HYPEROCTAHEDRAL SCHUR ALGEBRAS

R.M. Green

Mathematical Institute Oxford University 24–29 St. Giles’ Oxford OX1 3LB England E-mail: greenr@maths.ox.ac.uk

Typeset by AMS-TEX

The author was supported in part by an E.P.S.R.C. postdoctoral research assistantship.

Proposed running head: Hyperoctahedral Schur Algebras

Current correspondence address:

Dr. R. M. Green, Mathematical Institute, 24–29 St. Giles’, Oxford OX1 3LB, U.K.

Dedicated to Professor J.A. Green on his 70th birthday

Abstract. We study the centralising algebra of a natural action of the hyperoctahedral group (i.e. a finite Weyl group of type B_{r}) on the r-th tensor power of a 2n-dimensional space. The centralising algebra of this is shown to have a product rule similar to Schur’s product rule in type A. We deform this action to an action of the Hecke algebra of type B and study the associated centralising algebra of type B and its dual. We introduce and study q-permutation modules for the algebra.

0. Introduction

The classical Schur algebra, S(n, r), is a finite-dimensional associative algebra which has a basis consisting of certain elements ξ_{i,j}, as described in [7, §2.3]. The product of two such elements can be computed using Schur’s Product Rule ([7, 2.3b]). This is an explicit but elegant formula based on the combinatorics of certain symmetric group actions. A natural question to ask is: what happens if we replace the symmetric group in the definition of Schur’s Product Rule by another group? In Proposition 2.2.7, we investigate this in the case where the group is a hyperoctahedral group, i.e. a Weyl group of type B, with a certain natural action reminiscent of the action of the symmetric group. We call this object the hyperoctahedral Schur algebra.

In §3, the situation is generalised from Weyl groups to Hecke algebras. We thus find an analogue of the q-Schur algebra in type B, which was first introduced by Geck and Hiss in [5, pp. 173–227]. We show in §4 that our algebra is Morita equivalent to an algebra appearing in the work of Gruber and Hiss ([6] §7.2]), and to the original algebra of [5].

It should be remarked that our algebra arises as the q-analogue of a combinatoric object, but the algebra of [6] arises as an endomorphism ring of a certain sum of “qpermutation modules” for the Hecke algebra of type B. The hyperoctahedral Schur algebra of the title is not discussed in [6], although it can of course be recovered by setting the parameters in the q-Schur algebra (of type B) to 1.

We also show that the q-Schur algebra of type B is related to an algebra appearing in [4]. The basis we give for our algebra is naturally compatible with the bases

The author was supported in part by an E.P.S.R.C. postdoctoral research assistantship.

4 R.M. GREEN

in [4]. This leads to an IC (“intersection cohomology”) basis for our algebra, in the case where the parameters are equal.

Finally we show in §4.5 that, subject to certain restrictions on the algebra (similar to the restrictions required in type A), we have a double centraliser property involving the q-Schur algebra of type B and the Hecke algebra of type B.

1. Finite Weyl groups of type B

1.1 Basic properties of hyperoctahedral groups.

The hyperoctahedral groups which we use in this paper are the finite Weyl groups W = W (B_{r}) of type B_{r}. These are given (as in [8, §1]) by generators s_{1}, s_{2}, …, s_{r} and defining relations

\begin{aligned}s_{i}^{2} & = 1, \\ s_{i}s_{j} & = s_{j}s_{i} if |i − j| > 1, \\ s_{i}s_{j}s_{i} & = s_{j}s_{i}s_{j} if |i − j| = 1 and i, j < r, \\ s_{r−1}s_{r}s_{r−1}s_{r} & = s_{r}s_{r−1}s_{r}s_{r−1}. \end{aligned}

(The last three relations are known as braid relations.)

We see from [8, §2] that W (B_{r}) is isomorphic to the wreath product ℤ_{2} ≀ S_{r}. Once this isomorphism is fixed, we see that W acts naturally and faithfully on the right on the ordered set of symbols \{1, 2, …, r, r̄, \overline{r − 1}, …, ¯1\}. We use the representation of W implicit in [8, §2], where the generators s_{i} act as follows. If i < r, then s_{i} exchanges i with i + 1 and exchanges ī with \overline{i + 1}, leaving the other symbols fixed. The generator s_{r} exchanges the symbols r and r̄, leaving the others fixed. We thus have the following well-known lemma:

Lemma 1.1.1. The group W (B_{r}) embeds in the group W (A_{2r−1}) as follows. The generator s_{i} of W (B_{r}) (for i < r) is sent to the length two element s′_{i}s′_{2r−i} of W (A_{2r−1}) which is identified with the double transposition (i, i+1)(2r−i, 2r−i+1). The generator s_{r} of W (B_{r}) is sent to the simple transposition s′_{r} of W (A_{2r−1}) which is identified with the simple transposition (r, r + 1).

Proof. This is clear from the preceding discussion. □

HYPEROCTAHEDRAL SCHUR ALGEBRAS 5

We now define a set similar to the set I(n, r) which is used in the theory of ordinary Schur algebras.

Definition 1.1.2. We define the set I_{B}(2n, r) to be the set of r-tuples

(i_{1}, i_{2}, …, i_{r})

from the alphabet \{1, 2, …, n, n̄, \overline{n − 1}, …, ¯1\}. We call the elements \{1, 2, …, n\} unbarred elements, and we call the elements \{n̄, \overline{n − 1}, …, ¯1\} barred elements. We write |i| = i if i is unbarred, and |ī| = i if ī is barred.

Let V_{2n} be a vector space of dimension 2n with ordered basis

\{e_{1}, e_{2}, …, e_{n}, en̄, e\overline{n−1}, …, e¯1\}.

By analogy with the situation in type A, one would like to be able to define a natural action of W (B_{r}) on I_{B}(2n, r), and hence on the tensor space V^{⊗r}, which resembles the action of W on the symbols {1, …, n, n̄, …, ¯1}.

A natural question now presents itself: is there an analogue of Schur–Weyl duality involving the group W (B_{r}) and its centralising algebra with respect to the module V_{2n}^{⊗r}? One would hope that the centralising algebra would be an object similar to the ordinary Schur algebra which was presented in [7]. One may also ask whether there exists a hyperoctahedral version of the q-Schur algebra (introduced in [2]). We address this question later in the paper.

1.2 Permutation notation and length criteria.

First, we give a permutation representation for an element of the Weyl group W = W (B_{r}). This is well known.

Lemma 1.2.1. The elements w ∈ W are naturally parametrised by the elements of I_{B}(2r, r) of the form (i_{1}, …, i_{r}), where {|i_{m}| : 1 ≤ m ≤ r} = {1, …, r}. The correspondence is given by the criterion that i_{m} = (m)w.

Proof. It is immediate from the realisation of W as a wreath product that (m̄)w = \overline{(m)w} for all w. Thus w is determined by the images (m)w for m ∈ \{1, …, r\}.

6 R.M. GREEN

We now need to show that any element of the given form turns up as the image of an element of W. Since the size of the parametrising set is 2^{r}.r!, this follows by an easy counting argument. □

Note. Note that this notation does not correspond to the disjoint cycle notation for elements of the symmetric group, despite superficial similarities.

The length ℓ(w) of an element w ∈ W is the length of the shortest word in the Coxeter generators s_{i} which is equal to it. We now present a length criterion for elements of W in terms of the permutation representation described above.

Lemma 1.2.2. Let a be an element of I_{B}(2r, r) representing an element of w. Clearly ℓ(ws_{i}) = ℓ(w) ± 1. If i < r then ℓ(ws_{i}) = ℓ(w) + 1 if and only if a_{i} < a_{i+1}. Furthermore, ℓ(ws_{r}) = ℓ(w) + 1 if and only if a_{r} is unbarred.

Proof. This follows from the corresponding well-known length identities in type A. Let us write ι(w) for the element of W (A_{2r−1}) corresponding to w, as in Lemma 1.1.1.

If i < r, then we see that

ℓ(ι(w)ι(s_{i})) = ℓ(ι(w)) ± 2,

where the ± corresponds to the ± in the statement of the lemma. This means that

ℓ(ι(w)s_{i}) = ℓ(ι(w)) ± 1.

We obtain a plus if (i)ι(w) < (i + 1)ι(w), and a minus otherwise, as required. The same applies to w because of the way W (B_{r}) is embedded in the symmetric group on the letters {1, …, n, n̄, …, ¯1}. (This is essentially the observation made in the first paragraph of the proof of Lemma 1.2.1.) Using the fact that a_{i} = (i)w we complete the proof of the case i < r.

If i = r, we find that

ℓ(ι(w)s_{i}) = ℓ(ι(w)) ± 1,

because ι(s_{r}) = s_{r}. The plus occurs if and only if (r).ι(w) < (r̄).ι(w), which is equivalent to saying that (r)ι(w) is unbarred. This happens if and only if (r)w is unbarred, i.e. a_{r} is unbarred, as required. □

HYPEROCTAHEDRAL SCHUR ALGEBRAS 7

We now give a criterion for identifying the shortest right coset representatives with respect to certain parabolic subgroups of W.

Lemma 1.2.3. Let P be a parabolic subgroup of W which does not contain the generator s_{r}. Consider an interval 1 ≤ a < b ≤ r such that (m, m + 1) ∈ P for each a ≤ m < b, but where a is minimal with this property and b is maximal with this property.

Then d ∈ W is a distinguished right coset representative (i.e. one of minimal length) of P in W if for all pairs (a, b) as above, there exists c with a ≤ c ≤ b such that the representation for d (in the sense of Lemma 1.2.1) has the property that the numbers a, a + 1, …, c appear unbarred in ascending order from left to right, and the numbers b̄, \overline{b − 1}, …, \overline{c + 1} appear barred in the order given.

Proof. We prove this by embedding W (B_{r}) in W (A_{2r−1}). The subgroup P corresponds to a parabolic subgroup of W (A_{2r−1}) in an obvious way which preserves the notion of distinguished coset representatives. The group of type A has a well-known criterion for testing for distinguished coset representatives, and our criterion follows easily from it. □

2. Tensor space

2.1 Action of W (B_{r}) on tensor space.

In order to give an action of W = W (B_{r}) on tensor space, we first introduce a permutation representation for W, as follows.

Lemma 2.1.1. We have a natural action of W on I_{B}(2n, r), defined by the condition that i.s is such that

(i_{1}, …, i_{r}).s = \{ \begin{matrix}(i_{(1)s}, …, i_{(r)s}) & if s ≠ s_{r}, \\ (i_{1}, …, i_{r−1}, \overline{i_{r}}) & if s = s_{r}, \end{matrix}

where we interpret ¯ ī to mean i and where s is a generating involution. This gives rise to an action on I_{B}(2n, r) × I_{B}(2n, r), given by (i, j).π = (i.π, j.π). We write a ∼ b to mean that a and b are in the same W -orbit, where a and b are both elements of I_{B}(2n, r) or both elements of I_{B}(2n, r) × I_{B}(2n, r).

8 R.M. GREEN

Proof. It is an easy matter to check that the defining relations of W (B_{r}) are satisfied. □

Next, we transfer this action of W to the tensor power V^{⊗r} = V_{2n}^{⊗r}, where V is a ℚ-vector space. (The symbols n and r will be reserved with this meaning.) Lemma 2.1.2. There is a right action of W on V^{⊗r} in which the generator s = s_{a} acts on the natural basis vectors as follows:

(e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}).s_{a} = \{ \begin{matrix}e_{i_{(1)s}} ⊗ ⋯ ⊗ e_{i_{(r)s}} & if s ≠ s_{r}, \\ e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r−1}} ⊗ e\overline{i_{r}} & if s = s_{r}, \end{matrix}

where we interpret ¯ ī to mean i. Proof. This is simply the permutation module corresponding to the permutation representation of Lemma 2.1.1. □

We can relate this action to the results in §1.2. Lemma 2.1.3. Let a represent an element w ∈ W. Suppose n ≥ r. Then

(e_{1} ⊗ e_{2} ⊗ ⋯ ⊗ e_{r}).w = e_{a_{1}} ⊗ ⋯ ⊗ e_{a_{r}}.

Proof. It is enough to show that

(e_{1} ⊗ e_{2} ⊗ ⋯ ⊗ e_{r}).w = e_{(1)w} ⊗ ⋯ ⊗ e_{(r)w}.

This is proven by an easy induction on ℓ(w), using the definition of the action in Lemma 2.1.2. □

2.2 The centralising algebra of W (B_{r}).

We now consider the centralising algebra of the action of W on tensor space as in §2.1. Definition 2.2.1. Let η ∈ End(V^{⊗r}). For i, j ∈ I_{B}(2n, r), we define the coordinate functions c_{i,j} via

η(e_{j}) = ∑_{i∈I_{B}(2n,r)} c_{i,j}e_{i}.

Here, e_{i} denotes the tensor e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} where i = (i_{1}, …, i_{r}).

HYPEROCTAHEDRAL SCHUR ALGEBRAS 9

Definition 2.2.2. We denote the centralising algebra of the right action of W on V_{2n}^{⊗r} by H(2n, r).

In order for an element η ∈ End(V^{⊗r}) to lie in H(2n, r), the coordinate functions are required to satisfy certain relations. We now determine these relations.

Lemma 2.2.3. The coordinate functions on H(2n, r) satisfy the relations

c_{i,j} = c_{i.s_{a},j.s_{a}}

for all generating involutions s_{a}. These relations are necessary and sufficient.

Proof. To say that η ∈ H(2n, r) is equivalent to saying that its action commutes with the action of all the s_{a}.

This is essentially known for the case a < r, because such conditions are known to be necessary and sufficient to ensure that elements of End(V^{⊗r}) commute with the action of the symmetric group by place permutation of the tensors. (This fact is known and obvious from the situation in type A.) So we only need to check that each element h ∈ H(2n, r) commutes with the action of s_{r}.

Comparing the action of h on two basis vectors e_{i} and e_{i.s_{r}} and using the facts that e_{i.s_{r}} = e_{i}.s_{r} (by definition) and that the left action of h commutes with the right action of s_{r}, we obtain the proof of the statement for a = r. □

Corollary 2.2.4. Let Ω be a transversal for the action of W on I_{B}(2n, r) × I_{B}(2n, r). For any w ∈ W, we have c_{i,j} = c_{i.w,j.w}, and the set {c_{i,j} : (i, j) ∈ Ω} is a basis for the coordinate ring H^{∗}(2n, r) of H(2n, r).

Proof. The fact that c_{i,j} = c_{i.w,j.w} follows by induction on ℓ(w) and Lemma 2.2.3. The second assertion now follows. □

Definition 2.2.5. We define the element η_{ij} ∈ End(V^{⊗r}) by the conditions

c_{ab}(η_{ij}) = \{ \begin{matrix}1 & if (a, b) ∼ (i, j), \\ 0 & otherwise. \end{matrix}

Note that these elements η_{ij} satisfy the condition c_{ab}(η_{ij}) = c_{a.w,b.w}(η_{ij}) for any w ∈ W and are therefore, by Corollary 2.2.4, elements of H(2n, r).

10 R.M. GREEN

Proposition 2.2.6. The set {η_{ij} : (i, j) ∈ Ω} is a basis for the algebra H(2n, r).

Proof. This follows from standard properties of permutation modules. □

The following proposition is the analogue of Schur’s product rule ([7, 2.3b]) for hyperoctahedral Schur algebras.

Proposition 2.2.7. For (i, j), (k, l) in Ω, we have

η_{i,j}η_{k,l} = ∑_{(p,q)∈Ω} Z(i, j, k, l, p, q)η_{p,q},

where Z(i, j, k, l, p, q) := |{s ∈ I_{B}(2n, r) : (i, j) ∼ (p, s) and (k, l) ∼ (s, q)}|.

Proof. This is a consequence of the theory of permutation modules: see for example [9, Lemma II.12.8]. □

2.3 Connection with general linear groups.

We note here an interesting connection between the algebra H(2n, r) and a certain matrix group.

Definition 2.3.1. We denote by J the element of GL_{2n} (over a given field) for which J_{ab} is 1 if a + b = 2n + 1 and 0 otherwise.

We denote by C(J) the centraliser of J in GL_{2n}. This consists of all nonsingular matrices invariant under a rotation of the entries by a half turn.

We index the rows and columns of the matrices by the ordered set

\{1, 2, …, n, n̄, \overline{n − 1}, …, ¯1\}.

Definition 2.3.2. The group C(J) acts on V_{2n}^{⊗r} via

g.e_{j} = ∑_{i∈I_{B}(2n,r)} c_{i,j}(g).e_{i},

where

c_{i,j}(g) := c_{i_{1}j_{1}}(g) ⋯ c_{i_{r}j_{r}}(g)

and c_{ij}(g) := g_{ij}.

HYPEROCTAHEDRAL SCHUR ALGEBRAS 11

Proposition 2.3.3. The action of C(J) on V^{⊗r} centralises the action of W given in §2.1.

Proof. This works because c_{i.w,j.w}(g) = c_{i,j}(g) for all w ∈ W. (The only nontrivial aspect of this is the case where w = s_{r}.) □

Note. This may be regarded as an analogue of the fact that in type A, the corresponding action of GL_{n} centralises the action of the symmetric group on tensor space.

3. The Hecke algebra and tensor space

3.1 Action of H(B_{r}) on tensor space.

The next aim is to generalise the results of §2 to the Hecke algebra H(B_{r}) corresponding to the Weyl group W (B_{r}). We change the base field of vector spaces appearing from ℚ to ℚ(q, Q), where q and Q are independent indeterminates. (Contrast this to the situation in type A, which is “simply-laced”, so only one indeterminate is required.) The integral forms of the objects appearing are now defined over the ring of Laurent polynomials A := ℤ[q, q^{−1}, Q, Q^{−1}].

Definition 3.1.1. The Hecke algebra H(B_{r}) is defined as the free A-module with basis {T_{w} : w ∈ W (B_{r})} and the following multiplicative relations:

\begin{aligned}T_{s_{i}}T_{w} & = T_{s_{i}w} if ℓ(s_{i}w) > ℓ(w), \\ T_{s_{i}}T_{w} & = qT_{s_{i}w} + (q − 1)T_{w} if i < r, ℓ(s_{i}w) < ℓ(w), \\ T_{s_{r}}T_{w} & = QT_{s_{r}w} + (Q − 1)T_{w} if ℓ(s_{r}w) < ℓ(w). \end{aligned}

Here, ℓ denotes the length function on the Coxeter group W.

Note that an equivalent definition (like the one in [3, §3]) would be to give the quadratic relations satisfied by the T_{s_{i}} and the braid relations of the Weyl group, similarly to the definition of the group W in §1. (This works because any two reduced words in W can be transformed into each other by means of a finite number of applications of the braid relations.) Also note that if the parameters q and Q are replaced by 1, one recovers the relations of the group algebra of W.

Lemma 3.1.2. There is a right action of H = H(B_{r}) on V^{⊗r} in which the generators T_{s_{a}} act on the basis {e_{i} : i ∈ I_{B}(2n, r)} (where i = (i_{1}, …, i_{r})) as follows:

e_{i}.T_{s_{a}} = \{ \begin{matrix}qe_{i.s_{a}} & if a < r, i_{a} ≤ i_{a+1}, \\ e_{i.s_{a}} + (q − 1)e_{i} & if a < r, i_{a} > i_{a+1}, \\ Qe_{i.s_{a}} & if a = r, i_{r} ∈ \{1, …, n\}, \\ e_{i.s_{a}} + (Q − 1)e_{i} & if a = r, i_{r} ∈ \{n̄, …, ¯1\}. \end{matrix}

Note. Note how we could have replaced ≤ by <, and > by ≥ above, and the definition would have been the same.

Proof. Using the results in [1, Lemma 3.1.4], we know that the braid relations involving only the generators T_{s_{1}}, …, T_{s_{r−1}} will hold, because they hold in type A. (Dipper and Donkin consider a left action, but this doesn’t matter since the relations are left-right symmetric.)

The quadratic relation for T_{s_{r}} is easily checked and is similar to that for the other T_{s_{i}}.

The commutation relations corresponding to the commutations between s_{r} and s_{i} (i < r) are clear.

The only nontrivial step is checking the relation stst = tsts, where we write s for T_{s_{r}} and t for T_{s_{r−1}} for brevity. It is not hard to see that it is sufficient to check this in the case r = 2.

A short calculation shows that if i and j are both unbarred and i ≤ j, then

(e_{i} ⊗ e_{j}).stst = (e_{i} ⊗ e_{j}).tsts = q^{2}Q^{2}(eī ⊗ ej̄).

We must verify a similar statement in eight cases in total (one of which is the above calculation), according as i, j, neither, or both are barred, and according as |i| ≤ |j| or |i| > |j|. This can be deduced from the calculation in the above equation, together with the fact that there are eight equivalent formulations of the braid relation stst = tsts. These are the following:

\begin{aligned}stst & = tsts, \\ s^{−1}tst & = tsts^{−1}, \\ s^{−1}t^{−1}st & = tst^{−1}s^{−1}, \\ s^{−1}t^{−1}s^{−1}t & = ts^{−1}t^{−1}s^{−1}, \\ s^{−1}t^{−1}s^{−1}t^{−1} & = t^{−1}s^{−1}t^{−1}s^{−1}, \\ st^{−1}s^{−1}t^{−1} & = t^{−1}s^{−1}t^{−1}s, \\ sts^{−1}t^{−1} & = t^{−1}s^{−1}ts, \\ stst^{−1} & = t^{−1}sts. \end{aligned}

Note that the use of inverses is valid, since we have verified all the quadratic relations and all the T_{s_{i}} are invertible. In each case, the application of the correct form of the above relation to e_{i} ⊗ e_{j} will yield the desired result very easily.

We illustrate this with an example. Consider eī ⊗ e_{j}, where i > j, ī is barred, and j is not. Then

eī ⊗ e_{j} = q^{−1}Q^{−1}(e_{j} ⊗ e_{i}).st.

Since j < i and j and i are unbarred, we use the fact proved above that

(e_{j} ⊗ e_{i}).stst = (e_{j} ⊗ e_{i}).tsts

to show that

(eī ⊗ e_{j}).sts^{−1}t^{−1} = (eī ⊗ e_{j}).t^{−1}s^{−1}ts,

as required. The other cases are very similar to this. □

Definition 3.1.3. Let w ∈ W (B_{r}). We write q_{w} = Q^{c}q^{ℓ(w)−c} where c is the number of occurrences of s_{r} in a reduced expression for w.

Note. The number c is independent of the reduced expression chosen for w. This follows from the defining relations for W.

Lemma 3.1.4. Let a = (a_{1}, …, a_{r}) represent an element of w ∈ W (B_{r}). Then

(e_{1} ⊗ ⋯ ⊗ e_{r}).T_{w} = q_{w}e_{a_{1}} ⊗ ⋯ ⊗ e_{a_{r}}.

14 R.M. GREEN

Proof. This follows by induction on the length of w, Lemma 1.2.2, Lemma 3.1.2 and Lemma 2.1.3. (The induction hypothesis is that if it holds for w, then it holds for ws_{i} where ℓ(ws_{i}) = ℓ(w) + 1.) □

We may give a criterion to decide whether the above action of H is faithful, similar to the situation in type A.

Proposition 3.1.5. The right action of H(B_{r}) on V_{2n}^{⊗r} is faithful if and only if n ≥ r. The same is true for the action of the group algebra of W (B_{r}) on V_{2n}^{⊗r}.

Proof. We prove the statement for H. The statement for W may be obtained by replacing q and Q by 1 in the argument which follows.

Suppose n ≥ r. We consider the dimension of the H-module generated by the vector e_{1} ⊗ ⋯ ⊗ e_{r}. One finds from Lemma 3.1.4 that it is of dimension |ℤ_{2} ≀ S_{r}|, i.e. 2^{r}.r!, from which it follows that the action is faithful.

Suppose that n < r. Consider the element y ∈ H given by

∑_{w∈W}(−1)^{ℓ(w)}.q_{w}^{−1}T_{w}.

By an induction on ℓ(w), we see that T_{w}.y = (−1)^{ℓ(w)}y for all w ∈ W; this follows by standard properties of Coxeter groups. Consider a typical basis element e_{i_{1}}⊗⋯⊗e_{i_{r}} of V^{⊗r}. It follows from elementary properties of the action and Lemma 3.1.4 that this can be expressed in the form

e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.h

where h ∈ H (in fact, h may be taken to be a unit multiple of one of the basis elements of H) and the indices j_{m} all come from the set {1, …, n} and satisfy j_{1} ≤ ⋯ ≤ j_{r}. We now observe from the theory of Coxeter groups that the element y_{i} ∈ H given by T_{1} − q_{s_{i}}^{−1}T_{s_{i}} divides y on the left. Since the element e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}} contains a repeated index, one of the elements y_{i} for 1 ≤ i < r acts as zero on it, and hence so do y and h.y. We deduce that y acts trivially on V^{⊗r}, despite being nonzero. This completes the proof. □

HYPEROCTAHEDRAL SCHUR ALGEBRAS 15

3.2 The centralising algebra of H(B_{r}).

We now generalise the results of §2.2 to the Hecke algebra situation. Not surprisingly, the results are more complicated. Definition 3.2.1. We denote by H := H_{q,Q}(2n, r) the centralising algebra of H_{q,Q}(B_{r}) on V_{2n}^{⊗r}. We denote its dual by H^{∗} := H_{q,Q}^{∗}(2n, r). Proposition 3.2.2. The coordinate functions c_{i,j} in H_{q,Q}^{∗}(2n, r) satisfy the following relations:

c_{i,j} = \{ \begin{matrix}qc_{i.s_{a},j.s_{a}} & (a < r, i_{a} > i_{a+1}, j_{a} ≤ j_{a+1}) \\ c_{i.s_{a},j.s_{a}} + (q − 1)c_{i.s_{a},j} & (a < r, i_{a} > i_{a+1}, j_{a} > j_{a+1}) \\ c_{i.s_{a},j.s_{a}} & (a < r, i_{a} = i_{a+1}) \\ Qc_{i.s_{r},j.s_{r}} & (i_{r} barred, j_{r} unbarred) \\ c_{i.s_{r},j.s_{r}} + (Q − 1)c_{i.s_{r},j} & (i_{r} barred, j_{r} barred) \end{matrix}

Proof. The first three relations come from [1, Definition 1.1.1], and are necessary because H(B_{r}) contains the subalgebra H(A_{r−1}) generated by T_{s_{1}}, …, T_{s_{r−1}}, with the same action on tensor space.

We also need conditions corresponding to the fact that elements of H_{q,Q}(2n, r) commute with the action of T_{s_{r}}. Since the action of T_{s_{r}} on an element e_{j} essentially involves only the entry j_{r}, we will abuse notation and only write out the last place in the tensor for clarity. Thus, T_{s_{r}} is thought of as acting on V itself.

Let η lie in the coordinate ring of End(V) and let j be unbarred, and write

η.e_{j} = ∑_{i unbarred}(a_{i}e_{i} + b_{i}eī).

We know that e_{j}.T_{s_{r}} = Qej̄, so if the actions of η and T_{s_{r}} on V commute, we find that

η.Qej̄ = (η.e_{j}).T_{s_{r}} = ∑_{i unbarred} (b_{i}e_{i} + (Qa_{i} + (Q − 1)b_{i})eī).

Comparing coefficients proves that

c_{īj} = Qc_{ij̄}

and

cī ; j̄ = c_{ij} + (Q − 1)c_{ij̄},

where i and j are both unbarred. Knowing these facts makes the proof of the remaining relations easy. □

16 R.M. GREEN

Definition 3.2.3. Let I_{B}^{2} = I_{B}^{2}(2n, r) be that subset of I_{B}(2n, r) × I_{B}(2n, r) consisting of elements (i, j) satisfying the following two conditions:

(i) All the elements of i are unbarred, and i_{1} ≤ ⋯ ≤ i_{r}. (ii) If i_{m} = i_{m+1} then j_{m} ≤ j_{m+1}.

Remark. Notice that I_{B}^{2} is in canonical bijection with monomials of degree r in 2n^{2} indeterminates, and therefore has cardinality

\binom{2n^{2} + r − 1}{r}.

Also note that I_{B}^{2} is a transversal for the W -orbits on I_{B}(2n, r) × I_{B}(2n, r).

Theorem 3.2.4. The set {c_{i,j} : (i, j) ∈ I_{B}^{2}} is a basis for H_{q,Q}^{∗}(2n, r).

Proof. First we show that the given set is a spanning set. Consider an arbitrary element c_{a,b}. Using the first and second relations in Proposition 3.2.2, we can express this as a sum of elements c_{k,l} where k = (k_{1}, …, k_{r}), l = (l_{1}, …, l_{r}) and k_{1} ≤ ⋯ ≤ k_{r}, but where the k_{m} are not necessarily all unbarred. (In fact, the k_{m} are a rearrangement of the a_{m}.) If there are barred entries k_{m}, then k_{r} is one of them, and we apply the fourth or fifth relation according as l_{r} is unbarred or barred. This expresses c_{a,b} as a sum of elements c_{a′,b′} where all the a′ have one fewer barred entry than a. By iterating this process, we reduce to the case where a has no barred entries. After further applications of the first and second relations, we express c_{a,b} as a sum of elements satisfying condition (i). We then apply the third relation until condition (ii) is satisfied as well. This proves that the given set is a spanning set.

We now prove independence. Suppose

∑_{(i,j)∈I_{B}^{2}} α_{i,j}c_{i,j} = 0  (1)

for some scalars α. By clearing denominators and negative powers, we may assume all the α lie in ℤ[q, Q], and that the largest monomial involved is of minimal possible degree. Suppose that after substituting Q = 1, equation (1) becomes identically zero. This means that (Q−1) divides the equation over the field ℚ(q), and thus, by Gauss’ Lemma, over ℤ[q], because the latter is a unique factorisation domain with

HYPEROCTAHEDRAL SCHUR ALGEBRAS 17

field of fractions ℚ(q). This contradicts the minimality assumption, so equation (1) is nonzero after substituting Q = 1.

Now suppose we can find another equation satisfying the hypotheses of equation (1), but this time in the ℤ[q, q^{−1}]-module H_{q,1}^{∗}(2n, r). If this becomes zero after setting q = 1, we may use a similar (but easier) argument to the above to show that (q − 1) divides the equation. So if the equation is of minimal degree, it is nonzero after setting q = Q = 1.

This contradicts Corollary 2.2.4, since I_{B}^{2} is a transversal of W orbits and the natural ℤ[q, q^{−1}, Q, Q^{−1}]-form of H(B_{r}) becomes the natural ℤ-form of W (B_{r}) when q and Q are replaced by 1. This means that no equation of the form (1) exists, and that the given elements are independent. □

One of the consequences of Theorem 3.2.4 is that it gives us a basis for H. Definition 3.2.5. The dual basis {η_{i,j} : (i, j) ∈ I_{B}^{2}} is a basis for H_{q,Q}(2n, r).

Proposition 3.1.5 tells us whether the right action of H on tensor space is faithful. We now show that the left action of H = H_{q,Q}(2n, r) on tensor space is always faithful. Remark. We end this section with a remark on the possibility of setting one or both of the parameters to 0. Because the relations on the c_{i,j} do not involve negative powers of q or Q, the basis of Theorem 3.2.4 still makes sense and is still a basis even when q and/or Q is set to zero. Using the comultiplication

Δ(c_{p,q}) = ∑_{s∈I_{B}(2n,r)} c_{p,s} ⊗ c_{s,q}.

(which is a q-analogue of the map Δ appearing in §2.2, even though it has exactly the same form), we find that the basis in Proposition 2.2.6 also makes sense at zero values of the parameters, and the algebra structure remains.

4. Quantized Permutation Modules

In §4, we consider certain natural “permutation” modules for the Hecke algebra H(B_{r}). It is important to do this in order to relate the algebra H_{q,Q}(2n, r) to other algebras which appear in the literature.

18 R.M. GREEN

4.1 Tensor space as a q-Permutation Module.

We start by defining a certain subset of parabolic subgroups of W (B_{r}). Definition 4.1.1. We denote by P the set of parabolic subgroups generated by a set

S ⊆ {s_{1}, …, s_{r−1}},

i.e. parabolic subgroups of W which do not involve the last reflection, s_{r}.

The next brace of definitions gives us the analogue of the theory of weight spaces in type A. Definition 4.1.2. Let e = e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} ∈ V^{⊗r}. Then the weight, λ(e) of e is defined to be the n-tuple

(λ_{1}, …, λ_{r}),

where λ_{i} is the total number of occurrences of the indices i and ī in e. We denote the set of weights by Λ_{B} = Λ_{B}(2n, r).

The element u_{λ} of I_{B}(2n, r) is defined to be the r-tuple (i_{1}, …, i_{r}), where i_{1} ≤ ⋯ ≤ i_{r} ≤ n and e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} has weight λ. We say an element of I_{B}(2n, r) conjugate to u_{λ} (under the action of W (B_{r})) has content λ.

The space V_{λ} is the span of all the basis vectors e of weight λ.

The vector e_{λ} is the unique basis vector e ∈ V_{λ} whose indices satisfy i_{1} ≤ ⋯ ≤ i_{r} ≤ n.

The subgroup P_{λ} is the element of P for which s_{i} ∈ P_{λ} if and only if the i-th and (i + 1)-th entries of e_{λ} are equal.

The elements x_{λ} and y_{λ} of H are defined by

x_{λ} = ∑_{w∈P_{λ}} T_{w}

and

y_{λ} = ∑_{w∈P_{λ}}(−1)^{ℓ(w)}q_{w}^{−1}T_{w}.

Lemma 4.1.3. The space V_{λ} is a right H-module. Proof. This is immediate from the action of H on V_{λ} which arises from the statement of Lemma 3.1.2. □

HYPEROCTAHEDRAL SCHUR ALGEBRAS 19

Proposition 4.1.4. As H-modules, V_{λ} is isomorphic to x_{λ}H. The isomorphism may be chosen to send e_{λ} to x_{λ}. Proof. It follows from Lemma 3.1.4 that the dimension of V_{λ} is equal to the index of P_{λ} (which is the stabiliser of e_{λ}), and that V_{λ} is generated as an H-module by e_{λ}. We now set up a map from x_{λ}H to V_{λ} defined by

x_{λ}h ↦ e_{λ}.h.

This is well-defined because x_{λ} acts as a scalar on e_{λ}. It is clearly surjective, and it is clearly a homomorphism of right H-modules. Injectivity follows by a comparison of dimensions: the dimension of x_{λ}H is also the index of the parabolic subgroup P_{λ}. □

4.2 Endomorphism Rings of Quantized Permutation Modules.

Next we compare our algebra with certain algebras appearing in [6]. Lemma 4.2.1. There is an automorphism of H taking x_{λ} to y_{λ} for all λ. Therefore there is an algebra isomorphism

End_{H} ( ⨁_{P_{λ}∈P} x_{λ}H) ≅ End_{H} ( ⨁_{P_{λ}∈P} y_{λ}H) .

Proof. The required automorphism is a generalisation of that in [2, 2.1]. It sends T_{s_{i}} to −q_{s_{i}}T_{s_{i}}^{−1}; it is straightforward to check that this is an automorphism. We see from [2, 2.1] that x_{λ} and y_{λ} are exchanged by this automorphism.

A standard argument now shows that

Hom_{H}(x_{λ}H, x_{μ}H) ≅ Hom_{H}(y_{λ}H, y_{μ}H),

from which the lemma follows. □ Proposition 4.2.2. The algebra H_{q,Q}(2n, r) is Morita equivalent to the algebra S(H) of [6] in the case where H = H(B_{r}). Proof. The algebra S(H) is defined in [6, §7.2] as

End_{H} ( ⨁_{P_{λ}∈P} y_{λ}H) .

It follows easily from the definitions that

H_{q,Q}(2n, r) ≅ End_{H} ( ⨁_{λ∈Λ_{B}} V_{λ}) .

A standard argument shows that this is Morita equivalent to

H′_{q,Q}(2n, r) := End_{H} ( ⨁_{P_{λ}∈P} V_{λ})

The proof now follows from Proposition 4.1.4 and Lemma 4.2.1. □

4.3 Du’s Generalised Schur Algebras.

In §4.3, we assume that Q = q, partly for compatibility with the Kazhdan– Lusztig polynomials, P_{y,w}(q), where y, w ∈ W.

The following definition comes from [4].

Definition 4.3.1 (Du). The generalised Schur algebra of type B_{r} is given by

S_{q}(W (B_{r})) = End_{H} ( ⨁_{P_{λ}∈P′} x_{λ}H) ,

where P′ is the set of all parabolic subgroups of W.

This is a free ℤ[q, q^{−1}] module with basis ϕ_{I,J}^{d} as I and J range over P′ and d is a distinguished double W_{I}–W_{J} coset respresentative.

The effect of ϕ_{I,J}^{d} on ⨁_{J′∈P′} x_{J′}H is given by

ϕ_{I,J}^{d} : x_{J′}T_{w} ↦ δ_{J,J′}T_{W_{I}dW_{J}} T_{w},

where T_{C} in general stands for ∑_{w∈C} T_{w}.

Proposition 4.3.2. The elements ϕ_{I,J}^{d} appearing in Definition 4.3.1 which satisfy I, J ∈ P span a subalgebra isomorphic to the algebra H′_{q,q}(2n, r) appearing in Proposition 4.2.2.

Proof. This follows immediately from Definition 4.3.1 and the proof of Proposition 4.2.2. □

Remark. Du also defines a basis θ_{I,J}^{d}, ranging over the same indexing set as the ϕ basis. This also has the property that the elements for which I, J ∈ P form a basis

HYPEROCTAHEDRAL SCHUR ALGEBRAS 21

for the algebra H′. This basis is related to the ϕ basis via the Kazhdan–Lusztig polynomials.

It is convenient to extend the basis ϕ_{I,J}^{d} of H′_{q,q}(2n, r) to one of H_{q,q}(2n, r). This is achieved as follows.

Definition 4.3.3. The algebra S̃_{q}(2n, r) is defined to be

End_{H} ( ⨁_{λ∈Λ_{B}(2n,r)} x_{λ}H) .

This is a free ℤ[q, q^{−1}] module with basis ϕ_{λ,μ}^{d} for all λ, μ ∈ Λ_{B}(2n, r) and d a distinguished double P_{λ}–P_{μ} coset respresentative.

The effect of ϕ_{λ,μ}^{d} on ⨁_{λ∈Λ_{B}} x_{ν}H is given by

ϕ_{λ,μ}^{d} : x_{ν}T_{w} ↦ δ_{μ,ν}T_{P_{λ}dP_{μ}}T_{w}.

Lemma 4.3.4. We have

S̃_{q}(2n, r) ≅ H_{q,q}(2n, r).

Proof. This is clear from the proof of Proposition 4.2.2. □

4.4 Agreement of bases.

A natural question to ask is whether the basis η_{i,j} is compatible with the basis ϕ_{λ,μ}^{d}. This was proved in type A by Dipper and Donkin (see [1, Theorem 3.2.5]). It turns out to be true in type B as well, and the techniques in [1] are extendable to type B. We continue to assume that Q = q in §4.4.

First, we need to find an identification between the two bases, but this turns out not to be difficult.

Lemma 4.4.1. There is a bijection between W -orbits of I_{B}(2n, r) × I_{B}(2n, r) and basis elements ϕ_{λ,μ}^{d} of H_{q,Q}(2n, r) which maps ϕ_{λ,μ}^{d} to the orbit containing the element (u_{λ}.d, u_{μ}).

Proof. The proof follows from an elementary argument. □

We can now prove a result analogous to [1, Lemma 3.2.3].

22 R.M. GREEN

Lemma 4.4.2. Consider c_{i,j}, where (i, j) is an element of I_{B}^{2}, and i and j have contents λ and μ respectively. Let d be a distinguished right coset representative for P_{λ} in W such that (i, j) ∼ (i.d, u_{μ}). Then c_{i.d,u_{μ}} = q_{d}c_{i,j}, and

c_{i.d,u_{μ}}(η_{i,j}) = \{ \begin{matrix}q_{d} & if (i.d, u_{μ}) ∼ (i, j), \\ 0 & otherwise. \end{matrix}

Proof. We apply a “straightening procedure” to the element c_{i.d,u_{μ}} to express it as a sum of basis elements. Each step of this involves replacing an element c_{a,b} by a combination of others by using one of the five rules of Proposition 3.2.2.

As in the discussion following [1, Corollary 3.2.2], the fact that

u_{μ} = (u_{1}, …, u_{r}),

where u_{1} ≤ ⋯ ≤ u_{r} ≤ n means that during the straightening procedure, we never have to apply the second and fifth relations, which are the ones that produce extra terms. (This is a consequence of Lemma 1.2.2.) A multiple of q appears for each adjacent pair of indices sorted into order on the i-side, and a multiple of Q appears for each entry on the i-side as it is unbarred. We know from Lemma 1.2.3 that each simple reflection in a fixed reduced expression of d will cause one of these events to happen, so by the time the straightening is complete, we end up with

c_{i.d,u_{μ}} = q_{d}c_{i,j}.

The second assertion now follows immediately. □

Definition 4.4.3. Let W_{I}, W_{J} ∈ P and w ∈ W. We define the subset

C(W_{I}, w, W_{J})

of W to be the set of distinguished right coset representatives for W_{I} in W which lie in the double coset W_{I}wW_{J}. Thus we have

W_{I}wW_{J} = ⋃_{d∈C(W_{I},w,W_{J})}^{.} W_{I}d.

HYPEROCTAHEDRAL SCHUR ALGEBRAS 23

Theorem 4.4.4. As an endomorphism, η_{i,j} is equal to ϕ_{λ,μ}^{d}, under the identifications given in Lemma 4.4.1.

Proof. Let λ and μ be the contents of i and j, respectively. We wish to show that

η_{i,j}.x_{μ} = T_{W_{I}dW_{J}} = ∑_{w∈C(W_{I},w,W_{J})} x_{λ}.T_{w}.

Now we use the identification of Proposition 4.1.4, and the fact that e_{u_{μ}}.T_{w} = q_{w}e_{u_{μ}.w}; the latter follows easily by induction. This shows that the coefficient of e_{a} in η_{i,j}e_{u_{μ}} is q_{d} if (a, u_{μ}) ∼ (i, j) (where a = i.d and d is a distinguished right coset representative for P_{λ}) and 0 otherwise. Lemma 4.4.2 now completes the proof. □

4.5 The Double Centraliser Property.

This set-up makes it relatively easy to prove a double centraliser property analogous to Schur–Weyl duality in type A. We assume n ≥ r throughout §4.5.

Lemma 4.5.1. Any endmorphism of V^{⊗r} which commutes with the action of H_{q,Q}(2n, r) is determined by its effect on the vector

e_{1} ⊗ ⋯ ⊗ e_{r}.

Proof. Let ω be the weight

(\underbrace{1, 1, …, 1}_{r}, 0, 0, …, 0).

The set

{ϕ_{λ,ω}^{d}.e_{ω} : λ ∈ Λ_{B}, d ∈ D_{λ}}

(where D_{λ} is the set of distinguished right coset representatives of P_{λ}) is a basis for V_{λ}. (This is essentially just another formulation of Lemma 3.1.4, using Proposition 4.1.4.) The statement now follows from the fact that the actions of H and H on V^{⊗r} commute. □

Lemma 4.5.2. Each element η_{i,i} in H is idempotent and has the property that

η_{i,i}.e_{j} = \{ \begin{matrix}e_{j} & if i ∼ j, \\ 0 & otherwise. \end{matrix}

24 R.M. GREEN

Proof. The conditions

c_{a,b}(η) = \{ \begin{matrix}1 & if a = b ∼ i, \\ 0 & otherwise, \end{matrix}

where a and b range over all elements of I_{B}(2n, r), satisfy the relations on the c_{a,b} given in Proposition 3.2.2. Furthermore, the only element c_{a,b} with (a, b) ∈ I_{B}^{2} which does not annihilate η is c_{i,i}, and this takes value 1 on η. Thus η = η_{i,i}. The action of η on e_{j} is now clearly as claimed, from which it follows that η is idempotent. □

Lemma 4.5.3. Let ψ be an endomorphism of V^{⊗r} which commutes with the action of H. Then, writing ψ(e_{i}) = ∑_{k} α_{k}e_{k}, we have α_{k} ≠ 0 ⇒ k ∼ i.

Proof. Suppose α_{k} ≠ 0 for some k ≁ i. Let j be such that j ∼ k and j = (j_{1}, …, j_{r}) where j_{1} ≤ ⋯ ≤ j_{r} and all the j_{m} are unbarred. Then by Lemma 4.5.2, η_{j,j}.e_{k} = e_{k}, from which it follows that η_{j,j}(ψ(e_{i})) ≠ 0. However, η_{j,j}(e_{i}) = 0 because i ≁ j ∼ k. It is therefore not possible that the action of η_{j,j} commutes with that of ψ, which is a contradiction. □

Proposition 4.5.4. The algebras H_{q,Q}(2n, r) and H_{q,Q}(B_{r}) are centralising algebras of each other on tensor space.

Proof. Firstly, note that by Proposition 3.1.5, the action of each algebra is faithful, since the action of H is faithful by construction.

We know that H is the centralising algebra of H, by construction of H. Lemma 4.5.3 shows that if ψ commutes with the action of H, then it maps each weight space V_{λ} to itself. Lemma 4.5.1 now shows that the centralising algebra of H is of dimension at most dim H. Conversely, since H acts faithfully on the weight space containing the tensor e_{1} ⊗ ⋯ ⊗ e_{r}, this bound is achieved, which completes the proof. □

Acknowledgements

The author would like to thank Michael Collins, Karin Erdmann and Sandy Green for some helpful discussions and comments. The author is also grateful to the referee for some helpful suggestions.

HYPEROCTAHEDRAL SCHUR ALGEBRAS 25

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