W_{r}, we have

e_{i}w = \begin{cases}e_{iw} & if iw > 0 \\ −e_{−iw} & if iw < 0. \end{cases}

The roots for W_{r} are

±e_{i} and ± e_{i} ± e_{j} (i, j ∈ r^{+}, i ≠ j).

The positive roots are

e_{i}, e_{i} + e_{j}, e_{j} − e_{i} (i, j ∈ r^{+}, i < j),

and all other roots are negative roots. The length of w is equal to the number of positive roots changed to negative roots by w.

The simple roots

α_{0} = e_{1}, α_{1} = e_{2} − e_{1}, α_{2} = e_{3} − e_{2}, …, α_{r−1} = e_{r} − e_{r−1}

form a basis Δ_{0} of V, and every positive root is a non–negative linear combination of simple roots. Moreover, for i = 0, 1, …, r − 1, s_{i} acts on V as the reflection in the hyperplane orthogonal to α_{i}. If α is a positive root we sometimes write s_{α} for the reflection in the hyperplane orthogonal to α; in particular, s_{i} = s_{α_{i}}. Note that s_{α} ∈ W_{r} for all positive roots α.

Let R be a commutative ring with 1, and let q and Q be invertible elements of R. The Hecke algebra H_{R,q,Q}(W_{r}) of type B_{r} is defined to be the free R–module with basis {T_{w} | w ∈ W_{r}} and multiplication defined as below. If e is the identity element of W then T_{e} is the multiplicative identity for H_{R,q,Q}(W_{r}) and for ρ ∈ R we abbreviate ρT_{e} as ρ. We often write T_{s_{i}} as T_{i} for 0 ≤ i ≤ r − 1 and H_{R,q,Q}(W_{r}) as H(W_{r}) or as H. Then

(i) if w = v_{1}v_{2} ⋯ v_{l} is a reduced expression for w ∈ W_{r} where each v_{i} belongs to {s_{0}, s_{1}, …, s_{r−1}}, then T_{w} = T_{v_{1}}T_{v_{2}} ⋯ T_{v_{l}};

(ii) (T_{i})^{2} = q + (q − 1)T_{i} for 1 ≤ i ≤ r − 1; (iii) (T_{0})^{2} = Q + (Q − 1)T_{0}.

Let H(𝔖_{r}) denote the subalgebra of H generated by T_{1}, T_{2}, …, T_{r−1}.

For each pair of integers i, j in r^{+} define s_{i,j} ∈ W_{r} by

s_{i,j} = \{ \begin{matrix}s_{i}s_{i+1} … s_{j−1} & if i ≤ j \\ s_{i−1}s_{i−2} … s_{j} & if i > j \end{matrix}

For example, s_{1,r} is the permutation of r^{±} which is given in its cycle decomposition as

(r, r − 1, …, 2, 1)(−r, −r + 1, …, −2, −1).

For 0 ≤ a ≤ r, let w_{a,r−a} = (s_{1,r})^{a}. As in [10, 2.4–2.9] we have the following.

2.2 Let i, j ∈ r^{+} and 0 ≤ a ≤ r. Then

(i) ℓ(s_{i,j}) = |j − i|. (ii) ℓ(w_{a,r−a}) = a(r − a). (iii) s_{a+1,1}s_{a+2,2} ⋯ s_{r,r−a} gives a reduced expression for w_{a,r−a}.

For typographical reasons, let T_{i,j} = T_{s_{i,j}} and h_{a,r−a} = T_{w_{a,r−a}}.

Definition 2.3 For 0 ≤ a ≤ r, let the elements u_{a}^{+} and u_{a}^{−} of H be given by

u_{a}^{+} = ∏_{i=1}^{a}(q^{i−1} + T_{i,1}T_{0}T_{1,i})

u_{a}^{−} = ∏_{i=1}^{a}(Qq^{i−1} − T_{i,1}T_{0}T_{1,i})

It is easy to prove the following (compare [10, 3.3 and 3.4]).

2.4 Let 0 ≤ a ≤ r.

(i) If 0 ≤ b ≤ r − 1 with b ≠ a, then u_{a}^{+} and u_{a}^{−} commute with T_{b}. (ii) If a > 0 then u_{a}^{+}T_{0} = Qu_{a}^{+} and u_{a}^{−}T_{0} = −u_{a}^{−}.

An a-bicomposition of r is an ordered pair (λ^{(1)}, λ^{(2)}) of compositions, where λ^{(1)} is a composition of a and λ^{(2)} is a composition of r−a; if both λ^{(1)} and λ^{(2)} are partitions, then (λ^{(1)}, λ^{(2)}) is an a-bipartition of r.

Let Λ_{2}(n, r) denote the set of bicompositions λ = (λ^{(1)}, λ^{(2)}) of r with the property that the sum of the number of parts of λ^{(1)} and the number of parts of λ^{(2)} is n. (Note that we allow a composition to have zero parts.)

We order the bipartitions of r by letting all a-bipartitions precede all b-bipartitions if a > b, and by saying that the a-bipartition λ = (λ^{(1)}, λ^{(2)}) precedes the a-bipartition μ = (μ^{(1)}, μ^{(2)}) if λ^{(1)} precedes μ^{(1)} lexicographically or λ^{(1)} = μ^{(1)} and λ^{(2)} precedes μ^{(2)} lexicographically. We call this the lexicographic order on bipartitions.

Suppose that λ = (λ^{(1)}, λ^{(2)}) and μ = (μ^{(1)}, μ^{(2)}) are bicompositions of r. We say that λ and μ are associated if μ^{(i)} can be obtained from λ^{(i)}, i = 1, 2, by reordering the parts.

From λ we obtain a corresponding diagram [λ] which consists of crosses in the plane, as in the example λ = ((4, 3, 1), (3, 2)), where

[λ] = (\begin{matrix}× & × & × & × \\ × & × & × \\ × \end{matrix} , \begin{matrix}× & × & × \\ × & × \end{matrix})

A λ-bitableau is obtained from [λ] by replacing each cross by one of the numbers from r^{±} in such a way that for every i ∈ r^{+}, precisely one of {i, −i} is used. If 𝔱 is a λ–bitableau then 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}), where 𝔱^{(1)} is a λ^{(1)}-tableau and 𝔱^{(2)} is a λ^{(2)}-tableau.

Definition 2.5 Suppose that λ = (λ^{(1)}, λ^{(2)}) ∈ Λ_{2}(n, r), and that 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) is a λ– bitableau. Assume that i ∈ 𝔱. Let row_{𝔱}(i) = j if i belongs to row j of 𝔱^{(1)} and let row_{𝔱}(i) = n+j if i belongs to row j of 𝔱^{(2)}.

Definition 2.6 Let λ be a bicomposition.

(i) A λ–bitableau 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) is row standard if the entries increase from left to right in each row of 𝔱^{(1)} and in each row of 𝔱^{(2)}, and all entries in 𝔱^{(1)} belong to r^{+}.

(ii) The λ–bitableaux 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) and 𝔰 = (𝔰^{(1)}, 𝔰^{(2)}) are row equivalent if |𝔱^{(1)}| and |𝔰^{(1)}| are row equivalent, and 𝔱^{(2)} and 𝔰^{(2)} are row-equivalent. Here the entries of |𝔱^{(1)}| are the absolute values of the entries of 𝔱^{(1)} and |𝔰^{(1)}| is defined similarly. A row equivalence class of the λ–bitableaux is a λ–bitabloid and the λ-bitabloid containing 𝔱 is denoted by {𝔱}.

(iii) A λ–bitableau is standard if all its entries belong to r^{+}, and it is row standard, and all the entries increase down each column of 𝔱^{(1)} and each column of 𝔱^{(2)}.

We remark that every λ–bitabloid contains exactly one row standard λ–bitableau. When dealing with λ-bitabloids {𝔱}, we often find it convenient to specify that 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) is row standard; this ensures that all the entries in 𝔱^{(1)} are positive.

Here is an example of a row standard bitableau:

(\begin{matrix}2 & 8 \\ 3 & 4 \end{matrix} , \begin{matrix}−5 & −1 & 6 \\ −7 & 9 \end{matrix}) .

Next, we wish to specify, for a given λ = (λ^{(1)}, λ^{(2)}), several special standard λ–bitableaux. The λ–bitableaux which we wish to define are most easily understood with an example.

Example 2.7 Suppose that λ = ((4, 3, 1), (3, 2)). We shall define the λ–bitableaux 𝔱^{λ}, 𝔱̂^{λ}, 𝔱_{λ} and 𝔱̂_{λ} so that

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

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

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

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

Definition 2.8 Suppose that λ is an a-bicomposition of r.

(i) Let 𝔱^{λ} = (𝔱^{λ^{(1)}}, 𝔱^{λ^{(2)}}) be the standard λ–bitableau in which the numbers 1, 2, …, a appear in order by rows in 𝔱^{λ^{(1)}} and the numbers a + 1, a + 2, …, r appear in order by rows in 𝔱^{λ^{(2)}}.

(ii) Let 𝔱̂^{λ} = (𝔱̂^{λ^{(1)}}, 𝔱̂^{λ^{(2)}}) be the standard λ–bitableau in which the numbers 1, 2, …, r − a appear in order by rows in 𝔱̂^{λ^{(2)}} and the numbers r − a + 1, r − a + 2, …, r appear in order by rows in 𝔱̂^{λ^{(1)}}.

(iii) Let 𝔱_{λ} = (𝔱_{λ}^{(1)}, 𝔱_{λ}^{(2)}) be the standard λ–bitableau in which the numbers 1, 2, …, a appear in order by columns in 𝔱_{λ}^{(1)} and the numbers a + 1, a + 2, …, r appear in order by columns in 𝔱_{λ}^{(2)}.

(iv) Let 𝔱̂_{λ} = (𝔱̂_{λ}^{(1)}, 𝔱̂_{λ}^{(2)}) be the standard λ–bitableau in which the numbers 1, 2, …, r − a appear in order by columns in 𝔱̂_{λ}^{(2)} and the numbers r − a + 1, r − a + 2, …, r appear in order by columns in 𝔱̂_{λ}^{(1)}.

Note that W_{r} acts on the set of λ–bitableaux. If 𝔱 is a λ–bitableau and w ∈ W_{r}, then we obtain the λ–bitableau 𝔱w by replacing each i in 𝔱 by iw. For example, if λ is an abicomposition of r, then 𝔱^{λ}w_{a,r−a} = 𝔱̂^{λ}. It is easy to see that W_{r} acts transitively on the set of λ–bitableaux.

One easily checks that the action of W_{r} preserves the row equivalence classes of λ– bitableaux. We therefore have a transitive action of W_{r} on the set of λ–bitabloids. If λ is an a-bicomposition, then the stabilizer of the bitabloid {𝔱^{λ}} is the subgroup

(2.9)  W_{λ} = ((\underbrace{C_{2} × ⋯ × C_{2}}_{a factors}) ⋊ 𝔖_{λ^{(1)}}) × 𝔖_{λ^{(2)}}

of W_{r}. Thus the permutation representation of W_{r} on the set of λ–bitabloids is equivalent to the representation of W_{r} on the cosets of W_{λ}.

A reflection subgroup W_{J} of W_{r} is a subgroup generated by a set of reflections {s_{α} | α ∈ J} for some subset J of the positive roots. In particular, if J ⊆ Δ_{0} then W_{J} is a parabolic subgroup of W_{r}. If λ is an a–bicomposition then W_{λ} is a parabolic subgroup of W_{r} if and only if λ^{(1)} = (a); in this case W_{λ} is a direct product of a Weyl group W_{a} of type B and several Weyl groups of type A. In general the reflection group W_{λ} has precisely one factor of type B for each nonzero part of λ^{(1)} .

Associated with the parabolic subgroup W_{J}, J ⊆ Δ_{0}, is the parabolic subalgebra H_{J} = ∑_{w∈W_{J}} RT_{w} of H. This algebra is isomorphic to the Hecke algebra H_{R,Q,q}(W_{J}) of W_{J}. If W_{λ} is a reflection subgroup of W_{r}, which is not a parabolic subgroup, then ∑_{w∈W_{λ}} RT_{w} is not a subalgebra of H in general.

Definition 2.10 Suppose that λ ∈ Λ_{2}(n, r). The elements x_{λ}, x̂_{λ}, y_{λ} and ŷ_{λ} of H are defined as follows.

x_{λ} = ∑{T_{w} | w ∈ W_{r} and w stabilizes the rows of 𝔱^{λ}}
x̂_{λ} = ∑{T_{w} | w ∈ W_{r} and w stabilizes the rows of 𝔱̂^{λ}}
y_{λ} = ∑{(−q)^{−ℓ(w)}T_{w} | w ∈ W_{r} and w stabilizes the columns of 𝔱_{λ}}
ŷ_{λ} = ∑{(−q)^{−ℓ(w)}T_{w} | w ∈ W_{r} and w stabilizes the columns of 𝔱̂_{λ}}

Example 2.11 Suppose that λ = ((2, 1), (1)). Then

𝔱^{λ} = (\begin{matrix}1 & 2 \\ 3 \end{matrix} , 4)  𝔱_{λ} = (\begin{matrix}1 & 3 \\ 2 \end{matrix} , 4)
𝔱̂^{λ} = (\begin{matrix}2 & 3 \\ 4 \end{matrix} , 1)  𝔱̂_{λ} = (\begin{matrix}2 & 4 \\ 3 \end{matrix} , 1)

and

x_{λ} = 1 + T_{(1,2)}  y_{λ} = 1 − q^{−1}T_{(1,2)}
x̂_{λ} = 1 + T_{(2,3)}  ŷ_{λ} = 1 − q^{−1}T_{(2,3)}.

We remark that all four elements defined in (2.10) belong to H(𝔖_{r}) and that x̂_{λ} = xλ̂ and ŷ_{λ} = yλ̂, where λ̂ is the (r − a)-bicomposition (λ^{(2)}, λ^{(1)}).

From (2.4) we obtain the following.

2.12 Assume that λ is an a-bicomposition. Then

(i) x_{λ} and y_{λ} commute with u_{a}^{+}; (ii) x̂_{λ} and ŷ_{λ} commute with u_{r−a}^{−}.

As we noted in [10, (2.5)], we also have the following (see the remark after (2.2)).

2.13 If λ is an a-bicomposition then x_{λ}h_{a,r−a} = h_{a,r−a}x̂_{λ}.

Definition 2.14 Let π_{λ} be the element of W_{r} which is given by 𝔱^{λ}π_{λ} = 𝔱_{λ} and π̂_{λ} be the element of W_{r} which is given by 𝔱̂^{λ}π̂_{λ} = 𝔱̂_{λ}

We note that π_{λ} and π̂_{λ} belong to 𝔖_{r} ≤ W_{r}, since the entries of the bitableaux involved are all positive. Moreover π̂_{λ} = πλ̂, and h_{a,r−a}Tπ̂_{λ} = T_{π_{λ}}h_{a,r−a}. Note too that

Tπ̂_{λ}u_{r−a}^{−} = u_{r−a}^{−}Tπ̂_{λ}

by (2.4) since π̂_{λ} ∈ 𝔖_{(r−a,a)}.

Now, [10, (3.11)] and [7, (4.1)] imply the following fundamental result.

Theorem 2.15 Suppose that λ is an a-bipartition. Then

u_{a}^{+}x_{λ}Hu_{r−a}^{−}ŷ_{λ}

is a one-dimensional R–module. It is spanned by the vector

u_{a}^{+}x_{λ}h_{a,r−a}Tπ̂_{λ}u_{r−a}^{−}ŷ_{λ}.

Definition 2.16 Suppose that λ is an a-bipartition. Let

z_{λ} = u_{a}^{+}x_{λ}h_{a,r−a}Tπ̂_{λ}u_{r−a}^{−}ŷ_{λ}.

Note that, in the light of (2.12) and (2.13), we have several alternative expressions for z_{λ}; for example,

(2.17)  \begin{aligned}z_{λ} & = u_{a}^{+}x_{λ}T_{π_{λ}}h_{a,r−a}u_{r−a}^{−}ŷ_{λ} \\ & = u_{a}^{+}h_{a,r−a}u_{r−a}^{−}x̂_{λ}Tπ̂_{λ} ŷ_{λ} \\ & = u_{a}^{+}x_{λ}h_{a,r−a}u_{r−a}^{−}Tπ̂_{λ} ŷ_{λ}. \end{aligned}

We call the right ideal z_{λ}H of H the (Q, q)–Specht module S^{λ}. It is straightforward to show that S^{λ} is the dual of the module S̃^{λ} which we introduced in [11, §4]; in particular S^{λ} has dimension equal to the number of standard λ–bitableaux. Moreover there exists an H– invariant bilinear form on M^{λ} = u_{a}^{+}x_{λ}H and a submodule theorem holds for M^{λ} (compare Theorem 5.15 and [7, 4.8]). When R is a field, the submodule theorem allows us to construct irreducible H–modules, and so we can recover the main results of [11]. We do not wish to pursue this here; instead, we turn to the construction of the (Q, q)-Schur algebra.

3 The H–module M^{λ}

Definition 3.1 Suppose that λ is an a-bicomposition . Let M^{λ} be the right ideal of H which is given by M^{λ} = u_{a}^{+}x_{λ}H.

If λ is an a–bicomposition then M^{λ} contains the (Q, q)-Specht module S^{λ}, and postmultiplication by u_{r−a}^{−}ŷ_{λ} maps M^{λ} on to the one-dimensional R–module spanned by the generator z_{λ} of S^{λ}, by Theorem 2.15.

An argument similar to that of [9, (1.1)] shows the following.

3.2 If λ and μ are associated bicompositions of r, then M^{λ} and M^{μ} are isomorphic H– modules.

We next construct a basis of M^{λ}.

Theorem 3.3 Assume that λ is an a-bicomposition of r. Then M^{λ} is a free R–module with basis

{u_{a}^{+}x_{λ}T_{w} | w ∈ W_{r} and 𝔱^{λ}w is row standard}.

Proof: Note first, that u_{a}^{+}x_{λ}H(W_{a}) = u_{a}^{+}x_{λ}H(𝔖_{a}). This is immediate if a = 0 since H(W_{0}) = H(𝔖_{0}) = {1}. If a > 0 then from (2.4), for h ∈ H(𝔖_{a}),

u_{a}^{+}x_{λ}hT_{0} = x_{λ}hu_{a}^{+}T_{0} = Qx_{λ}hu_{a}^{+} = Qu_{a}^{+}x_{λ}h.

Next, [7, (3.2)(i)] implies that a basis of u_{a}^{+}x_{λ}H(𝔖_{a}) is given by

{u_{a}^{+}x_{λ}T_{w} | w ∈ 𝔖_{a} and 𝔱^{λ}w is row standard}.

Let 𝔖_{λ^{(2)}} denote the subgroup of 𝔖_{{a+1,... ,r}} which stabilizes the rows of 𝔱^{λ^{(2)}}. Then W_{a} × 𝔖_{λ^{(2)}} is a parabolic subgroup of W_{r}. Let D denote the set of distinguished right coset representatives of W_{a} × 𝔖_{λ^{(2)}} in W_{r}. We claim that B is a basis of M^{λ}, where

(3.4)  B = {u_{a}^{+}x_{λ}T_{w}T_{d} | w ∈ 𝔖_{a}, d ∈ D and 𝔱^{λ}w is row standard}.

To see this notice that we can write any element in W_{r} as w_{1}w_{2}d where w_{1} ∈ W_{a}, w_{2} ∈ 𝔖_{λ^{(2)}} and d ∈ D; but

u_{a}^{+}x_{λ}T_{w_{1}}T_{w_{2}}T_{d} = q^{ℓ(w_{2})}u_{a}^{+}x_{λ}T_{w_{1}}T_{d} ∈ u_{a}^{+}x_{λ}H(𝔖_{a})T_{d},

so B spans M^{λ}. Secondly, if d and d′ are distinct elements of D then corresponding elements in B have distinct supports, and hence B is linearly independent.

Since T_{w}T_{d} = T_{wd} for w ∈ 𝔖_{a} and d ∈ D, the proof of the theorem will be complete if we show that

(3.5) {wd | w ∈ 𝔖_{a}, d ∈ D and 𝔱^{λ}w is row standard}

= {v ∈ W_{r} | 𝔱^{λ}v is row standard}.

Suppose that w ∈ 𝔖_{a}, d ∈ D and 𝔱^{λ}w is row standard. Since d is a distinguished coset representative, d sends the positive roots for W_{a} × 𝔖_{λ^{(2)}} into positive roots. The positive roots for W_{a} × 𝔖_{λ^{(2)}} are

e_{i} (1 ≤ i ≤ a), e_{i} + e_{j}, e_{j} − e_{i} (1 ≤ i < j ≤ a),

together with

e_{j} − e_{i} (i < j with i, j in the same row of 𝔱^{λ^{(2)}}).

If 1 ≤ i ≤ a then e_{i}d is a positive root, so id ∈ r^{+}; thus all the entries in the first component of 𝔱^{λ}wd must be positive. If i < j and i and j are in the same row of 𝔱^{λ}w then (e_{j} − e_{i})d is a positive root, so id < jd. Therefore, 𝔱^{λ}wd is row standard (see Definition 2.6).

Finally, note that the sets on the two sides of equation (3.5) have the same size. This completes the proof of (3.5) and hence of the theorem. □

Definition 3.6 Let J ⊆ Δ_{0}. The subalgebra of H corresponding to the parabolic subgroup W_{J} of W_{r} is denoted by H_{J}. The induction functor from H_{J} to H_{K} for J ⊆ K ⊆ Δ_{0} is denoted by Ind_{J}^{K}. Similarly the restriction functor from H_{K} to H_{J} is denoted by Res_{J}^{K}.

Recall that H_{J} is free as an R–module with basis {T_{w} | w ∈ W_{J}}. Moreover, H is free as a left H_{J}–module with basis {T_{d} | d ∈ D_{J}}, where D_{J} denotes the set of distinguished right coset representatives of W_{J} in W_{r} which is the set of all elements of W_{r} which map the roots in J to positive roots. Note too, that H(𝔖_{r}) = H_{Δ} with Δ = {α_{1}, …, α_{r}} = Δ_{0}\{α_{0}}.

Lemma 3.7 Let λ = (λ^{(1)}, λ^{(2)}) be an a-bicomposition, and let

A = {α_{0}, α_{1}, …, α_{a−1}} ∪ J_{2}

where J_{2} is the subset of {α_{a+1}, α_{a+2}, …, α_{r−1}} corresponding to λ^{(2)}. Then u_{a}^{+}x_{λ} ∈ H_{A} and M^{λ} = Ind_{A}^{Δ_{0}} u_{a}^{+}x_{λ}H_{A}.

Proof: This follows by general arguments since H_{A} = H(W_{a} × 𝔖_{λ^{(2)}}). □

We find it convenient to adopt a shorthand for the basis elements of M^{λ} which appear in Theorem 3.3.

Notation 3.8 Suppose that λ is an a-bicomposition. Let 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) be a λ–bitableau. We identify the λ–bitabloid {𝔱} and u_{a}^{+}x_{λ}T_{w}, where 𝔱^{λ}w is the unique row standard λ–bitableau which is row equivalent to 𝔱.

This identification gives us an action of H on the λ–bitabloids. The discussion after Definition 2.8 shows that this is a (Q, q)–analogue of the permutation action of W_{r} on the cosets of W_{λ}.

Theorem 3.3 says that M^{λ} has a basis which consists of the distinct λ–bitabloids. The action of H on a λ–bitabloid {𝔱} is determined once we know {𝔱}T_{i} for i = 0, 1, …, r − 1. From (3.8) we have the following.

Lemma 3.9 Suppose that 0 ≤ i ≤ r − 1 and that both 𝔱 and 𝔱s_{i} are row standard (say 𝔱 = 𝔱^{λ}w). Then

\{𝔱\}T_{i} = \{ \begin{matrix}\{𝔱s_{i}\}, & & if ℓ(ws_{i}) & = ℓ(w) + 1, \\ q\{𝔱s_{i}\} & + (q − 1)\{𝔱\}, & if ℓ(ws_{i}) & = ℓ(w) − 1 and i > 0, \\ Q\{𝔱s_{0}\} & + (Q − 1)\{𝔱\}, & if ℓ(ws_{i}) & = ℓ(w) − 1 and i = 0. \end{matrix}

By considering roots, it is straightforward to determine from the tableau 𝔱 which case of the lemma applies.

Lemma 3.10 Maintain the notation of Lemma 3.9. Then ℓ(ws_{i}) = ℓ(w) + 1 if and only if one of the following holds.

(i) i, i + 1 ∈ 𝔱 and row_{𝔱}(i) < row_{𝔱}(i + 1); (ii) −i, −i − 1 ∈ 𝔱 and row_{𝔱}(−i − 1) < row_{𝔱}(−i); (iii) −i, i + 1 ∈ 𝔱; or, (iv) i = 0 and 1 ∈ 𝔱^{(2)}.

The cases where 𝔱s_{i} is not row standard are covered by the next result.

Lemma 3.11 Suppose that 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) is row standard and that 𝔱s_{i} is not. Then either

(i) i = 0 and 1 ∈ 𝔱^{(1)} and {𝔱}T_{0} = Q{𝔱}; or (ii) i > 0 and i and i + 1 belong to the same row of 𝔱 and {𝔱}T_{i} = q{𝔱}; or (iii) i > 0 and −i and −i − 1 belong to the same row of 𝔱 and {𝔱}T_{i} = q{𝔱}.

Proof: It is clear that the only cases where 𝔱s_{i} is not row standard are covered by the statement of the lemma.

Consider first the case where i = 0 and 1 ∈ 𝔱^{(1)}. We know that {𝔱} = u_{a}^{+}x_{λ}T_{w}T_{d}, as in (3.5). Moreover, 1 is fixed by d because 1 ∈ 𝔱^{(1)}. Therefore, T_{0} commutes with T_{d}, and using (2.4) we get

{𝔱}T_{0} = x_{λ}T_{w}u_{a}^{+}T_{0}T_{d} = Qx_{λ}T_{w}u_{a}^{+}T_{d} = Q{𝔱}.

This completes the proof of part (i) of the lemma. The proof of the other parts is similar to that of [7, 3.2(ii)]. □

For future reference, we next collect several results concerning the action of H on M^{λ}.

Lemma 3.12 Suppose that 𝔱 is row standard. Assume that i is a positive integer in 𝔱^{(2)}. Then {𝔱}T_{i,1}T_{0}T_{1,i} is a linear combination of λ–bitabloids {𝔰} such that 𝔰 is row standard and (i) −i ∈ 𝔰; and (ii) for all integers j with j ≠ ±i, we have j ∈ 𝔰 if and only if j ∈ 𝔱.

Proof: If i = 1 then the claim follows immediately from Lemmas 3.9 and 3.10. Thus let i > 1. Using (3.9), (3.10) and (3.11), we see that {𝔱}T_{i−1} is a linear combination of one or two λ–bitabloids {𝔰} where 𝔰 is row standard and

(i) i − 1 ∈ 𝔰^{(2)}. (ii) i ∈ 𝔰 ⇔ i − 1 ∈ 𝔱. (iii) j ∈ 𝔰 ⇔ j ∈ 𝔱 for |j| ≠ i − 1, i.

Hence, by induction we may assume that

{𝔱}T_{i,1}T_{0}T_{1,i−1} = {𝔱}T_{i−1}T_{i−1,1}T_{0}T_{1,i−1}

is a linear combination of λ–bitabloids {𝔰} such that 𝔰 is row standard and

(i) −(i − 1) ∈ 𝔰^{(2)}. (ii) i ∈ 𝔰 ⇔ i − 1 ∈ 𝔱. (iii) j ∈ 𝔰 ⇔ j ∈ 𝔱 for |j| ≠ i − 1, i.

Therefore, {𝔱}T_{i,1}T_{0}T_{1,i} = {𝔱}T_{i−1}T_{i−1,1}T_{0}T_{1,i−1}T_{i−1} is a linear combination of λ–bitabloids {𝔰} as in the statement of the lemma. □

Definition 3.13 Let M_{−}^{λ} denote the R–submodule of M^{λ} spanned by those λ–bitabloids {𝔱} where 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) and 𝔱^{(2)} contains a negative integer.

Note that Lemmas 3.9 and 3.11 show that

3.14 M_{−}^{λ} is an H(𝔖_{r})–submodule of M^{λ}.

Lemma 3.12 now has the following corollary.

Corollary 3.15 Suppose that {𝔱} is a λ–bitabloid. Assume that 1, 2, …, b ∈ 𝔱^{(2)}. Then

{𝔱}u_{b}^{−} ≡ Q^{b}q^{b(b−1)/2}{𝔱} (mod M_{−}^{λ}).

Proof: Note that Q^{b}q^{b(b−1)/2} is the coefficient of the identity in u_{b}^{−}. The corollary follows from multiplying out u_{b}^{−} and applying Lemma 3.12. □

Lemma 3.16 Assume that {𝔱} is a λ–bitabloid and that all the entries in 𝔱 are positive. Suppose that 1 ≤ i < j ≤ r. Then for some integer k

{𝔱}T_{j,i} = q^{k}{𝔱s_{j,i}} + m,

where m is a linear combination of λ–bitabloids {𝔰} such that 𝔰 is row standard and row_{𝔰}(i) > row_{𝔱s_{j,i}}(i) = row_{𝔱}(j).

Proof: Note first that js_{j,i} = i, so row_{𝔱s_{j,i}}(i) = row_{𝔱}(j). Now,

{𝔱}T_{j−1} = q^{ϵ}{𝔱s_{j−1}} + m_{1}

where ϵ = 0 or 1, and either m_{1} = 0 or row_{𝔱}(j − 1) > row_{𝔱}(j) and m_{1} = (q − 1){𝔱}. This proves the lemma in the case where i = j − 1. Assume, therefore, that i < j − 1.

By induction

{𝔱s_{j−1}}T_{j−1,i} = q^{k_{2}}{𝔱s_{j,i}} + m_{2}

for some integer k_{2}, where m_{2} is a linear combination of λ–bitabloids {𝔰} such that 𝔰 is row standard and row_{𝔰}(i) > row_{𝔱s_{j−1,i}}(i) = row_{𝔱}(j). Furthermore, if m_{1} ≠ 0 then m_{1} = (q − 1){𝔱} and induction shows that {𝔱}T_{j−1,i} is a linear combination of λ–bitabloids {𝔰} where 𝔰 is row standard and

row_{𝔰}(i) ≥ row_{𝔱s_{j−1,i}}(i) = row_{𝔱}(j − 1) > row_{𝔱}(j).

Therefore,

{𝔱}T_{j,i} = {𝔱}T_{j−1}T_{j−1,i} = q^{k}{𝔱s_{j,i}} + m,

for some integer k, where m is a linear combination of λ–bitabloids {𝔰} where 𝔰 is row standard and row_{𝔰}(i) > row_{𝔱s_{j,i}}(i). □

Corollary 3.17 Assume that {𝔱} is a λ–bitabloid and that all the entries in 𝔱 are positive. Suppose that 0 ≤ a ≤ r. Then for some integer k,

{𝔱}h_{a,r−a} = q^{k}{𝔱w_{a,r−a}} + m,

where m is a linear combination of λ–bitabloids {𝔰} (𝔰 row standard) such that for all i with 1 ≤ i ≤ r − a,

row_{𝔰}(i) ≥ row_{𝔱}(a + i)

with the inequality being strict for at least one i.

Proof: We claim that the following holds. For 0 ≤ j ≤ r − a

(3.18)  {𝔱}T_{a+1,1}T_{a+2,2} ⋯ T_{a+j,j} = q^{k_{1}}{𝔱s_{a+1,1} ⋯ s_{a+j,j}} + m

where k_{1} is an integer, and m is a linear combination of λ–bitabloids {𝔰} where 𝔰 is row standard and, for 1 ≤ i ≤ j, row_{𝔰}(i) ≥ row_{𝔱}(a + i), with strict inequality for some i with 1 ≤ i ≤ j, and

row_{𝔰}(i) = row_{𝔱}(i) for i > a + j.

Equation (3.18) certainly holds if j = 0. Since s_{a+j+1,j+1} permutes only numbers i with j + 1 ≤ i ≤ a + j, Lemma 3.16 provides the induction step, so (3.18) holds.

Since h_{a,r−a} = T_{a+1,1}T_{a+2,2} ⋯ T_{r,r−a}, by (2.2), the corollary is the special case of (3.18) when j = r − a. □

4 The (Q, q)-Schur algebra

The q–Schur algebra [8, 9] associated with the Hecke algebra H(𝔖_{r}) is defined as the endomorphism ring

End_{H(𝔖_{r})} ( ⨁_{λ∈Λ(n,r)} M^{λ})

where Λ(n, r) is the set of compositions of r into n parts and M^{λ} is the H(𝔖_{r})–module induced from the trivial module of the parabolic subalgebra H(𝔖_{λ}). Because each partition in Λ(n, r) corresponds to a parabolic subgroup of 𝔖_{r}, a natural generalization to a Schur algebra of type B is the algebra

(4.1)  S̃ = End_{H} ( ⨁_{J⊆Δ_{0}} M^{J}),

where M^{J} = Ind_{J}^{Δ_{0}} x_{J}R and x_{J} = ∑_{w∈W_{J}} T_{w}. The algebra S̃ has been investigated in [12, 15].

We wish to consider a larger algebra, which contains S̃ as a subalgebra, in which we take endomorphisms of induced modules which correspond to arbitrary reflection subgroups of W_{r}, rather than just the parabolic subgroups.

Definition 4.2 The (Q, q)-Schur algebra is the endomorphism ring

S_{R}(n, r) = End_{H} ( ⨁_{λ∈Λ_{2}(n,r)} M^{λ}).

If n ≥ r then all bipartitions of r belong to Λ_{2}(n, r), and we see from (3.2) that the next result holds.

Lemma 4.3 If n ≥ r then the (Q, q)-Schur algebra is Morita equivalent to End_{H} ( ⨁_{λ} M^{λ}) where the direct sum is now over all bipartitions λ of r.

If n ≤ r the (Q, q)-Schur algebra S_{R}(n, r) is Morita equivalent to an algebra of the form eS_{R}(r, r)e for some idempotent e in S_{R}(r, r). Similarly, the algebra S̃ defined in (4.1) is Morita equivalent to an algebra of the form eS_{R}(n, r)e.

Henceforth, we fix n and r and let S = S_{R}(n, r).

Definition 4.4 For subsets K, L, M of Δ_{0} we set

D_{K,L}^{M} = D_{K} ∩ D_{L}^{−1} ∩ W_{M}.

Thus, if K ⊆ M and L ⊆ M then D_{K,L}^{M} is the set of distinguished double (W_{K}, W_{L})– coset representatives in W_{M}. Similarly, we define D_{K}^{M} = D_{K} ∩ W_{M}. If M = Δ_{0} we usually omit the superscript Δ_{0}.

The next lemma summarizes the properties of the distinguished (double) coset representatives which we need; the proofs of these results can be found in [3, §2.7].

Lemma 4.5 Suppose that K, L ⊆ Δ_{0} and let d ∈ D_{K,L}.

(i) Every element of W_{K}dW_{L} is uniquely expressible in the form xdw where x^{−1} ∈ D_{K∩Ld^{−1}}^{K} and w ∈ W_{L}; moreover, ℓ(xdw) = ℓ(x) + ℓ(d) + ℓ(w).

(ii) W_{Kd∩L} = d^{−1}W_{K}d ∩ W_{L}; consequently, T_{d}^{−1}H_{K}T_{d} ∩ H_{L} = H_{Kd∩L}. (iii) If K ⊆ L then D_{K} = D_{K}^{L}D_{L}; in particular, D_{L} ⊆ D_{K}.

As in [7, 3.4] (compare [9, 1.4]) we have the following theorem.

Theorem 4.6 The algebra S̃ is free as an R–module with basis

{ψ_{I,J}^{d} | I, J ⊆ Δ_{0}, d ∈ D_{I,J}},

where ψ_{I,J}^{d} : x_{J}H → x_{I}H is the homomorphism given by

ψ_{I,J}^{d}(x_{J}h) = ∑_{w∈W_{I}dW_{J}} T_{w}h, (h ∈ H).

We now want to exhibit a generic basis of S, that is a basis which is independent of the choice of the ring R and the values of the parameters Q and q. By construction it is enough to show that

(4.7)  ℌ_{λ,μ} = Hom_{H}(M^{λ}, M^{μ})

for λ, μ ∈ Λ_{2}(n, r) is free as an R–module and has a generic basis.

First, from the defining relations (2.1) for W_{r} we see that for any w ∈ W_{r} the number of factors equal to s_{0} is the same in every reduced expression of w. Thus the following definition makes sense.

Definition 4.8 Suppose that w ∈ W_{r} and ℓ(w) = a + b where a reduced expression for w has exactly b factors equal to s_{0}. We set

\underline{q}^{\underline{ℓ}(w)} = q^{a}Q^{b} and \underline{q}^{−\underline{ℓ}(w)} = q^{−a}Q^{−b}.

Next we state Frobenius reciprocity and a result which shows that, as for Hecke algebras of type A, induction from parabolic subalgebras is not only a left adjoint functor to restriction, but a right adjoint functor as well.

Theorem 4.9 Let J ⊆ Δ_{0}, and let M be an H_{J}–module, N an H–module. Then the following hold.

(i) Hom_{H} ( Ind_{J}^{Δ_{0}} M , N) ≅ Hom_{H_{J}} (M , Res_{J}^{Δ_{0}} N) where an isomorphism is given by re-

stricting a map in Hom_{H} ( Ind_{J}^{Δ_{0}} M , N) to the H_{J}-subspace M⊗1 of Ind_{J}^{Δ_{0}} M = M ⊗_{H_{J}} H.

(ii) Hom_{H} (N , Ind_{J}^{Δ_{0}} M) ≅ Hom_{H_{J}} ( Res_{J}^{Δ_{0}} N , M) where an isomorphism given by send-

ing ϕ ∈ Hom_{H_{J}} ( Res_{J}^{Δ_{0}} N , M) to the map ϕ̂ ∈ Hom_{H} (N , Ind_{J}^{Δ_{0}} M) where ϕ̂ is defined on x ∈ N by

ϕ̂(x) = ∑_{d∈D_{J}} \underline{q}^{−\underline{ℓ}(d)}ϕ(xT_{d}^{∗}) ⊗ T_{d}.

Proof: Part (i) is Frobenius reciprocity which holds for arbitrary rings and subrings. For part (ii) we refer to [7, 2.6]; the proof there can be adjusted easily. □

We remark that the inverse to the map ϕ ↦ ϕ̂ in part (ii) of the theorem is given by

θ ↦ θ_{1} ∈ Hom_{H_{J}} ( Res_{J}^{Δ_{0}} N , M)

for θ ∈ Hom_{H}(N , Ind_{J}^{Δ_{0}} M), where θ_{1} is defined as follows. For x ∈ N we have x_{d} ∈ M uniquely defined by

θ(x) = ∑_{d∈D_{J}} x_{d} ⊗ T_{d} ∈ Ind_{J}^{Δ_{0}} M.

The map θ_{d} : N → M : x ↦ x_{d} is easily seen to be R–linear and, for d = 1, even H_{J}–linear, (in general, θ_{d} is T_{d}^{−1}H_{J}T_{d}–linear). Then θ_{1} is the desired map.

We shall also use the Mackey Decomposition Theorem [7, 2.7].

Theorem 4.10 Let I, J be subsets of Δ_{0} and let M be an H_{J}–module. Then

Res_{I}^{Δ_{0}} Ind_{J}^{Δ_{0}} M = ⨁_{d∈D_{J,I}} Ind_{Jd∩I}^{I} Res_{Jd∩I}^{Jd} M ⊗ T_{d},

considering M ⊗ T_{d} as an H_{Jd}–module, where H_{Jd} = T_{d}^{−1}H_{J}T_{d}.

Now this result taken in conjunction with Theorem 4.9 produces the following intertwining number theorem (compare [7, 2.8]).

Corollary 4.11 Let I, J and M be as in Theorem 4.10 and let N be an H_{I}–module. Let ℌ = Hom_{H} ( Ind_{J}^{Δ_{0}} M , Ind_{I}^{Δ_{0}} N). Then

(i) ℌ ≅ ⨁_{d∈D_{I,J}} Hom_{H_{J}} (M , Ind_{Id∩J}^{J} Res_{Id∩J}^{Id} N ⊗ T_{d})

≅ ⨁_{d∈D_{I,J}} Hom_{H_{Id∩J}} ( Res_{Id∩J}^{J} M , Res_{Id∩J}^{Id} N ⊗ T_{d}),

(ii) ℌ ≅ ⨁_{d∈D_{J,I}} Hom_{H_{I}} ( Ind_{Jd∩I}^{I} Res_{Jd∩I}^{Jd} M ⊗ T_{d} , N)

≅ ⨁_{d∈D_{J,I}} Hom_{H_{Jd∩I}} ( Res_{Jd∩I}^{Jd} M ⊗ T_{d} , Res_{Jd∩I}^{I} N).

Remark 4.12 We shall apply Corollary 4.11 in the special case where the modules M and N are the trivial modules x_{J}H_{J} and x_{I}H_{I} respectively. Note that the restriction of the trivial module to a parabolic subalgebra is again the trivial module. Moreover, the identity map generates the endomorphism ring of the trivial module. Corollary 4.11 provides ways to exhibit a basis of ℌ, by choosing the identity maps in each summand. Using Theorem 4.9 the basis elements in parts (i) and (ii) which are labelled by d ∈ D_{I,J} and d^{−1} ∈ D_{J,I} differ by a factor of \underline{q}^{−\underline{ℓ}(d)}.

It is an immediate consequence of the way that W_{r} acts on V that the next result holds.

4.13 Suppose that w ∈ W_{r} and J ⊆ Δ_{0}. Then the following are equivalent.

(i) α_{0} ∈ Jw. (ii) α_{0} ∈ J and α_{0}w = α_{0}. (iii) α_{0} ∈ J and s_{0}w = ws_{0} and ℓ(s_{0}w) = ℓ(w) + 1.

From now on we fix the following notation.

Notation 4.14 Let a and b be integers with 0 ≤ a, b ≤ r and fix an a–bicomposition λ = (λ^{(1)}, λ^{(2)}) and a b–bicomposition μ = (μ^{(1)}, μ^{(2)}) with λ, μ ∈ Λ_{2}(n, r).

(i) Let J_{1} be the subset of {α_{1}, …, α_{a−1}} corresponding to λ^{(1)} and J_{2} be the subset of {α_{a+1}, …, α_{r−1}} corresponding to λ^{(2)}. Similarly, define subsets I_{1} and I_{2} corresponding to μ^{(1)} and μ^{(2)} respectively. Let J = J_{1} ∪ J_{2} and I = I_{1} ∪ I_{2}.

(ii) Let Ã = Δ_{0} \ {α_{a}} if a ≠ r, and Ã = Δ_{0} if a = r. Similarly, B̃ = Δ_{0} \ {α_{b}} is b ≠ r, and B̃ = Δ_{0} otherwise. Let A = Ã \ {α_{0}} and B = B̃ \ {α_{0}}.

Note that

x_{J} = ∑_{w∈W_{J}} T_{w} = x_{J_{1}}x_{J_{2}} = x_{J_{2}}x_{J_{1}} = x_{λ}.

We also remark that J ⊆ A ⊆ Ã ⊆ Δ_{0} and that I ⊆ B ⊆ B̃ ⊆ Δ_{0}.

Note that we write D_{B̃,Ã} instead of D_{B̃,Ã}^{Δ_{0}}.

Lemma 4.15 Let d ∈ D_{B̃,Ã}, and suppose that α_{0} ∈ B̃d ∩ Ã. Then a ≥ 1, b ≥ 1 and T_{d}T_{0} = T_{0}T_{d}. Moreover for h ∈ HÃ we have

u_{a}^{+}x_{J}hT_{0} = Qu_{a}^{+}x_{J}h

and for h ∈ HB̃T_{d} we have

u_{b}^{+}x_{I}hT_{0} = Qu_{b}^{+}x_{I}h.

Proof: Since α_{0} ∈ B̃d ∩ Ã, we have α_{0} ∈ B̃ ∩ Ã and s_{0}d = ds_{0} by (4.13). Therefore, a ≥ 1 and b ≥ 1 by (4.14). Further, since d ∈ D_{B̃,Ã}, we have T_{d}T_{0} = T_{ds_{0}} = T_{s_{0}d} = T_{0}T_{d}.

Let h ∈ HÃ. Now Ã = {α_{0}, …, α_{a−1}} ∪ {α_{a+1}, … α_{r−1}} and hence we may write h = h_{1}h_{2} = h_{2}h_{1} with h_{1} ∈ H_{{α_{0},... ,α_{a−1}}} and h_{2} ∈ H_{{α_{a+1},... ,α_{r−1}}}. By (2.4),

u_{a}^{+}x_{J}hT_{0} = x_{J}u_{a}^{+}h_{1}h_{2}T_{0} = x_{J}h_{1}u_{a}^{+}T_{0}h_{2} = Qx_{J}h_{1}u_{a}^{+}h_{2} = Qu_{a}^{+}x_{J}h.

Similarly, u_{b}^{+}x_{I}h′T_{0} = Qu_{b}^{+}x_{I}h′ for any h′ ∈ HB̃. Since T_{d} commutes with T_{0} we have u_{b}^{+}x_{I}hT_{0} = Qu_{b}^{+}x_{I}h for any h ∈ HB̃T_{d}. □

Definition 4.16 We say that a triple (d, v, u) is admissible for (λ, μ) if d ∈ D_{B̃,Ã} and v ∈ D_{J,Bd∩A}^{A} and u ∈ D_{I,B∩Jvd^{−1}}^{B}.

Our current aim is to show that ℌ_{λ,μ} has a generic basis indexed by the set of admissible triples for (λ, μ). We first prove three technical lemmas about admissible triples; after that, the next three lemmas introduce in turn the elements d, v and u of an admissible triple and show how they determine a basis of the R–module ℌ_{λ,μ}.

Lemma 4.17 Let (d, v, u) be an admissible triple. Then B ∩ Jvd^{−1} ⊆ Ad^{−1}.

Proof: Let α ∈ B ∩ Jvd^{−1}. Then d^{−1}s_{α}d = v^{−1}s_{β}v for some β ∈ J. Now, v^{−1}s_{β}v ∈ W_{A} so

ℓ(d) + ℓ(v^{−1}s_{β}v) = ℓ(dv^{−1}s_{β}v) = ℓ(s_{α}d) = ℓ(s_{α}) + ℓ(d) = 1 + ℓ(d).

Therefore, ℓ(v^{−1}s_{β}v) = 1 and so βv ∈ A; consequently, α = βvd^{−1} ∈ Ad^{−1} proving the lemma. □

Ultimately we are interested in bases which are indexed not by admissible triples but by elements of D_{I,J}; this is the point of part (iv) below.

Lemma 4.18 Suppose that (d, v, u) is an admissible triple. Then

(i) ℓ(udv^{−1}) = ℓ(u) + ℓ(d) + ℓ(v^{−1}). (ii) ud ∈ D_{I,A∩Jv}. (iii) dv^{−1} ∈ D_{B,J}. (iv) udv^{−1} ∈ D_{I,J}.

Proof: Note that d ∈ D_{B̃,Ã} ⊆ D_{B,A} and that u ∈ W_{B} and v^{−1} ∈ D_{Bd∩A}^{A}; hence (i) follows from Lemma 4.5(i).

To prove (ii), first note that ud ∈ D_{I}^{B}D_{B} = D_{I} by Lemma 4.5(iii) because u ∈ D_{I}^{B} and d ∈ D_{B̃,Ã} ⊆ DB̃ ⊂ D_{B}. On the other hand, by Lemma 4.17, u ∈ (D_{B∩(A∩Jv)d^{−1}}^{B})^{−1} and d ∈ D_{B,A∩Jv} so, by taking K = B and L = A ∩ Jv in Lemma 4.5(i),

ℓ(udw) = ℓ(u) + ℓ(d) + ℓ(w) = ℓ(ud) + ℓ(w)

for any w ∈ W_{A∩Jv}. Therefore, ud ∈ (D_{A∩Jv})^{−1} proving (ii). A similar argument proves (iii).

Finally, we prove (iv). By (iii), dv^{−1} ∈ D_{B} so udv^{−1} ∈ D_{I}^{B}D_{B} = D_{I} by Lemma 4.5(iii). Furthermore, if w ∈ W_{J} then

ℓ(udv^{−1}w) = ℓ(u) + ℓ(dv^{−1}) + ℓ(w) = ℓ(udv^{−1}) + ℓ(w)

by Lemma 4.5(i), since u^{−1} ∈ D_{B∩Jvd^{−1}}^{B} and dv^{−1} ∈ D_{B,J} by (iii). Therefore, udv^{−1} ∈ D_{J}^{−1} and the proof is complete. □

Lemma 4.19 Let (d, v, u) be an admissible triple. Then I ∩ Jvd^{−1}u^{−1} ⊆ Bu^{−1}.

Proof: Suppose that α ∈ I ∩ Jvd^{−1}u^{−1}. Then u^{−1}s_{α}u = dv^{−1}s_{β}vd^{−1} for some β ∈ J. Since u^{−1}s_{α}u ∈ W_{B} and dv^{−1} ∈ D_{B,J} by Lemma 4.18(iii),

ℓ(vd^{−1}) + ℓ(u^{−1}s_{α}u) = ℓ(vd^{−1}u^{−1}s_{α}u) = ℓ(s_{β}vd^{−1}) = 1 + ℓ(vd^{−1}).

Therefore, αu ∈ B and so the lemma is proved. □

We are now ready to investigate the Hom–space ℌ_{λ,μ} = Hom_{H}(M^{λ}, M^{μ}).

Lemma 4.20 Suppose that λ and μ are bicompositions in Λ_{2}(n, r). Then

ℌ_{λ,μ} ≅ ⨁_{d∈D_{B̃,Ã}} Hom_{H_{B̃d∩Ã}} ( ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ , Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d}).

Proof: Note that {α_{0}, …, α_{a−1}} ∪ J_{2} ⊆ Ã. Hence, by Lemma 3.7 and transitivity of induction, M^{λ} ≅ IndΔ_{0} ; Ã u_{a}^{+}x_{J}HÃ. Similarly, we have M^{μ} ≅ IndΔ_{0} ; B̃ u_{b}^{+}x_{I}HB̃. This is the key observation which allows us to apply Theorems 4.9 and 4.10 to ℌ_{λ,μ}.

ℌ_{λ,μ} = Hom_{H}(M^{λ}, M^{μ})

≅ Hom_{H} ( IndΔ_{0} ; Ã u_{a}^{+}x_{J}HÃ , IndΔ_{0} ; B̃ u_{b}^{+}x_{I}HB̃)

≅ Hom_{HÃ} (u_{a}^{+}x_{J}HÃ , ResΔ_{0} ; Ã IndΔ_{0} ; B̃ u_{b}^{+}x_{I}HB̃)

≅ ⨁_{d∈D_{B̃,Ã}} Hom_{HÃ} (u_{a}^{+}x_{J}HÃ , IndÃ ; B̃d∩Ã Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d})

≅ ⨁_{d∈D_{B̃,Ã}} Hom_{H_{B̃d∩Ã}} ( ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ , Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d})

□

Recall that Δ = Δ_{0} \ {α_{0}}.

Lemma 4.21 Suppose that d ∈ D_{B̃,Ã} and let

ℌ_{λ,μ}^{(d)} = Hom_{H_{B̃d∩Ã}} ( ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ , Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d}).

Then

ℌ_{λ,μ}^{(d)} ≅ ⨁_{v∈D_{J,Bd∩A}^{A}} Hom_{H_{S(v)}} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , Res_{S(v)}^{Bd}(Ind_{I}^{B} x_{I}R)T_{d}),

where S(v) = Bd ∩ A ∩ Jv ⊆ Δ.

Proof: If α_{0} ∈ B̃d ∩ Ã then T_{0} acts on the modules u_{a}^{+}x_{J}HÃ and u_{b}^{+}x_{I}HB̃T_{d} as multiplication by Q, by Lemma 4.15; so, every H_{Bd∩A}–linear map between these modules is automatically H_{B̃d∩Ã}–linear.

If α_{0} ∉ B̃d ∩ Ã then B̃d ∩ Ã = Bd ∩ A ⊆ Δ by (4.13). Therefore,

ℌ_{λ,μ}^{(d)} = Hom_{H_{Bd∩A}} ( ResÃ ; Bd∩A u_{a}^{+}x_{J}HÃ , Res_{Bd∩A}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d}).

Now, by [10, 3.6], u_{a}^{+}x_{J}HÃ ≅ x_{J}H_{A} as H_{A}–modules and, similarly, u_{b}^{+}x_{I}HB̃T_{d} ≅ x_{I}H_{B}T_{d} as T_{d}^{−1}H_{B}T_{d}–modules. Therefore, by Theorems 4.10 and 4.9,

ℌ_{λ,μ}^{(d)} ≅ Hom_{H_{Bd∩A}} ( Res_{Bd∩A}^{A} x_{J}H_{A} , Res_{Bd∩A}^{Bd} x_{I}H_{B}T_{d})

= Hom_{H_{Bd∩A}} ( Res_{Bd∩A}^{A} Ind_{J}^{A} x_{J}R , Res_{Bd∩A}^{Bd} x_{I}H_{B}T_{d})

≅ ⨁_{v∈D_{J,Bd∩A}^{A}} Hom_{H_{Bd∩A}} ( Ind_{S(v)}^{Bd∩A} Res_{S(v)}^{Jv} x_{J}T_{v}R , Res_{Bd∩A}^{Bd} x_{I}H_{B}T_{d}),

≅ ⨁_{v∈D_{J,Bd∩A}^{A}} Hom_{S(v)} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , Res_{S(v)}^{Bd∩A} Res_{Bd∩A}^{Bd} x_{I}H_{B}T_{d}).

The lemma now follows from the transitivity of restriction and the observation that Ind_{I}^{B} x_{I}R ≅ x_{I}H_{B}. □

Lemma 4.22 Suppose that d ∈ D_{B̃,Ã} and v ∈ D_{J,Bd∩A}^{A} and let

ℌ_{λ,μ}^{(d,v)} = Hom_{H_{S(v)}} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , Res_{S(v)}^{Bd}(Ind_{I}^{B} x_{I}R)T_{d}).

Then

ℌ_{λ,μ}^{(d,v)} ≅ ⨁_{u∈D_{I,B∩Jvd^{−1}}^{B}} Hom_{H_{S[u]}} ( Res_{S[u]}^{Jv} x_{J}T_{v}R , Res_{S[u]}^{Iud} x_{I}T_{ud}R),

where S[u] = S(v) ∩ Iud.

Proof: We observe that S(v)d^{−1} = B ∩ Ad^{−1} ∩ Jvd^{−1} ⊆ Δ determines the parabolic subgroup W_{B} ∩ d(W_{A} ∩ v^{−1}W_{J}v)d^{−1} of W_{r} by Lemma 4.5(ii) since d^{−1} ∈ D_{Ã,B̃} ⊆ D_{A∩Jv,B}. Therefore,

ℌ_{λ,μ}^{(d,v)} = Hom_{H_{S(v)}} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , Res_{S(v)}^{Bd}(Ind_{I}^{B} x_{I}R)T_{d})

≅ Hom_{H_{S(v)}} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , (Res_{S(v)d^{−1}}^{B} Ind_{I}^{B} x_{I}R)T_{d})

≅ ⨁_{u} Hom_{H_{S(v)}} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , (Ind_{S[u]d^{−1}}^{S(v)d^{−1}} Res_{S[v]d^{−1}}^{Iu} x_{I}T_{u}R)T_{d}),

observing that S[u]d^{−1} = S(v)d^{−1} ∩ Iu, where the last isomorphism follows by the Mackey theorem 4.10, and u runs through D_{I,S(v)d^{−1}}^{B}. Now

S(v)d^{−1} = B ∩ Ad^{−1} ∩ Jvd^{−1} = B ∩ Jvd^{−1}

by Lemma 4.17 and ud ∈ D_{I,S(v)}^{B} by Lemma 4.18(ii). Therefore

d^{−1}W_{S(v)d^{−1}}d = W_{S(v)}.

Moreover x_{I}T_{u}T_{d}R is a T_{d^{−1}}T_{u^{−1}}H_{I}T_{u}T_{d}-module, and we have

(Ind_{S[u]d^{−1}}^{S(v)d^{−1}} Res_{S[v]d^{−1}}^{Iu} x_{I}T_{u}R)T_{d} = Ind_{S[u]}^{S(v)} Res_{S[u]}^{Iud}(x_{I}T_{ud}R),

since T_{u}T_{d} = T_{ud} by Lemma 4.18(i). We have shown

ℌ_{λ,μ}^{(d,v)} ≅ ⨁_{u∈D_{I,B∩Jvd^{−1}}^{B}} Hom_{H_{S(v)}} ( Res_{S(v)}^{Jv} x_{J}T_{v}R , Ind_{S[u]}^{S(v)} Res_{S[u]}^{Iud} x_{I}T_{ud}R).

So, Frobenius reciprocity and the transitivity of restriction complete the proof. □

Combining the last three lemmas we have shown that ℌ_{λ,μ} ≅ ⨁ ℌ_{λ,μ}^{(d,v,u)} where the sum is over all admissible triples (d, v, u) and

ℌ_{λ,μ}^{(d,v,u)} = Hom_{H_{S[u]}} ( Res_{S[u]}^{Jv} x_{J}T_{v}R , Res_{S[u]}^{Iud} x_{I}T_{ud}R).

Now, ℌ_{λ,μ}^{(d,v,u)} is a free R–module of rank 1, since both modules involved are one dimensional. As a nonzero element in ℌ_{λ,μ}^{(d,v,u)} we may choose

ϕ_{I,J}^{(d,v,u)} : x_{J}T_{v} ↦ x_{I}T_{ud}.

By Theorem 4.9, in the R–module ℌ_{λ,μ}^{(d)} of Lemma 4.21, this map corresponds to the map (compare also [7, 3.4])

ϕ̂_{I,J}^{(d,v,u)} ∈ Hom_{H_{Bd∩A}} ( Res_{Bd∩A}^{A} x_{J}H_{A} , Res_{Bd∩A}^{Bd} x_{I}H_{B}T_{d}),

given by (noting that ud ∈ D_{I,S(v)}),

(4.23)  ϕ̂_{I,J}^{(d,v,u)}(x_{J}T_{v}h) = ∑_{w∈W_{I}udW_{S(v)}} T_{w}h

where h ∈ H_{Bd∩A}. The H_{Bd∩A}–direct summands of

(4.24)  Res_{Bd∩A}^{A} x_{J}H_{A} = ⨁_{v′∈D_{J,Bd∩A}^{A}} Ind_{S(v′)}^{Bd∩A} Res_{S(v′)}^{Jv′} x_{J}T_{v′}R

with v′ ≠ v are taken to zero by ϕ̂_{I,J}^{(d,v,u)}.

We denote the image of q^{ℓ(v)}ϕ̂_{I,J}^{(d,v,u)} in ℌ_{λ,μ} given by Lemmas 4.21 and 4.20 by φ_{μ,λ}^{(d,v,u)}. (The power of q is inserted here because in Lemma 4.21 we used part (ii) of Corollary 4.11; see Remark 4.12.) We have proved the following.

Theorem 4.25 Let λ, μ ∈ Λ_{2}(n, r). Then ℌ_{λ,μ} is a free R–module with basis

𝔅_{λ,μ} = {φ_{μ,λ}^{(d,v,u)} | (d, v, u) an admissible triple for (λ, μ)}.

Corollary 4.26 The (Q, q)-Schur algebra S = S_{R}(n, r) is a free R–module with basis

𝔅 = {φ_{μ,λ}^{(d,v,u)} | λ, μ ∈ Λ_{2}(n, r) , and (d, v, u) an admissible triple for (λ, μ)}

which does not depend on the commutative ring R or the choice of the parameter values Q and q.

We call the basis 𝔅 the standard basis of S.

We next describe the μ–bitabloids which do not belong to M_{−}^{μ} (see Definition 3.13) and are in the support of the image of {𝔱^{λ}} = u_{a}^{+}x_{J} under the homomorphism φ_{μ,λ}^{(d,v,u)} for an admissible triple (d, v, u) for (λ, μ).

Theorem 4.27 Let (d, v, u) be an admissible triple for (λ, μ) and let c = udv^{−1}. Then c ∈ D_{I,J} and

φ_{μ,λ}^{(d,v,u)}({𝔱^{λ}}) ≡ u_{b}^{+}ψ_{I,J}^{c}(x_{J}) ≡ ∑_{w∈W_{I}cW_{J}} u_{b}^{+}T_{w} (mod M_{−}^{μ}).

Proof: We have already seen in Lemma 4.18 that c = udv^{−1} is an element of D_{I,J}.

In (4.23) we constructed the H_{Bd∩A}–linear map ϕ̂_{I,J}^{(d,v,u)} in ℌ_{λ,μ}^{(d)}, and determined the image of x_{J}T_{v}h for h ∈ H_{Bd∩A} under this map.

We apply Lemma 4.15 (compare Lemma 4.21) to get an H_{B̃d∩Ã}–linear map φ̂_{I,J}^{(d,v,u)} :

ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ ⟶ Res_{B̃d∩Ã}^{B̃.d} u_{b}^{+}x_{I}HB̃T_{d} such that

(4.28)  φ̂_{I,J}^{(d,v,u)}(u_{a}^{+}x_{J}T_{v}h) = ∑_{w∈W_{I}udW_{S(v)}} u_{b}^{+}T_{w}h

for h ∈ H_{B̃d∩Ã}.

According to Lemma 4.20 we have to trace up this map using Theorem 4.9(ii), to get a map

φ̂_{μ,λ}^{(d,v,u)} ∈ Hom_{HÃ} (u_{a}^{+}x_{J}HÃ , IndÃ ; B̃d∩Ã Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d})

≅ Hom_{H_{B̃d∩Ã}} ( ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ , Res_{B̃d∩Ã}^{B̃.d} u_{b}^{+}x_{I}HB̃T_{d}).

Let m ∈ u_{a}^{+}x_{J}HÃ. Then by Theorem 4.9 we have

φ̂_{μ,λ}^{(d,v,u)}(m) = ∑_{f∈DÃ ; B̃d∩Ã} \underline{q}^{−\underline{ℓ}(f)}φ̂_{I,J}^{(d,v,u)}(mT_{f}^{∗})T_{f}.

By (4.13), B̃d ∩ Ã ∩ Δ = Bd ∩ A; so DÃ ; B̃d∩Ã ∩ W_{Δ} = D_{Bd∩A}^{A}. Therefore, we may rewrite the last equation as

(4.29)  φ̂_{μ,λ}^{(d,v,u)}(m) = ∑_{f∈D_{Bd∩A}^{A}} q^{−ℓ(f)}φ̂_{I,J}^{(d,v,u)}(mT_{f}^{∗})T_{f} + \underline{z}.

where

\underline{z} = ∑_{f̃∈DÃ ; B̃d∩Ã\W_{Δ}} \underline{q}^{−\underline{ℓ}(f̃)}φ̂_{I,J}^{(d,v,u)}(mT ∗ ; f̃)Tf̃.

We claim that \underline{z} ∈ M_{−}^{μ}. Observe that f̃ ∈ WÃ and d ∈ (DÃ)^{−1}, so ℓ(df̃) = ℓ(d) + ℓ(f̃). Moreover by (2.4), u_{b}^{+}x_{I}HB̃ = u_{b}^{+}x_{I}H_{B}; therefore

\underline{q}^{−\underline{ℓ}(f̃)}φ̂_{I,J}^{(d,v,u)}(mT ∗ ; f̃)Tf̃ ∈ u_{b}^{+}x_{I}HB̃T_{df̃} = u_{b}^{+}x_{I}H_{B}T_{df̃}.

Now f̃ ∈ DÃ ; B̃d∩Ã but f̃ ∉ W_{Δ}; so, s_{0} is involved in f̃, and in df̃ as well, since ℓ(df̃) = ℓ(d) + ℓ(f̃). We apply [7, 1.4] and Lemmas 3.9 and 3.10 to conclude that \underline{q}^{−\underline{ℓ}(f̃)}φ̂_{I,J}^{(d,v,u)}(mT ∗ ; f̃)Tf̃ ∈ M_{−}^{μ}. Consequently, \underline{z} ∈ M_{−}^{μ} and (4.29) becomes

(4.30)  φ̂_{μ,λ}^{(d,v,u)}(m) ≡ ∑_{f∈D_{Bd∩A}^{A}} q^{−ℓ(f)}φ̂_{I,J}^{(d,v,u)}(mT_{f}^{∗})T_{f} (mod M_{−}^{μ}).

We next investigate φ̂_{I,J}^{(d,v,u)}(mT_{f}^{∗}) when m = u_{a}^{+}x_{J}. Fix f ∈ D_{Bd∩A}^{A} and note that f^{−1} ∈ D_{∅,Bd∩A}^{A}. Let v′ be the distinguished double coset representative in W_{J}f^{−1}W_{Bd∩A}. Then v′ ∈ D_{J,Bd∩A}^{A} and f^{−1} = gv′ for some g ∈ D_{(Bd∩A)v′^{−1}∩J}^{J}. However, by (4.24),

φ̂_{I,J}^{(d,v,u)}(u_{a}^{+}x_{J}T_{f}^{∗}) = 0 unless v′ = v; therefore we assume that f^{−1} = gv for some g ∈ D_{(Bd∩A)v^{−1}∩J}^{J}. By Lemma 4.5(i), we have a disjoint union

W_{J}f^{−1}W_{Bd∩A} = W_{J}vW_{Bd∩A} = ⋃_{g^{−1}∈D_{S(v)v^{−1}}^{J}} gvW_{Bd∩A},

recalling that v ∈ D_{J,Bd∩A}^{A} and S(v) = Bd∩A∩Jv. Hence, modulo M_{−}^{μ}, we can rewrite (4.30) when m = u_{a}^{+}x_{J} as

φ̂_{μ,λ}^{(d,v,u)}(u_{a}^{+}x_{J}) ≡ ∑_{g^{−1}∈D_{S(v)v^{−1}}^{J}} q^{−ℓ(gv)}φ̂_{I,J}^{(d,v,u)}(u_{a}^{+}x_{J}T_{gv})T_{gv}^{∗}

≡ ∑_{g^{−1}∈D_{S(v)v^{−1}}^{J}} q^{−ℓ(v)}φ̂_{I,J}^{(d,v,u)}(u_{a}^{+}x_{J}T_{v})T_{gv}^{∗}

≡ ∑_{g^{−1}∈D_{S(v)v^{−1}}^{J}} q^{−ℓ(v)}u_{b}^{+} ∑_{w∈W_{I}udW_{S(v)}} T_{w}T_{gv}^{∗}

where the second equality comes from the fact that g ∈ W_{J} and the third equality from (4.28).

Using Theorem 4.9 we can lift the homomorphism q^{ℓ(v)}φ̂_{μ,λ}^{(d,v,u)} to the homomorphism φ_{μ,λ}^{(d,v,u)} in Hom_{H}(M^{λ}, M^{μ}) by extending it to the module M^{λ} = IndΔ_{0} ; Ã u_{a}^{+}x_{J}HÃ. In particular the image of the generator u_{a}^{+}x_{J} remains unchanged. Consequently, in order to complete the proof it suffices to prove the next lemma.

Lemma 4.31 Suppose that (d, v, u) is an admissible triple and let c = udv^{−1}. Then

∑_{x∈W_{I}cW_{J}} T_{x} = ∑_{g^{−1}∈D_{S(v)v^{−1}}^{J}} ∑_{y∈W_{I}udW_{S(v)}} T_{y}T_{gv}^{∗}.

Proof: Since c ∈ D_{I,J}, by Lemma 4.5(i)

∑_{w∈W_{I}cW_{J}} T_{w} = ∑_{x^{−1}∈D_{I∩Jc^{−1}}^{I}^{y∈W_{J}}} T_{x}T_{c}T_{y} = ∑_{x^{−1}∈D_{I∩Jc^{−1}}^{I}} T_{x}T_{ud}T_{v}^{∗} ∑_{y∈W_{J}} T_{y}.

Recall that S(v) = Bd ∩ A ∩ Jv. Suppose that f ∈ W_{S(v)v^{−1}} ⊆ W_{J}, and let e = v^{−1}fv. Then e ∈ W_{S(v)} ⊆ W_{Bd∩A}. Since v ∈ D_{J,Bd∩A} we have ℓ(ev^{−1}) = ℓ(e) + ℓ(v^{−1}) and ℓ(v^{−1}f) = ℓ(v^{−1}) + ℓ(f); so,

T_{e}T_{v}^{∗} = T_{e}T_{v^{−1}} = T_{v^{−1}}T_{f} = T_{v}^{∗}T_{f}.

Therefore,

T_{v}^{∗} ∑_{y∈W_{J}} T_{y} = ∑_{g∈D_{S(v)v^{−1}}^{J}^{f∈W_{S(v)v^{−1}}}} T_{v}^{∗}T_{f}T_{g}^{∗} = ∑_{g∈D_{S(v)v^{−1}}^{J}^{e∈W_{S(v)}}} T_{e}T_{v}^{∗}T_{g}^{∗},

and consequently,

\begin{aligned}∑_{w∈W_{I}cW_{J}} T_{w} & = ∑_{x^{−1}∈D_{I∩Jc^{−1}}^{I}} T_{x}T_{ud} ∑_{g∈D_{S(v)v^{−1}}^{J}^{e∈W_{S(v)}}} T_{e}T_{gv}^{∗} \\ & = ∑_{g∈D_{S(v)v^{−1}}^{J}} ( ∑_{x^{−1}∈D_{I∩Jc^{−1}}^{I}^{e∈W_{S(v)}}} T_{x}T_{ud}T_{e})T_{gv}^{∗}. \end{aligned}

Now ud ∈ D_{I,S(v)} by Lemma 4.18(ii) so by Lemma 4.5(i) it suffices to show that I ∩ S(v)d^{−1}u^{−1} = I ∩ Jc^{−1}. However,

I ∩ S(v)d^{−1}u^{−1} = I ∩ (B ∩ Ad^{−1} ∩ Jvd^{−1})u^{−1}
= I ∩ (B ∩ Jvd^{−1})u^{−1}
= I ∩ Jvd^{−1}u^{−1},

by Lemma 4.17 and Lemma 4.19 respectively. □

Remark 4.32 In the group algebra case, that is specializing both Q and q to 1, one sees easily that

u_{a}^{+}x_{J} = ∑_{w∈W_{λ}} w ∈ RW_{r}.

In this case φ_{μ,λ}^{(d,v,u)} takes the generator u_{a}^{+}x_{J} of the module M^{λ} to

φ_{μ,λ}^{(d,v,u)}(u_{a}^{+}x_{J}) = ∑_{w∈W_{λ}cW_{μ}} w,

where c = udv^{−1} and the subgroups W_{λ} and W_{μ} are defined in (2.9).

By (4.13), D_{B̃,Ã} ∩ W_{Δ} = D_{B,A}^{Δ}. Therefore, ℌ_{λ,μ} in (4.20) splits into two R–modules ℌ_{λ,μ}^{+} and ℌ_{λ,μ}^{−} where

ℌ_{λ,μ}^{+} = ⨁_{d∈D_{B,A}^{Δ}} Hom_{H_{B̃d∩Ã}} ( ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ , Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d})

and

ℌ_{λ,μ}^{−} = ⨁_{d∈D_{B,A}\W_{Δ}} Hom_{H_{B̃d∩Ã}} ( ResÃ ; B̃d∩Ã u_{a}^{+}x_{J}HÃ , Res_{B̃d∩Ã}^{B̃d} u_{b}^{+}x_{I}HB̃T_{d}).

We apply Lemma 4.15 and Theorem 4.9 again to get

ℌ_{λ,μ}^{+} ≅ ⨁_{d∈D_{B,A}^{Δ}} Hom_{H_{Bd∩A}} ( Res_{Bd∩A}^{A} u_{a}^{+}x_{J}H_{A} , Res_{Bd∩A}^{Bd} u_{b}^{+}x_{I}H_{B}T_{d})

≅ ⨁_{d∈D_{B,A}^{Δ}} Hom_{H_{A}} (u_{a}^{+}x_{J}H_{A} , Ind_{Bd∩A}^{A} Res_{Bd∩A}^{Bd} u_{b}^{+}x_{I}H_{B}T_{d})

≅ Hom_{H_{Δ}} (x_{J}H_{Δ} , x_{I}H_{Δ}).

The last Hom-set has basis 𝔅_{Δ} = {φ̃_{I,J}^{c} | c ∈ D_{I,J}^{Δ}}, where the map φ̃_{I,J}^{c} is defined by

φ̃_{I,J}^{c}(x_{J}h) = ∑_{w∈W_{I}cW_{J}} T_{w}h (h ∈ H_{Δ}).

We observe that the map φ̃_{I,J}^{c} involves the same double coset W_{I}cW_{J} as the map ψ_{I,J}^{c} in Theorem 4.27. Moreover, it follows that

𝔅_{λ,μ}^{+} = {φ_{μ,λ}^{(d,v,u)} ∈ 𝔅_{λ,μ} | d ∈ D_{B,A}^{Δ} and (d, v, u) admissible for (λ, μ)}

is a basis of ℌ_{λ,μ}^{+} since 𝔅_{λ,μ}^{+} and 𝔅_{Δ} have the same cardinality. This proves the next lemma (it is also easily checked directly).

Lemma 4.33 Let λ, μ ∈ Λ_{2}(n, r) and let I and J be as in (4.14). Then

D_{I,J}^{Δ} = {udv^{−1} | (d, v, u) an admissible triple for (λ, μ) and d ∈ D_{B,A}^{Δ}}

In view of the last result we make the following definition.

Definition 4.34 Given c ∈ D_{I,J}^{Δ} let φ_{μ,λ}^{c} ∈ 𝔅_{λ,μ} be the map φ_{μ,λ}^{(d,v,u)} where (d, v, u) is the admissible triple such that c = udv^{−1}.

Observe that Theorem 4.27 implies that for an admissible triple (d, v, u) and c = udv^{−1}

(4.35)  φ_{μ,λ}^{(d,v,u)}({𝔱^{λ}}) ≡ ∑_{b∈D_{Ic∩J}^{J}}{𝔱^{μ}}T_{cb} (mod M_{−}^{μ}).

From this the first part of the next result follows easily. The second part is a special case of Theorem 4.27.

Corollary 4.36 Let (d, v, u) be admissible, and let c = udv^{−1}.

(i) If c ∉ D_{I,J}^{Δ} then φ_{μ,λ}^{(d,v,u)}({𝔱^{λ}}) ≡ 0 (mod M_{−}^{μ}). (ii) If c ∈ D_{I,J}^{Δ}, then

φ_{μ,λ}^{c}({𝔱^{λ}}) ≡ u_{b}^{+} ∑_{w∈W_{I}cW_{J}} T_{w} (mod M_{−}^{μ}).

5 (Q, q)–tensor space and (Q, q)–Weyl modules

In view of Lemma 4.3 we assume hereafter that n ≥ r. We denote the bicomposition ((−), (1^{r})) by ω. Thus ω ∈ Λ_{2}(n, r), since n ≥ r.

Definition 5.1 (Q, q)-tensor space is the R–module

E = E_{R}(n, r) = ⨁_{λ∈Λ_{2}(n,r)} M^{λ}.

Definition 5.2 Let λ be an a–bicomposition in Λ_{2}(n, r).

(i) The homomorphism φ_{λ,λ} is the identity map on M^{λ} and maps M^{μ} to zero when λ ≠ μ. (ii) The homomorphism φ_{λ,ω} is premultiplication of H = M^{ω} by u_{a}^{+}x_{λ} and maps M^{μ} to

zero when μ ≠ ω.

Thus φ_{λ,λ} is the projection of E onto M^{λ} and φ_{λ,ω} is the canonical epimorphism of H onto the cyclic H-module M^{λ}. In particular φ_{ω,ω} is the identity on H and maps M^{λ} to zero for λ ≠ ω. From the proof of Theorem 4.27, using in particular (4.29), one sees easily that the following holds.

Lemma 5.3 Let λ be a bicomposition.

(i) The restriction of φ_{λ,λ} to M^{λ} equals φ_{λ,λ}^{(1,1,1)}. (ii) The restriction of φ_{λ,ω} to H equals φ_{λ,ω}^{(1,1,1)}.

By construction {φ_{λ,λ} | λ ∈ Λ_{2}(n, r)} is a set of orthogonal idempotents of S whose sum is the identity element of S. Observe that

φ_{λ,λ}φ_{λ,ω} = φ_{λ,ω} = φ_{λ,ω}φ_{ω,ω}, (5.4)

hence, as in [9, 2.5], we see that postmultiplication by φ_{λ,ω} embeds the left ideal Sφ_{λ,λ} of S in the left ideal Sφ_{ω,ω}. Hence, S acts faithfully on Sφ_{ω,ω} and, when R is a field, every irreducible left S-module occurs as a composition factor of the left ideal Sφ_{ω,ω} of S. We have the following (compare [9, 2.10 and 2.6]).

Lemma 5.5 The Hecke algebra H is canonically isomorphic to φ_{ω,ω}Sφ_{ω,ω} and acts on Sφ_{ω,ω} as a set of S–linear maps. The H–submodule φ_{λ,ω}H of Sφ_{ω,ω} is isomorphic to M^{λ} and (Q, q)-tensor space E is isomorphic to the left ideal Sφ_{ω,ω} of S as an (S,H)–bimodule.

Proof: We have canonical isomorphisms

H ≅ End_{H}(H) ≅ Hom_{H}(x_{ω}H , x_{ω}H) ≅ φ_{ω,ω}Sφ_{ω,ω}.

Identifying H and φ_{ω,ω}Sφ_{ω,ω} we see that H acts on Sφ_{ω,ω} on the right as a set of S–linear maps. Also, φ_{λ,ω}h ↦ u_{a}^{+}x_{λ}h for h ∈ H gives an H–isomorphism between φ_{λ,ω}H and M^{λ}.

By premultiplying Sφ_{ω,ω} by ∑_{λ∈Λ_{2}(n,r)} φ_{λ,λ}, which is the identity of S, we obtain

Sφ_{ω,ω} = ⨁_{λ∈Λ_{2}(n,r)} M^{λ} = E.  □

Definition 5.6 Let U be a left S-module, and let λ ∈ Λ_{2}(n, r). Then the R–submodule U_{λ} = φ_{λ,λ}U of U is the weight space of U of weight λ.

Note that φ_{λ,λ}U is free as an R–module. We have

U = ⨁_{λ∈Λ_{2}(n,r)} φ_{λ,λ}U, (5.7)

since {φ_{λ,λ} | λ ∈ Λ_{2}(n, r)} is a set of orthogonal idempotents whose sum is the identity of S. This decomposition of U is called the weight space decomposition of U, (compare [9, 2.13]).

Lemma 5.8 The weight space decomposition of (Q, q)–tensor space E is given as

E = ⨁_{λ∈Λ_{2}(n,r)} M^{λ}.

More generally, if U is any R–submodule of E then its weight space decomposition is

U = ⨁_{λ∈Λ_{2}(n,r)} U ∩ M^{λ}.

Let λ ∈ Λ_{2}(n, r) and recall the definition of z_{λ} from (2.16). With the above identifications we have

(5.9)  z_{λ} = φ_{λ,ω}h_{a,r−a}Tπ̂_{λ}u_{r−a}^{−}ŷ_{λ}.

Contrast the next definition with our definition of the Specht module S^{λ} = z_{λ}H.

Definition 5.10 Suppose that λ is an a-bicomposition of r.

(i) Let W^{λ} = Sz_{λ}. (ii) Let L^{λ} = Su_{r−a}^{−}ŷ_{λ}.

We call W^{λ} a (Q, q)-Weyl module.

As in [9, 3.9] we have the following.

5.11 If λ and μ are associated bicompositions then W^{λ} ≅ W^{μ} and L^{λ} ≅ L^{μ}.

We next define a bilinear form ⟨ , ⟩_{λ} on M^{λ} by specifying its values at pairs of basis elements, which are given in (3.8) by λ–bitabloids. For such a pair {𝔱}, {𝔱′}, where 𝔱 = 𝔱^{λ}w is row standard and w ∈ W_{r}, we set

⟨\{𝔱\}, \{𝔱′\}⟩_{λ} = \begin{cases}\underline{q}^{\underline{ℓ}(w)} & if \{𝔱\} = \{𝔱′\} \\ 0 & if \{𝔱\} ≠ \{𝔱′\}. \end{cases} (5.12)

Extend these bilinear forms to a non-degenerate bilinear form ⟨ , ⟩ on (Q, q)-tensor space E = ⨁ M^{λ} by specifying that M^{λ} and M^{μ} are orthogonal when λ ≠ μ.

Recall that the anti–automorphism ^{∗} on H is defined by T_{w}^{∗} = T_{w^{−1}} for w ∈ W_{r}. As in [9, 4.1] for all h ∈ H, x, y ∈ E

⟨xh, y⟩ = ⟨x, yh^{∗}⟩, (5.13)

and consequently E is self–dual as an H-module. Here M^{∗} = Hom_{H}(M, R) is the dual of an H-module M, the right action of H on M^{∗} being given by fh(x) = f(xh^{∗}), for f ∈ M^{∗}, h ∈ H and x ∈ M. The discussion in [9, section 1] shows that ∗ extends to an anti–isomorphism of S, which we also denote by ∗. This allows us to define the dual of a left S–module. We then have the following.

Lemma 5.14 Let x, y ∈ E, h ∈ H and s ∈ S. Then

(i)  ⟨sx, y⟩ = ⟨x, s^{∗}y⟩
(ii)  ⟨xh, y⟩ = ⟨x, yh^{∗}⟩

In particular, (Q, q)–tensor space E is self–dual as a left S–module.

Since (u_{r−a}^{−}ŷ_{λ})^{∗} = u_{r−a}^{−}ŷ_{λ} we can contract the bilinear form which is given by restricting ⟨ , ⟩ to L^{λ} to obtain a bilinear form ≪ , ≫ on L^{λ} such that

\begin{aligned}≪s_{1}u_{r−a}^{−}ŷ_{λ} , s_{2}u_{r−a}^{−}ŷ_{λ}≫ & = ≪s_{1}φ_{ω,ω}u_{r−a}^{−}ŷ_{λ} , s_{2}φ_{ω,ω}u_{r−a}^{−}ŷ_{λ}≫ \\ & = ⟨s_{1}φ_{ω,ω} , s_{2}u_{r−a}^{−}ŷ_{λ}⟩, \end{aligned}

for all s_{1}, s_{2} ∈ S, (compare [14, section 5.5] and [9, 4.2]). The proof of [9, 4.4] now gives the following theorem.

Theorem 5.15 (The Submodule Theorem) Suppose that R is a field and λ ∈ Λ_{2}(n, r). Let U be an S–submodule of L^{λ}. Then W^{λ} ⊆ U or U ⊆ W^{λ⊥}.

Corollary 5.16 Suppose that R is a field and λ ∈ Λ_{2}(n, r). Then W^{λ} ∩ W^{λ⊥} is the unique maximal submodule of W^{λ}, and the quotient W^{λ}/(W^{λ} ∩ W^{λ⊥}) is an absolutely irreducible self–dual S-module.

Proof: The corollary will be an immediate consequence of the Submodule Theorem once we prove that the generator z_{λ} of W^{λ} is anisotropic. We have using (5.9) and (3.8)

\begin{aligned}≪z_{λ}, z_{λ}≫ & = ⟨φ_{λ,ω}h_{a,r−a}Tπ̂_{λ} , φ_{λ,ω}h_{a,r−a}Tπ̂_{λ}u_{r−a}^{−}ŷ_{λ}⟩ \\ & = ⟨\{𝔱\} , \{𝔱\}u_{r−a}^{−}ŷ_{λ}⟩, \end{aligned}

where 𝔱 = 𝔱̂_{λ} = 𝔱^{λ}w_{a,r−a}π̂_{λ} (see Definitions 2.8 and 2.14). Hence , by Corollary 3.15,

≪z_{λ}, z_{λ}≫ = Q^{r−a}q^{(r−a)(r−a−1)/2}⟨{𝔱}, {𝔱}ŷ_{λ}⟩.

But {𝔱}ŷ_{λ} = {𝔱} + v, where v is a linear combination of λ–bitabloids distinct from {𝔱} (cf. [7, 4.1]). Therefore

≪z_{λ}, z_{λ}≫ = Q^{r−a}q^{(r−a)(r−a−1)/2}⟨{𝔱}, {𝔱}⟩ ≠ 0.
□

Definition 5.17 Suppose that R is a field and λ ∈ Λ_{2}(n, r). Let F^{λ} be the irreducible S_{R}(n, r)-module W^{λ}/(W^{λ} ∩ W^{λ⊥}).

Theorem 5.18 Suppose that R is a field and that λ, μ ∈ Λ_{2}(n, r). We have F^{λ} ≅ F^{μ} if and only if λ and μ are associated bicompositions. Thus

{F^{λ} | λ is a bipartition of r}

is a set of non–isomorphic absolutely irreducible self–dual S_{R}(n, r)-modules.

If λ and μ are bipartitions of r then let d_{λμ} be the multiplicity of F^{μ} as a composition factor of W^{λ}. If the bipartitions of r are ordered lexicographically then the matrix (d_{λμ}) is upper unitriangular.

Proof: This theorem is proved in exactly the same way as [9, 4.11 and 4.13] (note [10, 3.7]). □

6 The semistandard basis theorem

Hereafter, we assume that λ, μ ∈ Λ_{2}(n, r), with λ being an a-bipartition.

Definition 6.1 (i) A λ–bitableau of type μ is an array T = (T^{(1)}, T^{(2)}) of integers obtained from the diagram [λ] by replacing each cross by a non-zero integer according to the following restrictions. For 1 ≤ i ≤ n, the number of entries j with j = i is equal to the ith part of μ^{(1)}; and the number of entries j with |j| = n + i is equal to the ith part of μ^{(2)}. For k = 1, 2, T^{(k)} denotes the array of integers replacing the crosses in λ^{(k)}. We denote the set of λ–bitableaux of type μ by 𝔗(λ, μ). Note that all entries less than n in T ∈ 𝔗(λ, μ) are positive.

(ii) An element T = (T^{(1)}, T^{(2)}) of 𝔗(λ, μ) is positive if all the entries in T^{(1)} are greater than 0 and all the entries in T^{(2)} are greater than n. We denote the set of positive elements of 𝔗(λ, μ) by 𝔗^{+}(λ, μ).

(iii) A semistandard λ–bitableau of type μ is an element of 𝔗^{+}(λ, μ) in which the entries are weakly increasing along each row and strictly increasing down each column. We denote the set of semistandard λ–bitableaux of type μ by 𝔗_{∘}(λ, μ).

Example 6.2 If r = n = 7 and λ = ((3, 2), (1, 1)) and μ = ((3, 1), (2, 1)) then

T_{1}= (\begin{matrix}1 & 8 & 9 \\ 2 & 1 \end{matrix} , \begin{matrix}1 \\ −8 \end{matrix}) ∈ 𝔗(λ, μ),
and  T_{2} = (\begin{matrix}1 & 1 & 1 \\ 2 & 8 \end{matrix} , \begin{matrix}8 \\ 9 \end{matrix}) ∈ 𝔗_{∘}(λ, μ).

Note that there are no positive λ–bitableaux of type μ if λ is an a-bipartition and μ is a b-bicomposition of r with a < b.

Definition 6.3 Given T ∈ 𝔗^{+}(λ, μ), let T(i) equal the integer which appears in T in the place which is occupied by i in 𝔱̂_{λ}. Choose a total order < on 𝔗^{+}(λ, μ) such that if A and B are elements of 𝔗^{+}(λ, μ) then A < B if one of the following holds.

(i) For all j, k.

#{i | A(i) = j and i belongs to column k of 𝔱̂_{λ}}

= #{i | B(i) = j and i belongs to column k of 𝔱̂_{λ}},

and A(i) < B(i) for the smallest integer i such that A(i) ≠ B(i).

(ii) ∑_{i=1}^{r−a} A(i) < ∑_{i=1}^{r−a} B(i).

(iii) ∑_{i=1}^{r−a} A(i) = ∑_{i=1}^{r−a} B(i), and for all j, k

#{i | A(i) ≤ j and i belongs to the first k columns of 𝔱̂_{λ}}

≥ #{i | B(i) ≤ j and i belongs to the first k columns of 𝔱̂_{λ}}.

Let V (λ, μ) be the free R-module spanned by all T ∈ 𝔗(λ, μ). Given T ∈ 𝔗^{+}(λ, μ), define E_{T} to be the element of V (λ, μ) obtained by summing over those T′ which are row equivalent to T, and then applying the signed column symmetrizer in a fashion similar to that in [17, 8.1.11].

Example 6.4 If

T = (\begin{matrix}1 & 1 \\ 2 \end{matrix} , \begin{matrix}5 \\ 6 \end{matrix} ) ,

then

E_{T} = (\begin{matrix}1 & 1 \\ 2 \end{matrix} , \begin{matrix}5 \\ 6 \end{matrix} ) − (\begin{matrix}2 & 1 \\ 1 \end{matrix} , \begin{matrix}5 \\ 6 \end{matrix} ) − (\begin{matrix}1 & 1 \\ 2 \end{matrix} , \begin{matrix}6 \\ 5 \end{matrix} ) + (\begin{matrix}2 & 1 \\ 1 \end{matrix} , \begin{matrix}6 \\ 5 \end{matrix} )

We shall need the following result, which follows easily from the corresponding theorem [17, 8.1.16] for Weyl modules for general linear groups.

Theorem 6.5 The R–submodule of V (λ, μ) spanned by {E_{T} | T ∈ 𝔗^{+}(λ, μ)} is a free R– module with basis {E_{T} | T ∈ 𝔗_{∘}(λ, μ)}.

Definition 6.6 (i) If T = (T^{(1)}, T^{(2)}) ∈ 𝔗(λ, μ) then let ε(T) = 1 if an even number of

entries in T^{(2)} are negative , and ε(T) = −1, otherwise. (ii) Let σ be the linear map on V (λ, μ) which sends each T ∈ 𝔗(λ, μ) to

σ(T) = ∑ ε(T′)T,

where the sum ranges over all those T′ ∈ 𝔗(λ, μ) which are obtained from T by changing the sign of some of the entries in T which are greater than n.

Example 6.7 If

T = (\begin{matrix}1 & 1 \\ 2 \end{matrix} , ^{^{5 6})} ,

then

σ(T) = (\begin{matrix}1 & 1 \\ 2 \end{matrix} , ^{^{5 6})} − (\begin{matrix}1 & 1 \\ 2 \end{matrix} , ^{^{−5 6})}

− (\begin{matrix}1 & 1 \\ 2 \end{matrix} , ^{^{5 −6})} + (\begin{matrix}1 & 1 \\ 2 \end{matrix} , ^{^{−5 −6})} .

Theorem 6.5 immediately gives the following corollary.

Corollary 6.8 The R–module generated by {σ(E_{T}) | T ∈ 𝔗^{+}(λ, μ)} is free with basis {σ(E_{T}) | T ∈ 𝔗_{∘}(λ, μ)}.

To facilitate our calculations, we introduce for fixed bicompositions λ, μ ∈ Λ_{2}(n, r) three linear maps, α, β, and γ from M^{μ} into V (λ, μ).

Definition 6.9 Let α, β, γ be the linear transformations from M^{μ} into V (λ, μ) which are given as follows. Suppose that 𝔱 = (𝔱^{(1)}, 𝔱^{(2)}) is a μ–bitableau. The λ–bitableau α{𝔱} of type μ is obtained from 𝔱^{λ} and 𝔱 as follows. For 1 ≤ i ≤ r

(i) replace the entry i in 𝔱^{λ} by j if i or −i occurs in row j of 𝔱^{(1)}, and (ii) replace the entry i in 𝔱^{λ} by n + j (respectively −n − j) if i (respectively −i) occurs in

row j of 𝔱^{(2)}.

The definitions of β{𝔱} and γ{𝔱} are obtained in a similar way, replacing 𝔱^{λ} by 𝔱̂^{λ} and 𝔱̂_{λ} respectively.

Observe that these maps are independent of the choice of tableau 𝔱 in {𝔱} and so are well defined.

Example 6.10 Assume that r = n = 6, and let λ = ((2^{2}), (2)), μ = ((3, 1), (1^{2})). Then

𝔱^{λ} = (\begin{matrix}1 & 2 \\ 3 & 4 \end{matrix} , ^{^{5 6})} 𝔱̂^{λ} = (\begin{matrix}3 & 4 \\ 5 & 6 \end{matrix} , ^{^{1 2})} 𝔱̂_{λ} = (\begin{matrix}3 & 5 \\ 4 & 6 \end{matrix} , ^{^{1 2})} .

If

𝔱 = (\begin{matrix}3 & 4 & 6 \\ 1 \end{matrix} , \begin{matrix}−2 \\ 5 \end{matrix})

then

α\{𝔱\} = (\begin{matrix}2 & −7 \\ 1 & 1 \end{matrix} , ^{^{8 1})} , β\{𝔱\} = (\begin{matrix}1 & 1 \\ 8 & 1 \end{matrix} , ^{^{2 −7})}

and

γ\{𝔱\} = (\begin{matrix}1 & 8 \\ 1 & 1 \end{matrix} , ^{^{2 −7})} .

Note that all three maps defined above have inverses. Given T ∈ 𝔗(λ, μ) we define the μ– bitabloid {𝔱} = α^{−1}T as follows. If the place occupied by i in 𝔱^{λ} is occupied by j (respectively by −j) in T put i (respectively −i) in row j of 𝔱, counting the rows of 𝔱^{(1)} as row 1, 2, … and the rows of 𝔱^{(2)} as row n + 1, n + 2, …, and then take the bitabloid {𝔱} containing the bitableau 𝔱. For the maps β and γ use 𝔱̂^{λ} and 𝔱̂_{λ} respectively instead of 𝔱^{λ}.

Lemma 6.11 The maps α, β and γ induce bijections between the set of μ–bitabloids and 𝔗(λ, μ).

The maps defined in (6.9) can be used to define an action of W_{r} on the set 𝔗(λ, μ) by taking the preimage under one of these maps, acting on the resulting bitabloid and taking the image under the same map again. The action of 𝔖_{r} on 𝔗(λ, μ) is given by place permutations, where the numbering of the places is determined by 𝔱^{λ} if we use α, by 𝔱̂^{λ} if we use β, and by 𝔱̂_{λ} if we use γ.

Our main aim in this section is to prove that the Weyl module W^{λ} has a basis which is indexed by the semistandard λ–bitableaux of various types. Indeed, we shall show that the weight space W^{λ} ∩ M^{μ} (see Definition 5.6) is free as an R–module with basis indexed by 𝔗_{∘}(λ, μ). We begin with a special case.

Theorem 6.12 Let R be a field of characteristic zero and Q = q = 1. Then

dim(W^{λ} ∩ M^{μ}) = |𝔗_{∘}(λ, μ)|.

Proof: In this case H is isomorphic to the group algebra RW_{r} so h_{a,r−a} = w_{a,r−a}, and

u_{a}^{+} = ∏_{i=1}^{a}(1 + (i, −i)) and u_{r−a}^{−} = ∏_{i=1}^{r−a}(1 − (i, −i)).

Since R is a field of characteristic zero, W^{λ} ∩ M^{μ} is spanned by the elements of the form {𝔱}z_{λ} where {𝔱} varies over the μ–bitabloids. This follows immediately from Definition 5.6 and Lemma 5.8.

Let 𝔱 be a row standard μ–bitableau. For i ∈ r^{+}, we have

\{𝔱\}(i, −i) = \begin{cases}\{𝔱\}, & if i is an entry of 𝔱^{(1)}, \\ \{𝔱(i, −i)\}, & if i or −i is an entry of 𝔱^{(2)}. \end{cases}

Let X = {i ∈ r^{+} | i or − i ∈ 𝔱^{(2)}} and X_{a} = X ∩ {1, 2, …, a}. Then

{𝔱}u_{a}^{+} = 2^{a−|X_{a}|} ∑{{𝔱_{1}} | 𝔱_{1} ∈ A_{𝔱}}, (6.13)

where A_{𝔱} is the set of row standard μ–bitableaux which agree with 𝔱, except that the integers in X_{a} are allowed to have either sign.

Next,

ũ_{r−a}^{−} := w_{a,r−a}u_{r−a}^{−}w_{a,r−a}^{−1} = ∏_{i=a+1}^{r}(1 − (i, −i)).

Therefore

(6.14) \{𝔱\}u_{a}^{+}ũ_{r−a}^{−} = \begin{cases}2^{a−|X_{a}|} ∑\{±\{𝔱_{1}\} | 𝔱_{1} ∈ B_{𝔱}\}, & if \{a + 1, …, r\} ⊆ X, \\ 0, & otherwise. \end{cases}

where the sum is over the set B_{𝔱} of the 2^{|X|} row standard μ–bitableaux 𝔱_{1} = (𝔱_{1}^{(1)}, 𝔱_{1}^{(2)}) which are the same as 𝔱 except that the integers in 𝔱_{1}^{(2)} are allowed to have either sign. (The coefficient of {𝔱_{1}} is +1 if and only if |{i ∈ 𝔱^{(2)} | a + 1 ≤ i ≤ r}| − |{i ∈ 𝔱_{1}^{(2)} | a + 1 ≤ i ≤ r}| is even.)

Assume now that {a + 1, …, r} ⊆ X. Then

{𝔱}u_{a}^{+}w_{a,r−a}u_{r−a}^{−} = 2^{a−|X_{a}|} ∑{±{𝔱_{1}} | 𝔱_{1} ∈ C_{𝔱}},

where the sum is over the set C_{𝔱} of row standard μ–bitableaux 𝔱_{1} = (𝔱_{1}^{(1)}, 𝔱_{1}^{(2)}) which are row equivalent to 𝔱w_{a,r−a}, except that the integers in 𝔱_{1}^{(2)} can have either sign. Let {𝔱^{+}} be the μ–bitabloid obtained from 𝔱w_{a,r−a} by changing the signs of all the negative entries and taking the row equivalence class. Since {a + 1, …, r} ⊆ X, we have

{a + 1, …, r}w_{a,r−a} = {1, 2, …, r − a} ⊆ 𝔱^{+(2)}.

Hence β({𝔱^{+}}) ∈ 𝔗^{+}(λ, μ) and 2^{|X_{a}|−a}β({𝔱}u_{a}^{+}w_{a,r−a}u_{r−a}^{−}) = ±σ(β{𝔱^{+}}). Therefore, by (2.17),

2^{|X_{a}|−a}β(z_{λ}) = ±σ(β({𝔱^{+}})x̂_{λ}Tπ̂_{λ} ŷ_{λ}).

If β({𝔱^{+}}) = T, then Tx̂_{λ} is a multiple of the sum over those T′ which are row equivalent to T (since x̂_{λ} is the sum over the row symmetrizer of T). Hence β(z_{λ}) is a non-zero multiple of σ(E_{T}). The theorem now follows from Corollary 6.8. □

We shall generalise Theorem 6.12, making use of the special case in the course of the proof. First we need to reformulate some results from section 4 in the language of λ– tableaux using the map α introduced in Definition 6.9. Define μ̃ to be the composition of r into 2n parts μ̃ = (μ̃_{1}, μ̃_{2}, …, μ̃_{2n}) as follows. Suppose that μ^{(1)} = (μ_{1}^{(1)}, …, μ_{n_{1}}^{(1)}) and μ^{(2)} = (μ_{1}^{(2)}, …, μ_{n_{2}}^{(2)}). Thus n_{1} + n_{2} = n. We define

(6.15)  μ̃_{i} = \begin{cases}μ_{i}^{(1)}, & for 1 ≤ i ≤ n_{1}, \\ μ_{i}^{(2)}, & for n + 1 ≤ i ≤ n + n_{2}, \\ 0, & otherwise. \end{cases}

Define λ̃ similarly.

Denote the set of λ̃–tableaux of type μ̃ by 𝔗(λ̃, μ̃). We can turn T ∈ 𝔗^{+}(λ, μ) into an element T̃ of 𝔗(λ̃, μ̃) by combining the two components of T.

Example 6.16 If r = n = 7 and λ, μ ∈ Λ_{2}(n, r) as in Example 6.2. Then

λ̃ = (3, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0)  and
μ̃ = (3, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0).

Moreover if T = (\begin{matrix}1 & 1 & 1 \\ 2 & 8 \end{matrix} , \begin{matrix}8 \\ 9 \end{matrix}) ∈ 𝔗^{+}(λ, μ), then

T̃  =  \begin{matrix}1 & 1 & 1 \\ 2 & 8 \\ − \\ − \\ − \\ − \\ − \\ 8 \\ 9 \\ − \\ − \\ − \\ − \\ − \end{matrix}

in 𝔗(λ̃, μ̃).

As in (4.14) we let I and J be the subsets of Δ such that x_{λ} = x_{J} and x_{μ} = x_{I}.

We have the following lemma. The first part is trivial; the second part follows from part one and the corresponding result for type A [7, 1.7(i)].

Lemma 6.17 (i) The map T ↦ T̃ defines a bijection between 𝔗^{+}(λ, μ) and 𝔗(λ̃, μ̃). (ii) There is a bijection between D_{I,J}^{Δ} and the set of positive row standard λ–bitableaux of

type μ given by c ↦ T_{c} for c ∈ D_{I,J}^{Δ}, where T_{c} = α{𝔱^{μ}c}.

Definition 6.18 Given T_{1}, T_{2} ∈ 𝔗(λ, μ) we write T_{1} ∼ T_{2} if T_{1} and T_{2} are row equivalent.

Corollary 4.36 (see also 4.35), taken in conjunction with [7, 1.7, 3.4] now gives the following theorem (note that the map A ↦ 1_{A} in [7, 1.7] is the analogue of the map α from Definition 6.9).

Theorem 6.19 Let c ∈ D_{I,J}^{Δ}. Then

φ_{μ,λ}^{c}({𝔱^{λ}}) ≡ ∑_{α{𝔱_{1}}∼T_{c}}{𝔱_{1}} (mod M_{−}^{μ})

where T_{c} = α{𝔱^{μ}c}.

We are now prepared to embark on the generalisation of Theorem 6.12.

Let T ∈ 𝔗_{∘}(λ, μ) and let 𝔱 be the row standard μ–bitableau such that α{𝔱} = T, which is given by Lemma 6.11.

The semistandard tableau T corresponds to an element c ∈ D_{I,J}^{Δ} by Lemma 6.17. By Theorem 6.19 the H-homomorphism φ_{μ,λ}^{c} maps the generator {𝔱^{λ}} of M^{λ} to an element v of M^{μ} which satisfies

v ≡ ∑_{α{𝔱_{1}}∼T}{𝔱_{1}} (mod M_{−}^{μ}).

Since h_{a,r−a} ∈ H(𝔖_{r}),

vh_{a,r−a} ≡ ∑_{α{𝔱_{1}}∼T}{𝔱_{1}}h_{a,r−a} (mod M_{−}^{μ})

by (3.14). Next, Corollary 3.17 gives

vh_{a,r−a} ≡ ∑_{α{𝔱_{1}}∼T}r_{𝔱_{1}}{𝔱_{1}w_{r,r−a}} + v_{1} (mod M_{−}^{μ}),

where each r_{𝔱_{1}} is a unit in R and v_{1} is a linear combination of μ–bitabloids {𝔱′_{1}} satisfying β{𝔱′_{1}} > β{𝔱w_{a,r−a}} (see part (ii) of Definition 6.3).

Now, α{𝔱_{1}} = β{𝔱_{1}w_{a,r−a}}, so

vh_{a,r−a} ≡ ∑_{β{𝔱_{2}}∼T}r_{𝔱_{2}}{𝔱_{2}} + v_{1} (mod M_{−}^{μ}), (6.20)

where each r_{𝔱_{2}} is a unit.

Note that the numbers 1, 2, …, r − a belong to 𝔱̂^{λ(2)}. On the other hand, since T is semistandard by assumption, all the numbers in T^{(2)} are greater than n. Thus for all β{𝔱_{2}} which are row equivalent to T, all of the entries in the second component of β{𝔱_{2}} are greater than n. By Definition 6.9 we conclude that all the μ–bitabloids {𝔱_{2}} which appear in (6.20) have 1, 2, …, r − a in 𝔱^{(2)}. A similar result applies to the μ–bitabloids which occur in v_{1}. Therefore, by Corollary 3.15,

vh_{a,r−a}u_{r−a}^{−} ≡ ∑_{β{𝔱_{2}}∼T}r′_{𝔱_{2}}{𝔱_{2}} + v_{2} (mod M_{−}^{μ}). (6.21)

where each r′_{𝔱_{2}} is a unit and v_{2} is a linear combination of μ–bitabloids {𝔱′_{1}} which are precisely the μ–bitabloids which are involved with nonzero coefficients in v_{1}. (Indeed Corollary 3.15 implies that the coefficients of {𝔱′_{1}} in v_{1} and v_{2} differ only by a unit.)

Next, let {𝔱^{∗}} be the μ–bitabloid such that γ{𝔱^{∗}} = T. Note that the row standard μ–bitableau 𝔱^{∗} in {𝔱^{∗}} is 𝔱π̂_{λ}.

We have that

vh_{a,r−a}u_{r−a}^{−}Tπ̂_{λ} ≡ r{𝔱^{∗}} + v_{3} (mod M_{−}^{μ})

where r is a unit and v_{3} is a linear combination of μ–bitabloids {𝔱_{3}} such that γ{𝔱_{3}} > T. To see this, note part (iii) of Definition 6.3 and compare with [9, 7.26]. The matrix χ(𝔱, 𝔱^{μ}w_{μ}) which appears in [9, 7.26] is defined in such a way that its (j, k)th entry is equal to the number of entries less than or equal to j in the first k columns of the μ–tableau of type λ obtained by replacing each entry i of 𝔱^{μ}w_{μ} by row_{𝔱}(i).

We arrive at the following element of W^{λ} ∩ M^{μ}.

φ_{μ,λ}^{c}(z_{λ}) = vh_{a,r−a}u_{r−a}^{−}Tπ̂_{λ} ŷ_{λ} ≡ r{𝔱^{∗}} + v_{4} (mod M_{−}^{μ}),

where v_{4} is a linear combination of μ–bitabloids {𝔱_{4}} such that

γ{𝔱_{4}} > T,

by part (i) of Definition 6.3. This is justified as follows. All of the μ–bitabloids {𝔱_{3}} involved in v_{3} when acted upon by the terms T_{w}, appearing in ŷ_{λ}, are linear combinations of μ– bitabloids {𝔱_{4}} which are obtained from {𝔱_{3}} by permuting the entries of the columns in 𝔱_{3}. Since γ{𝔱_{3}} > T, Definition 6.3 shows that γ{𝔱_{4}} > T. Also, {𝔱^{∗}}T_{w} is a linear combination of terms of the form {𝔱^{∗}w′} where w′ is an element in the column stabilizer of 𝔱̂_{λ}. From part (i) of Definition 6.3 we conclude that γ{𝔱^{∗}w′} > T for w ≠ 1 since T is semi-standard by assumption. We have shown the following.

Lemma 6.22 Let T ∈ 𝔗_{∘}(λ, μ) and let

v_{λ,μ}(T) = φ_{μ,λ}^{c}({𝔱})h_{a,r−a}u_{r−a}^{−}Tπ̂_{λ} ŷ_{λ},

where 𝔱 is the unique row standard μ–bitableau such that α{𝔱} = T. Let 𝔱^{∗} be the row standard μ–bitableau such that γ{𝔱^{∗}} = T. Then v_{λ,μ}(T) ∈ W^{λ} ∩ M^{μ} and v_{λ,μ}(T) is congruent modulo M_{−}^{μ} to a linear combination of μ–bitabloids {𝔱_{4}} such that

γ{𝔱_{4}} ≥ T

and the coefficient of {𝔱^{∗}} in v_{λ,μ}(T) is invertible.

Recall that by Lemma 5.8 W^{λ} ∩ M^{μ} is a weight space of the (Q, q)–Weyl module W^{λ}.

Corollary 6.23 Let λ, μ ∈ Λ_{2}(n, r). Then {v_{λ,μ}(T) | T ∈ 𝔗_{∘}(λ, μ)} is a linearly independent subset of the weight space W^{λ} ∩ M^{μ} of W^{λ}.

We now prove our main result.

Theorem 6.24 (The Semistandard Basis Theorem) Let λ, μ ∈ Λ_{2}(n, r) and let Q, q be invertible elements of R. Then the weight space W^{λ}∩M^{μ} of the Weyl module W^{λ} is free as an R–module with basis {v_{λ,μ}(T) | T ∈ 𝔗_{∘}(λ, μ)}. Consequently, W^{λ} is free as an R–module with basis

{v_{λ,μ}(T) | μ ∈ Λ_{2}(n, r), T ∈ 𝔗_{∘}(λ, μ)}.

Proof: Suppose, for the moment, that R = ℚ(q, Q) where Q and q are independent transcendentals. Then H_{R,q,Q}(W_{r}) is isomorphic to RW_{r} by [2]. Hence, in this case, dim(W^{λ}∩M^{μ}) = |𝔗_{∘}(λ, μ)| by Theorem 6.12, so {v_{λ,μ}(T) | T ∈ 𝔗_{∘}(λ, μ)} is a basis of W^{λ}∩M^{μ}.

Assume that m is a nonzero element of W^{λ} ∩ M^{μ} and that the coefficient of every μ– bitabloid which is involved in m belongs to ℤ[q, q^{−1}, Q, Q^{−1}]. We write

m = ∑_{T∈𝔗_{∘}(λ,μ)}r_{T}v_{λ,μ}(T)

with coefficients r_{T} in ℚ(q, Q). We claim that r_{T} ∈ ℤ[q, q^{−1}, Q, Q^{−1}]. In the total order of (6.3) let T_{1} be the first element of 𝔗_{∘}(λ, μ) such that r_{T_{1}} ≠ 0, and let 𝔱_{1} be the row standard μ–bitableau with γ{𝔱_{1}} = T_{1}. The coefficient of {𝔱_{1}} in m belongs to ℤ[q, q^{−1}, Q, Q^{−1}], so by Lemma 6.22

r_{T_{1}} ∈ ℤ[q, q^{−1}, Q, Q^{−1}].

Using Lemma 6.22, we see that

m − r_{T_{1}}v_{λ,μ}(T_{1}) = ∑_{T≠T_{1}^{T∈𝔗∘(λ,μ)}} r_{T}v_{λ,μ}(T)

has the property that the first element T_{2} of 𝔗_{∘}(λ, μ) such that r_{T_{2}} ≠ 0 satisfies T_{2} > T_{1}. Hence, by induction, every r_{T} belongs to ℤ[q, q^{−1}, Q, Q^{−1}].

Now assume that R is an arbitrary commutative ring and q, Q are invertible elements of R. The result of the last paragraph shows that every non-zero element of W^{λ} ∩ M^{μ} can be written as a linear combination of {v_{λ,μ}(T) | T ∈ 𝔗_{∘}(λ, μ)}. Taken in conjunction with (6.23), this proves that

{v_{λ,μ}(T) | T ∈ 𝔗_{∘}(λ, μ)}

is a basis of W^{λ} ∩ M^{μ} and concludes the proof of the theorem. □

References

[1] A.A. Beilinson, G. Lusztig, R. McPherson A geometric setting for the quantum deformation of GL_{n}, Duke Math. J. 61 (1990), 655–677.

[2] C.T. Benson, C.W. Curtis, On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251–273.

[3] R. Carter, Finite groups of Lie type, conjugacy classes and complex characters, John Wiley, New York, 1985.

[4] R. Dipper, Polynomial representations of finite general linear groups in the non-describing characteristic, Progress in Mathematics 95, Birkhäuser Verlag Basel (1991), 343-370.

[5] R. Dipper and S. Donkin, Quantum GL_{n}, Proc. London Math. Soc. (3) 63 (1991), 165-211.

[6] R. Dipper, J. Gruber, Generalized q-Schur algebras and modular representation theory of finite groups with split BN-pairs, preprint Stuttgart (1996).

40 R. Dipper, G. James, A. Mathas

[7] R. Dipper, G.D. James Representations of Hecke algebras of general linear groups, Proc. London Math.Soc.(3), 52 (1986), 20-52.

[8] R. Dipper, G.D. James, The q-Schur algebra, Proc. London Math. Soc. (3)59 (1989), 23-50.

[9] R. Dipper, G.D. James, q-Tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251–282.

[10] R. Dipper, G.D. James, Representations of Hecke algebras of type B_{n}, J. Algebra 146 (1992), 454–481.

[11] R. Dipper, G.D. James, G.E. Murphy, Hecke algebras of type B_{n} at roots of unity, Proc. London Math. Soc. (3), 70 (1995), 505–528.

[12] J. Du, B. Parshall, L. Scott, Stratifying endomorphism algebras associated to Hecke algebras, preprint Sydney (1996).

[13] M. Geck, G. Hiss, Modular representations of finite groups of Lie type in non-defining characteristic, in Finite reductive groups: Related structures and representations, M. Cabanes ed., Birkhäuser (1996), 173–227.

[14] J.A. Green, Polynomial representations of GL_{n}, Lecture notes in Math. 830 Springer-Verlag, Berlin, Heidelberg, New York (1980).

[15] R.Green, Hyperoctahedral Schur algebras, to appear in J. Algebra.

[16] J. Gruber, Cuspidale Untergruppen und Zerlegungszahlen klassischer Gruppen, Dissertation, Heidelberg (1995).

[17] G.D. James, A. Kerber The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley, Reading, Mass. 1981.

[18] A. Mathas,Canonical bases and the Ariki–Koike algebras, preprint, 1996.

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

g.james@ic.ac.uk a.mathas@ic.ac.uk