Gram Determinants of Type B_{n}

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^{†}

Eugene Murphy Department of Mathematical Sciences University of East London Romford Road London E15 4LZ England^{‡}

30th October 1996

Abstract

The Hecke algebra H = H_{R,Q,q}(W_{n}) over a commutative ring R with the two parameters Q, q associated with the Weyl group W_{n} of type B_{n} has certain distinguished representations afforded by modules S̃^{λ} labelled by bipartitions λ of n. These are precisely the irreducible H-modules, if H is semisimple. In general there is a symmetric Hinvariant bilinear form defined on S̃^{λ}. We present here a formula for the determinant of the Gram matrix of S̃^{λ}.

A. M. S. subject classification (1991): 16G99, 20C20, 20G05 This paper is a contribution to the DFG project on “Algorithmic number theory and algebra”. The authors acknowledge support from DFG.

The first two authors gratefully acknowledge support received from EPSRC Grant No.

GR/K41571

∗Email: rdipper@mathematik.uni-stuttgart.de †Email: g.james@ic.ac.uk ‡Email: g.e.murphy@uel.ac.uk

1 Introduction

We here continue our study of Hecke algebras of type B_{n}, which we began in [2, 3]. Our objective is to find a formula for the determinant of the Gram matrix with respect to the standard basis of the module S̃^{λ}, where λ is a bipartition of n and S̃^{λ} is the analogue of the dual of the Specht module which is familiar from the representation theory of symmetric groups. The usual Specht modules are indexed by partitions, and the determinant of the Gram matrix for the Specht modules for symmetric groups was given in [5]; the version for Hecke algebras of type A_{n} appears in [1, 4.11]. The situation for Hecke algebras of type B_{n} is made more complicated by the facts that there are two variables q and Q, and that we have to cope with bipartitions instead of partitions. None the less, the formula which we derive can still be described by manipulating sets of crosses in the plane (see the examples at the end of section 5).

In the case of the symmetric groups, all the entries in the Gram matrix are integers, but it is a mystery why the formula [5, p.224] for the determinant of the Gram matrix gives an integer. For Hecke algebras of type B_{n} we have an even more remarkable formula, which we know must yield a polynomial in the variables q and Q, since the determinant is such a polynomial, yet the formula involves a quotient of polynomials, and it is opaque that cancellation always occurs in a way which gives a polynomial.

2 Notation

In this section, we summarize some of the results from [3].

The Weyl group W_{n} of type B_{n} has generators s_{0}, s_{1}, …, s_{n−1}, and relations which are determined in the usual way by the Dynkin diagram below:

•_{s_{0}} = = = = = = = = = •_{s_{1}}  •_{s_{2}}  •_{s_{3}} . . . . . .  •_{s_{n−2}}  •_{s_{n−1}}

Let R be a commutative ring, containing elements q and Q, where q is a unit. Then H_{R,Q,q}(W_{n}) = H(W_{n}) is an R-algebra which is a free module over R, with basis {T_{w} | w ∈ W_{n}}. The multiplication in H(W_{n}) satisfies the following relations:

1. T_{1} is the identity 1_{H(W_{n})} of H(W_{n}).

2. If w = v_{1}v_{2} ⋯ v_{m} ∈ W_{n}, where each v_{i} belongs to {s_{0}, s_{1}, …, s_{n−1}} and if l(w) = m, then T_{w} = T_{v_{1}}T_{v_{2}} ⋯ T_{v_{m}}. Here l : W_{n} → ℕ is the usual length function on W_{n}.

3. (T_{s_{i}})^{2} = qT_{1} + (q − 1)T_{s_{i}} for 1 ≤ i ≤ n − 1.

4. (T_{s_{0}})^{2} = QT_{1} + (Q − 1)T_{s_{0}}.

The algebra H(W_{n}) is called the Hecke algebra of type B_{n}. We write λ = (λ^{(1)}, λ^{(2)}) and call λ a bipartition of n, if for some integer a with 0 ≤ a ≤ n, λ^{(1)} is a partition of a and λ^{(2)} is a partition of n−a. The diagram [λ] then consists of an ordered pair of arrays of crosses in the plane, the arrays being determined by λ^{(1)} and λ^{(2)} as in the following example, where n = 13 and λ = ((4, 3, 1), (3, 2)).

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

To obtain a λ-bitableau t, each cross in [λ] is replaced by one of the numbers 1, 2, …, n allowing no repeats; we write t = (t^{(1)}, t^{(2)}), where t^{(1)} is the first part of t (corresponding to λ^{(1)}) and t^{(2)} is the second part of t (corresponding to λ^{(2)}). In particular, let t^{λ} = (t^{λ^{(1)}}, t^{λ^{(2)}}) be the λ-bitableau in which 1, 2, …, a appear in order by rows in t^{λ^{(1)}} and a + 1, …, n appear in order by rows in t^{λ^{(2)}}, where λ^{(1)} is a partition of a, and λ^{(2)} is a partition of n − a. For example, if λ = ((4, 3, 1), (3, 2)), then

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

A λ-bitableau (t^{(1)}, t^{(2)}) is standard if the entries increase along every row and down every column of t^{(1)} and the same property holds for t^{(2)}.

By identifying s_{i} with the basic transposition (i, i + 1) for 1 ≤ i ≤ n − 1, we regard < s_{1}, …, s_{n−1} > as the symmetric group 𝔖_{n}, and we let 𝔖_{n} act upon the λ-bitableaux as the group of number permutations. Let < be a total order on the set of standard λ-bitableaux such that, if w_{1}, w_{2} ∈ 𝔖_{n} with l(w_{1}) < l(w_{2}), then t^{λ}w_{1} < t^{λ}w_{2}. Note that t^{λ} is the first standard λ-bitableau in this order.

Define the elements T_{0}, T_{i,1} and T_{1,i} of H(W_{n}) for 1 ≤ i ≤ n as follows:

T_{0} = T_{s_{0}}
T_{i,1} = T_{s_{i−1}}T_{s_{i−2}} ⋯ T_{s_{2}}T_{s_{1}}
T_{1,i} = T_{s_{1}}T_{s_{2}} ⋯ T_{s_{i−2}}T_{s_{i−1}}.

For 1 ≤ m ≤ n, let

L̃_{m} = q^{1−m}T_{m,1}T_{0}T_{1,m}.

Then the following properties are easy to check.

2.1 The operators L̃_{m} satisfy:

1. L̃_{1}, L̃_{2}, …, L̃_{m} commute.

2. Let v = (m − 1, m), and 1 ≤ u ≤ n. Then T_{v} commutes with L̃_{u} if u ≠ m − 1, m, and

T_{v}L̃_{m} = L̃_{m−1}T_{v} + (q − 1)L̃_{m}
L̃_{m}T_{v} = T_{v}L̃_{m−1} + (q − 1)L̃_{m}.

Let λ = (λ^{(1)}, λ^{(2)}) be a bipartition of n, where λ^{(1)} is a partition of a. Define

2.2

u_{a}^{+} = ∏_{i=1}^{a}(q^{i−1}T_{1} + T_{i,1}T_{0}T_{1,i}) = ∏_{i=1}^{a} q^{i−1}(T_{1} − L̃_{i})
x_{λ}^{(1)} = ∑{T_{w} | w ∈< s_{1}, … s_{a−1} > and w stabilizes the rows of t^{λ^{(1)}}}
x_{λ}^{(2)} = ∑{T_{w} | w ∈< s_{a+1}, … s_{n−1} > and w stabilizes the rows of t^{λ^{(2)}}}.

Note that the factors in the product which defines u_{a}^{+} commute with each other. Also, u_{a}^{+}, x_{λ}^{(1)} and x_{λ}^{(2)} commute. Let

x_{λ} = u_{a}^{+}x_{λ}^{(1)}x_{λ}^{(2)}.

We are going to define certain H(W_{n})-modules which depend upon the elements x_{λ}, as λ runs over bipartitions of n.

Given two bipartitions λ = (λ^{(1)}, λ^{(2)}) and μ = (μ^{(1)}, μ^{(2)}) of n, write μ ⊵ λ if

∑_{i=1}^{j} μ_{i}^{(1)} ≥ ∑_{i=1}^{j} λ_{i}^{(1)}

for all j and

∑_{i=1}^{∞} μ_{i}^{(1)} + ∑_{i=1}^{j} μ_{i}^{(2)} ≥ ∑_{i=1}^{∞} λ_{i}^{(1)} + ∑_{i=1}^{j} λ_{i}^{(2)}

for all j. Here μ_{i}^{(1)} denotes part i of μ^{(1)}, and so on.

Let \overline{N}^{λ} be the 2-sided ideal of H(W_{n}) which is generated by

{x_{μ} | μ ⊵ λ and μ ≠ λ},

and let M^{λ}, \overline{M}^{λ} and S̃^{λ} be the right H(W_{n})-modules which are given by

M^{λ} = x_{λ}H(W_{n}) \overline{M}^{λ} = M^{λ} ∩ \overline{N}^{λ} S̃^{λ} = M^{λ}/\overline{M}^{λ}.

The H(W_{n})-modules S̃^{λ} are important for many reasons. For example, if R is a field and H(W_{n}) is semisimple, then every S̃^{λ} is absolutely irreducible, and as λ runs over bipartitions of n, S̃^{λ} runs over a complete set of nonisomorphic simple H(W_{n})-modules [3, 4.22]. More generally, if R is a field, then the dimension of S̃^{λ} is independent of R [3, 4.20], and every simple H(W_{n})- module is a homomorphic image of some S̃^{λ} [3, 6.6]. We shall explain next the way in which these results come about.

For all h ∈ H(W_{n}), there exists α ∈ R such that

x_{λ}hx_{λ} ≡ αx_{λ} mod \overline{N}^{λ}.

(See [3, 5.5].) Hence, there is a bilinear form < , >_{λ} on M^{λ} where, for all h_{1}, h_{2} ∈ H(W_{n}), we have

< x_{λ}h_{1}, x_{λ}h_{2} >_{λ}= α if x_{λ}h_{1}h_{2}^{∗}x_{λ} ≡ αx_{λ} mod \overline{N}^{λ}.

Here ^{∗} is the antiautomorphism of H(W_{n}) given by T_{w}^{∗} = T_{w^{−1}}. The radical M^{λ⊥} of this bilinear form is an H(W_{n})-module, and \overline{M}^{λ} ≤ M^{λ⊥}. Therefore, we can define a bilinear form on S̃^{λ}, which we also denote by < , >_{λ}, by specifying that

< u + \overline{M}^{λ}, v + \overline{M}^{λ} >_{λ}=< u, v >_{λ} (u, v, ∈ M^{λ}).

The key property of our bilinear form is that either M^{λ⊥} = M^{λ} or M^{λ⊥}/\overline{M}^{λ} is the unique maximal submodule of S̃^{λ} = M^{λ}/\overline{M}^{λ} and the corresponding factor module, which is isomorphic to M^{λ}/M^{λ⊥}, is simple. Moreover every simple module is of this form for some bipartition λ of n and simple H(W_{n})- modules for different bipartitions are non-isomorphic.

Now, S̃^{λ} has a standard basis, which consists of

\{x_{λ}T_{w} + \overline{M}^{λ} | t^{λ}w is a standard λ-bitableau\},

by [3, 4.20]. The Gram matrix of the bilinear form < , >_{λ} with respect to the standard basis of S̃^{λ} has rows and columns indexed by the standard λ-bitableaux in our order <. We aim to calculate the determinant of this Gram matrix, and we denote the determinant by det(λ). The formula which we produce is not only interesting in its own right, but it also throws light on problems in the representation theory of H(W_{n}). For example, since the dimension of M^{λ}/M^{λ⊥} equals the rank of the Gram matrix, we obtain a lower bound for the dimension of each simple H(W_{n})-module (see [5, p.222]). Furthermore, S̃^{λ} is itself a simple module precisely if det(λ) is non-zero.

3 An orthogonal basis

In this section, we construct an orthogonal basis of S̃^{λ} from the standard basis. In order to carry out the construction, we assume hereafter that the ring R over which H(W_{n}) is defined is the field of fractions of ℤ[Q, q], where Q and q are independent transcendentals. It is clear that all the entries in our Gram matrix belong to ℤ[Q, q], and therefore our formula for det(λ) gives the correct answer when H(W_{n}) is defined over an arbitrary commutative ring R, where q is a unit.

We have said that a diagram consists of an ordered pair of arrays of crosses in the plane. Let a position be a place where a cross occurs in some diagram (allowing arbitrarily many crosses). Define the residue r(x) of a position x as follows:

3.1

r(x) = \{ \begin{matrix}Qq^{j−i} & if x is in row i and column j of the first set of positions \\ −q^{j−i} & if x is in row i and column j of the second set of positions \end{matrix}

For example,

(\begin{matrix}Q & Qq & Qq^{2} \\ Qq^{−1} & Q & Qq \\ Qq^{−2} \end{matrix} , \begin{matrix}−1 & −q & −q^{2} \\ −q^{−1} & −1 & −q \\ −q^{−2} \end{matrix})

illustrates some residues. If t is a λ-bitableau and 1 ≤ m ≤ n, then let r_{t}(m) denote the residue of the position occupied by m in t.

Henceforth, let λ = (λ^{(1)}, λ^{(2)}) be a bipartition of n, where λ^{(1)} is a partition of a. If t is a standard λ-bitableau, then let

x_{λt} = x_{λ}T_{w},

if w ∈ 𝔖_{n} and t^{λ}w = t.

The following is proved in [3, 8.6].

3.2 If t is a standard λ-bitableau, then

x_{λt}L̃_{m} ≡ r_{t}(m)x_{λt} + ∑_{s} α_{s}x_{λs} mod \overline{N}^{λ}

for certain α_{s} ∈ R, where s runs through the set of standard λ-bitableax with s < t.

For each standard λ-bitableau t, let

f_{t} = x_{λt}E_{t} + \overline{M}^{λ},

where

E_{t} = ∏_{m=1}^{n} ∏_{c∈ℜ(m)\\{r_{t}(m)\}} \frac{L̃_{m} − c}{r_{t}(m) − c}

and

ℜ(m) = {Qq^{i}, −q^{i} | − m + 1 ≤ i ≤ m − 1}.

Now, [3, 8.11] shows the following:

3.3 An orthogonal basis of S̃^{λ}, with respect to the bilinear form < , >_{λ} is given by

{f_{t} | t is a standard λ-bitableau}.

Moreover, the matrix transforming the standard basis of S̃^{λ} into this orthogonal basis is unitriangular and so has determinant 1.

From (3.3), we immediately deduce our next result:

3.4 The Gram determinant of S̃^{λ} with respect to the standard basis of S̃^{λ} is given by

det(λ) = ∏_{t} < f_{t}, f_{t} >_{λ},

the product being over standard λ-bitableaux t.

The elements E_{t} of H(W_{n}), which are used in the definition of f_{t}, are idempotents [3, 8.10], and the results (3.5) and (3.6), below, concerning these idempotents, are taken from [3, 8.10 and 8.11].

3.5 For all m with 1 ≤ m ≤ n and all standard bitableaux s, we have

E_{s}L̃_{m} = r_{s}(m)E_{s}.

3.6 If s and t are standard λ-bitableaux, with t < s, then x_{λt}E_{s} ∈ \overline{M}^{λ}.

We are going to use result (3.4) to calculate det(λ). First, we want to find < f_{t}, f_{t} >_{λ} when t = t^{λ}. To this end, we introduce some more notation. For each positive integer r, let

[r] = [r]_{q} = 1 + q + q^{2} + … + q^{r−1}

and

{r} = [1][2] ⋯ [r].

Also let {0} = 1. Then, for j = 1 or 2, we have

(x_{λ}^{(j)})^{2} = (∏_{i≥1}{λ_{i}^{(j)}}) x_{λ}^{(j)}.

Note, too, that, by (2.2),

u_{a}^{+} = ∏_{m=1}^{a} q^{m−1}(T_{1} + L̃_{m}).

For t = t^{λ}, we have

f_{t} ≡ x_{λt} = x_{λ} = u_{a}^{+}x_{λ}^{(1)}x_{λ}^{(2)} mod \overline{M}^{λ}

and, by (3.2),

(x_{λ})^{2} = q^{\frac{a(a−1)}{2}} ( ∏_{m=1}^{a}(1 + r_{t}(m))) (∏_{i≥1}\{λ_{i}^{(1)}\}\{λ_{i}^{(2)}\}) x_{λ}.

We have shown:

3.7 Let t = t^{λ}. Then

< f_{t}, f_{t} >_{λ}= q^{\frac{a(a−1)}{2}} ( ∏_{m=1}^{a}(1 + r_{t}(m))) (∏_{i≥1}\{λ_{i}^{(1)}\}\{λ_{i}^{(2)}\}) .

For example, if λ = ((2, 1), (3, 2)) and t = t^{λ}, then

< f_{t}, f_{t} >_{λ}= q^{3}(1 + Q)(1 + Qq)(1 + Qq^{−1})(1 + q)^{3}(1 + q + q^{2}).

We fix the bipartition λ and write simply < , > for our bilinear form < , >_{λ}. Having calculated < f_{t}, f_{t} > for the first standard λ-bitableau t^{λ}, our next step is to relate < f_{s}, f_{s} > and < f_{t}, f_{t} > when s and t are standard λbitableaux, and s is obtained from t by interchanging the numbers m−1 and m for some 2 ≤ m ≤ n.

Proposition 3.8 Assume that v = (m − 1, m) (2 ≤ m ≤ n), and let s and t be standard λ-bitableaux with t < s and s = tv. Let r_{1} = r_{t}(m − 1) and r_{2} = r_{t}(m). Then

< f_{s}, f_{s} >= \frac{(qr_{1} − r_{2})(r_{1} − qr_{2})}{(r_{1} − r_{2})^{2}} < f_{t}, f_{t} > .

Proof (cf. [6, 6.2]). Note that r_{1} = r_{s}(m) and r_{2} = r_{s}(m − 1). Hence E_{s} + E_{t} is symmetric in L̃_{m−1} and L̃_{m}, and therefore T_{v} commutes with E_{s} + E_{t}. Thus,

x_{λt}T_{v}(E_{s} + E_{t})(L̃_{m} − r_{2}T_{1}) = x_{λt}(E_{s} + E_{t})T_{v}(L̃_{m} − r_{2}T_{1}).

Using (3.5) and (2.1), we deduce that

\begin{aligned}(r_{1} − r_{2})x_{λt}T_{v}E_{s} & = x_{λt}(E_{s} + E_{t})(L̃_{m−1}T_{v} + (q − 1)L̃_{m} − r_{2}T_{v}) \\ & = x_{λt}((r_{1} − r_{2})E_{t}T_{v} + (q − 1)(r_{1}E_{s} + r_{2}E_{t})). \end{aligned}

Since x_{λt}T_{v} = x_{λs}, result (3.6) now yields

(r_{1} − r_{2})x_{λs}E_{s} ≡ (r_{1} − r_{2})x_{λt}E_{t}T_{v} + (q − 1)r_{2}x_{λt}E_{t} mod \overline{M}^{λ}.

That is,

f_{t}T_{v} = −ρf_{t} + f_{s}, where ρ = \frac{(q − 1)r_{2}}{r_{1} − r_{2}} .

Next,

q < f_{t}, f_{t} > +(q − 1) < f_{t}T_{v}, f_{t} > =< f_{t}T_{v}^{2}, f_{t} >
=< f_{t}T_{v}, f_{t}T_{v} >
= ρ^{2} < f_{t}, f_{t} > + < f_{s}, f_{s} >,

because f_{s} and f_{t} are orthogonal. Therefore,

< f_{s}, f_{s} >= (q − (q − 1)ρ − ρ^{2}) < f_{t}, f_{t} >= \frac{(qr_{1} − r_{2})(r_{1} − qr_{2})}{(r_{1} − r_{2})^{2}} < f_{t}, f_{t} > .

This proves the proposition. □

Since we can reach any standard λ-bitableau from t^{λ} by applying a series of basic transpositions, result (3.7) and Proposition 3.8, in principle, allow us to calculate < f_{s}, f_{s} > for all standard λ-bitableaux s, and hence, by the way of result (3.4), to calculate our desired determinant det(λ). In the next two sections, we improve upon this remark by deducing a closed formula for det(λ).

4 The branching theorem for determinants

We are going to derive an expression for < f_{t}, f_{t} >_{λ} from the recursive formula which we gave in the last section.

Suppose that μ is a bipartition. We say that a position x is removable from [μ] if there is a cross in position x of the diagram [μ] and this cross can be removed from [μ] to leave the diagram of a bipartition. We say that a position x is addable to [μ] if no cross of [μ] is at position x, but a cross can be added to [μ] at x to obtain the diagram of a bipartition.

For example, if μ = ((3, 2^{2}), (2, 1)) then the positions below marked “r” are removable, and those marked “a” are addable.

(\begin{matrix}× & × & r & a \\ × & × & a \\ × & r \\ a \end{matrix} , \begin{matrix}× & r & a \\ r & a \\ a \end{matrix})

Given two positions x and y, we write x < y if one of the following holds:

1. x is in the first set of positions and y is in the second set of positions;

2. x and y are in the same set of positions, and the column index of x is strictly bigger than the column index of y.

If the position x is removable from [μ] then let [μ]\{x} denote [μ] with the cross at x deleted, and let μ\{x} denote the corresponding bipartition. Define A_{μ}(x) and B_{μ}(x) as follows:

A_{μ}(x) = {positions y | x < y and y is addable to [μ]\{x}}
B_{μ}(x) = {positions y | x < y and y is removable from [μ]\{x}}

For example, if μ = ((5, 4^{2}, 1), (3, 2^{2})) and x is the position which we have circled, then the positions which we have marked “a” are the positions in A_{μ}(x) and the positions which we have marked “b” are the positions in B_{μ}(x).

(\begin{matrix}× & × & × & × & × \\ × & × & × & × \\ × & × & b & ⊗ \\ b & a \\ a \end{matrix} , \begin{matrix}× & × & b & a \\ × & × & a \\ × & b \\ a \end{matrix})

Note that the first set of positions always contains equally many positions from A_{μ}(x) and B_{μ}(x); the second set of positions has the same property, unless x is in the first set of positions, in which case the size of A_{μ}(x) exceeds the size of B_{μ}(x) by 1.

Definitions 4.1 1. Suppose that μ is a bipartition and x is a position which is removable from [μ]. Let

γ_{μ}(x) = \frac{∏_{y∈A_{μ}(x)}(r(x) − r(y))}{∏_{y∈B_{μ}(x)}(r(x) − r(y))}.

2. Assume that 1 ≤ u ≤ n and that t is a standard λ-bitableau. Let [μ] be the shape of the bitableau which is obtained from t by omitting all numbers which are strictly bigger than u. (Note that this procedure indeed produces the shape of a bitableau for some bipartition, since in a standard bitableau the position occupied by the largest integer is always removable). Suppose that x is the position of u in t. Define

γ_{u,t} = γ_{μ}(x).

Example 4.2 Let

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

Then

γ_{11,t} = \frac{(−q^{2} + 1)(−q^{2} + q^{−2})}{(−q^{2} + q)(−q^{2} + q^{−1})} = q^{−2} \frac{[2] [4]}{[1] [3]}

and

\begin{aligned}γ_{8,t} & = \frac{(Qq^{2} − Qq^{−2})(Qq^{2} + q^{2})(Qq^{2} + 1)(Qq^{2} + q^{−2})}{(Qq^{2} − Q)(Qq^{2} + q)(Qq^{2} + q^{−1})} \\ & = q^{−2} \frac{[4](Q + 1)(Qq^{2} + 1)(Qq^{4} + 1)}{[2](Qq + 1)(Qq^{3} + 1)}. \end{aligned}

Example 4.3 Let

t = (1 2  ,  3 4 5)

Then

γ_{1,t} = Q + 1
γ_{2,t} = \frac{(Qq − Qq^{−1})(Qq + 1)}{(Qq − Q)}
γ_{3,t} = 1
γ_{4,t} = \frac{−q + q^{−1}}{−q + 1}
γ_{5,t} = \frac{−q^{2} + q^{−1}}{−q^{2} + q} .

Therefore

∏_{u=1}^{5} γ_{u,t} = q^{−4}(Q + 1)(Qq + 1)([2])^{2}[3].

Theorem 4.4 For all standard λ-bitableaux t, there exists an integer v such that

< f_{t}, f_{t} >_{λ}= q^{v} ∏_{u=1}^{n} γ_{u,t}.

Proof Assume first that t = t^{λ}, where λ = (λ^{(1)}, λ^{(2)}) is a bipartition of n and λ^{(1)} is a partition of a. Then it is straightforward to verify that

γ_{n,t} = \{ \begin{matrix}q^{i}[λ_{k}^{(2)}] & if λ^{(2)} is non-empty and λ^{(2)} has k parts \\ q^{i}(1 + r_{t}(n))[λ^{(1)}] & if λ^{(2)} is empty and λ^{(1)} has k parts \end{matrix}

for some integer i, and hence that

∏_{u=1}^{n} γ_{u,t} = q^{j} ( ∏_{m=1}^{a}(1 + r_{t}(m))) (∏_{i≥1}{λ_{i}^{(1)}}{λ_{i}^{(2)}})

for some integer j (cf. Example 4.3). By result (3.7) this product equals q^{−v} < f_{t}, f_{t} >_{λ} for some integer v.

Assume next that the equality in the statement of the theorem is true for some standard λ-bitableau t, and that s = t(m − 1, m) is a standard λbitableau with t < s. Now γ_{u,s} = γ_{u,t} for all u ≠ m−1, m, and γ_{m−1,s} = γ_{m,t}.

Let [μ] be the shape of the bitableau which is obtained from t by omitting all numbers strictly bigger than m−1, and let [ν] be the shape of the bitableau which is obtained from t by omitting all numbers strictly bigger than m. Suppose that m − 1 occupies position x in t and m occupies position y in t. Let y_{1}, y_{2}, y_{3}, y_{4} be the positions left of, right of, above and below position y, respectively. Then

y_{1} ∈ B_{μ}(x) or y_{4} ∈ A_{ν}(x) and not both

and

y_{3} ∈ B_{μ}(x) or y_{2} ∈ A_{ν}(x) and not both .

For all positive z, except y_{1}, y_{2}, y_{3}, y_{4}, we have

z ∈ A_{μ}(x) ⇔ z ∈ A_{ν}(x)

and

z ∈ B_{μ}(x) ⇔ z ∈ B_{ν}(x).

Furthermore

r(y_{3}) = r(y_{2}) = qr(y)

and

r(y_{1}) = r(y_{4}) = q^{−1}r(y).

Therefore

\frac{γ_{m,s}}{γ_{m−1,t}} = \frac{(r(x) − qr(y))(r(x) − q^{−1}r(y))}{(r(x) − r(y))^{2}} = \frac{< f_{s}, f_{s}, >_{λ}}{q < f_{t}, f_{t} >_{λ}} ,

by Proposition 3.8. The result of the theorem now follows by induction. □

Although it is possible to find the exponent of q which occurs in Theorem 4.4, we do not bother to keep track of this exponent here or in later theorems, since q is assumed to be a unit.

For all bipartitions μ, let dim(μ) be equal to the number of standard μ-bitableaux. Thus dim(μ) is the dimension of S̃^{μ}. The next result is an immediate consequence of result (3.4) and Theorem 4.4.

Theorem 4.5 (Branching Theorem for Determinants) For a bipartition λ there exists an integer v such that

det(λ) = q^{v} ∏_{x} det(λ\{x})(γ_{λ}(x))^{dim(λ\{x})},

the product being taken over positions x which are removable from [λ].

Example 4.6 Suppose that λ = ((2^{2}, 1), (2)). Then

det(λ) = q^{v} det((2, 1^{2}), (2)) det((2^{2}), (2)) det((2^{2}, 1), (1))

× (\frac{(Q − Qq^{−3})(Q + q^{2})(Q + q^{−1})}{(Q − Qq^{−2})(Q + q)})^{dim((2,1^{2}),(2))}

× (\frac{(Qq^{−2} + q^{2})(Qq^{−2} + q^{−1})}{(Qq^{−2} + q)})^{dim((2^{2}),(2))}

× (\frac{−q + q^{−1}}{−q + 1} )^{dim((2^{2},1),(1))} .

In order to manipulate the Branching Theorem for Determinants, we introduce β-numbers. Recall from [4, p.77] that each partition can be described by a set of β-numbers, in many ways.

If

β_{1}^{(1)} > β_{2}^{(1)} > … > β_{m}^{(1)} ≥ 0 and β_{1}^{(2)} > β_{2}^{(2)} > … > β_{m}^{(2)} ≥ 0,

where every β_{j}^{(i)} is an integer, then let

d((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)})) = dim(μ^{(1)}, μ^{(2)}),

where β_{1}^{(1)}, … β_{m}^{(1)} are β-numbers for μ^{(1)} and β_{1}^{(2)}, …, β_{m}^{(2)} are β-numbers for μ^{(2)}. If π, σ ∈ 𝔖_{m}, then let

d((β_{1π}^{(1)}, …, β_{mπ}^{(1)}), (β_{1σ}^{(2)}, …, β_{mσ}^{(2)}))

= (sgn π)(sgn σ)d((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)})).

If β_{i}^{(1)} = β_{j}^{(1)} or β_{i}^{(2)} = β_{j}^{(2)} for some i ≠ j, or if some β_{i}^{(1)} < 0 or some β_{i}^{(2)} < 0, then let

d((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)})) = 0.

Note that

4.7 The following addition formula holds:

d((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)}))

= ∑_{k=1}^{m} d((β_{1}^{(1)}, …, β_{k}^{(1)} − 1, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)}))

+ ∑_{k=1}^{m} d((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{k}^{(2)} − 1, …, β_{m}^{(2)})).

Now let λ = (λ^{(1)}, λ^{(2)}) be a bipartition of n. Assume that m ≥ 2n. Let α_{1}, α_{2}, …, α_{m} be β-numbers for λ^{(1)} and β_{1}, β_{2}, …, β_{m} be β-numbers for λ^{(2)}, with

α_{1} > α_{2} > … > α_{m} and β_{1} > β_{2} > … > β_{m}.

Since m > n, we see that α_{m} = β_{m} = 0.

Our next lemma describes γ_{λ}(x) in terms of these β-numbers.

Lemma 4.8 Assume that x is a position which is removable from [λ].

(1) If x lies in row i of λ^{(1)} then

\begin{matrix}γ_{λ}(x) = & q^{v_{1}} ( ∏_{j=i+1}^{m}[α_{i} − α_{j} − 1]) ( ∏_{j=i+1}^{m−1}[α_{i} − α_{j}]^{−1}) \\ × ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−1} + 1) ∏_{j=1}^{m−1}(Qq^{α_{i}−β_{j}} + 1)^{−1}. \end{matrix}

(2) If x lies in row i of λ^{(2)} then

γ_{λ}(x) = q^{v_{2}} ( ∏_{j=i+1}^{m}[β_{i} − β_{j} − 1]) ( ∏_{j=i+1}^{m−1}[β_{i} − β_{j}]^{−1}) .

Here v_{1} and v_{2} are some integers.

Proof Assume that x lies in row i of λ^{(1)}. Suppose first that y lies in row j of λ^{(1)}. If y ∈ A_{λ}(x) then

r(x) − r(y) = (a power of q)Q(q^{α_{i}} − q^{α_{j}+1})
= (a power of q)Q(q − 1)[α_{i} − α_{j} − 1].

Similarly, if y ∈ B_{λ}(x) then

r(x) − r(y) = (a power of q)Q(q − 1)[α_{i} − α_{j}].

All the factors Q(q−1) cancel when we evaluate γ_{λ}(x), and, in the product

( ∏_{j=i+1}^{m}[α_{i} − α_{j} − 1]) ( ∏_{j=i+1}^{m−1}[α_{i} − α_{j}]^{−1})

we obtain only those terms which appear in γ_{λ}(x), since any term which comes from factors outside A_{λ}(x) and B_{λ}(x) cancel in pairs from numerator and denominator.

Next suppose that y lies in row j of λ^{(2)}. If y ∈ A_{λ}(x) then

r(x) − r(y) = (a power of q)(Qq^{α_{i}} + q^{β_{j}+1})
= (a power of q)(Qq^{α_{i}−β_{j}−1} + 1)

Similarly, if y ∈ B_{λ}(x) then

r(x) − r(y) = (a power of q)(Qq^{α_{i}−β_{j}} + 1).

In the product

∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−1} + 1) ∏_{j=1}^{m−1}(Qq^{α_{i}−β_{j}} + 1)^{−1}

we obtain only those terms which appear in γ_{λ}(x), since terms which come from factors outside A_{λ}(x) and B_{λ}(x) cancel in pairs in the numerator and denominator.

We have now proved part (1) of the lemma. The proof of part (2) is similar, but easier. □

In order to state the Branching Theorem for Determinants in terms of β-numbers, we make a final definition. Suppose that

β_{1}^{(1)} ≥ β_{2}^{(1)} ≥ … ≥ β_{m}^{(1)} = 0 and β_{1}^{(2)} ≥ β_{2}^{(2)} ≥ … ≥ β_{m}^{(2)} = 0,

where every β_{j}^{(i)} is an integer. If β_{i}^{(1)} = β_{j}^{(1)} for some i ≠ j or β_{i}^{(2)} = β_{j}^{(2)} for some i ≠ j, then let

det((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)})) = 1;

otherwise, let

det((β_{1}^{(1)}, …, β_{m}^{(1)}), (β_{1}^{(2)}, …, β_{m}^{(2)})) = det((μ^{(1)}, μ^{(2)})),

where β_{1}^{(1)}, …, β_{m}^{(1)} are β-numbers for μ^{(1)} and β_{1}^{(2)}, …, β_{m}^{(2)} are β-numbers for μ^{(2)}.

We may now use Lemma 4.8 to reformulate the Branching Theorem for Determinants in the following way.

Proposition 4.9 Let λ be a bipartition of n. Then

det(λ) = q^{v} ∏_{k=1}^{m−1} det((α_{1}, …, α_{k} − 1, …, α_{m}), (β_{1}, …, β_{m}))

× ∏_{k=1}^{m−1} det((α_{1}, …, α_{m}), (β_{1}, …, β_{k} − 1, …, β_{m}))

× ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{i}=α_{j} +h+1}}^{m}[h]^{d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{i}=α_{j} +h}}^{m−1}[h]^{−d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop β_{i}=β_{j} +h+1}}^{m}[h]^{d((α_{1},... ,α_{m}),(β_{1},... ,β_{i}−1,... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop β_{i}=β_{j} +h}}^{m−1}[h]^{−d((α_{1},... ,α_{m}),(β_{1},... ,β_{i}−1,... ,β_{m}))}

× ∏_{i=1}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−1} + 1)^{d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

× ∏_{i=1}^{m} ∏_{j=1}^{m−1}(Qq^{α_{i}−β_{j}} + 1)^{−d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}.

5 The determinant of the Gram matrix

In this final section, we present the culmination of our calculations, namely a formula for det(λ).

We need a preliminary lemma.

Lemma 5.1 Suppose that λ = (λ^{(1)}, λ^{(2)}) is a bipartition of n, and α_{1} > α_{2} > … > α_{m} = 0 are β-numbers for λ^{(1)} and β_{1} > β_{2} > … > β_{m} = 0 are β-numbers for λ^{(2)}, with m ≥ 2n.

(1) Assume that 1 ≤ i ≤ m − 1 and h > 1, and let α′_{i} = α_{i} − h > 0, α′_{m} = h − 1 and α′_{j} = α_{j} if j ≠ i, m. Then

d((α′_{1}, …, α′_{m}), (β_{1}, …, β_{m})) = 0.

(2) Assume that 1 ≤ i ≤ m and h ≥ 1, and let α′_{i} = α_{i} − h − 1 and α′_{j} = α_{j} if j ≠ i; also, β′_{m} = h and β′_{j} = β_{j} if j ≠ m. Then

d((α′_{1}, …, α′_{m}), (β′_{1}, …, β′_{m})) = 0.

Proof Assume the hypothesis of part (1) of the lemma. Let s be the largest integer such that α_{m−s} = s. Since m ≥ 2n, and λ^{(1)} is a partition of an integer which is at most n, it follows that n ≤ s and α_{j} ≤ s + n for all j.

It is sufficient to prove that α′_{j} = α′_{k} for some j and k with j ≠ k.

If i ≥ m − s then α_{i} − h = α_{k} for some k with i < k ≤ m − 1, so assume that i < m − s.

If h ≥ s + 1 then α_{i} − h < n ≤ s and α_{i} − h = α_{k} for some k with m − s ≤ k ≠ m. Hence i < k and α′_{i} = α′_{k}.

If h < s + 1 then h − 1 = α_{k} for some k with m − s ≤ k ≤ m − 1, so α′_{m} = α′_{k}.

The proof of part (1) of the lemma is now complete, and the proof of part (2) is similar. □

We can now state and prove our Main Theorem.

Theorem 5.2 Suppose that λ = (λ^{(1)}, λ^{(2)}) is a bipartition of n, and α_{1} > α_{2} > … > α_{m} = 0 are β-numbers for λ^{(1)} and β_{1} > β_{2} > … > β_{m} = 0 are β-numbers for λ^{(2)}, with m ≥ 2n. Then for some integer v,

det(λ) = q^{v}X_{λ}Y_{λ}Z_{λ},

where

X_{λ} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−h}}^{m}[h]^{−d((α_{1},... ,α_{i}−h,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}

Y_{λ} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop β_{j} <β_{i}−h}}^{m}[h]^{−d((α_{1},... ,α_{m}),(β_{1},... ,β_{i}−h,... ,β_{j}+h,... ,β_{m}))}

Z_{λ} = ∏_{h=1}^{∞} ∏_{i=1}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{j}+h,... ,β_{m}))}.

Proof Note first that if α_{1}, …, α_{m}, β_{1}, …, β_{m} are integers with

α_{1} ≥ α_{2} ≥ … ≥ α_{m} = 0
β_{1} ≥ β_{2} ≥ … ≥ β_{m} = 0

and α_{i} = α_{j} for some i ≠ j or β_{i} = β_{j} for some i ≠ j, then X_{λ} = Y_{λ} = Z_{λ} = 1. This remark allows us to apply induction, by the way of Proposition 4.9.

Proposition 4.9, and induction, give det(λ) as a product of several terms. First, we gather the terms which will give X_{λ}: We call these terms A_{1}, A_{2}, A_{3}, A_{4} and A_{5}, and they are given below. The terms A_{1}, A_{2}, A_{3} come from the first product occurring as a factor in det(λ) in Proposition 4.9, and induction; we distinguish the cases where k ≠ i, j or k = i or k = j in the formula for X_{λ}. The terms A_{4} and A_{5} are the expressions which occur as the third and fourth products in Proposition 4.9.

A_{1} = ∏_{h=1}^{∞} ∏_{k=1}^{m−1} ∏_{{i=1 \atop i≠k}}^{m−1} ∏_{j=i+1 ; {j≠k \atop α_{j} <α_{i}−h}}^{m} [h]^{−d((α_{1},... ,α_{k}−1,... ,α_{i}−h,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}

A_{2} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−1−h}}^{m}[h]^{−d((α_{1},... ,α_{i}−h−1,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}

A_{3} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} −1<α_{i}−h}}^{m−1}[h]^{−d((α_{1},... ,α_{i}−h,... ,α_{j}−1+h,... ,α_{m}),(β_{1},... ,β_{m}))}

A_{4} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{i}=α_{j} +h+1}}^{m}[h]^{d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

A_{5} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{i}=α_{j} +h}}^{m−1}[h]^{−d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}.

Now, by swapping α_{i} − 1 and α_{j} in A_{4}, we obtain

A_{4} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} +h=α_{i}−1}}^{m}[h]^{−d((α_{1},... ,α_{j},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

= ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} +h=α_{i}−1}}^{m}[h]^{−d((α_{1},... ,α_{i}−h−1,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}.

Therefore

A_{2}A_{4} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−h}}^{m}[h]^{−d((α_{1},... ,α_{i}−h−1,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}.

Also,

A_{5} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} −1+h=α_{i}−1}}^{m−1}[h]^{−d((α_{1},... ,α_{j},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

= ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} −1+h=α_{i}−1}}^{m−1}[h]^{d((α_{1},... ,α_{i}−h,... ,α_{j}−1+h,... ,α_{m}),(β_{1},... ,β_{m}))}.

Therefore

A_{3}A_{5} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−h}}^{m−1}[h]^{−d((α_{1},... ,α_{i}−h,... ,α_{j}−1+h,... ,α_{m}),(β_{1},... ,β_{m}))}

= ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−h}}^{m}[h]^{−d((α_{1},... ,α_{i}−h,... ,α_{j}−1+h,... ,α_{m}),(β_{1},... ,β_{m}))},

by Lemma 5.1(1).

We now have by (4.7) that A_{1}A_{2}A_{3}A_{4}A_{5} = A, where

A = ∏_{h=1}^{∞} ∏_{k=1}^{m−1} ∏_{{i=1 \atop i≠k}}^{m−1} ∏_{j=i+1 ; {j≠k \atop α_{j} <α_{i}−h}}^{m} [h]^{−d((α_{1},... ,α_{k}−1,... ,α_{i}−h,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−h}}^{m}[h]^{−d((α_{1},... ,α_{i}−h−1,... ,α_{j}+h,... ,α_{m}),(β_{1},... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{{j=i+1 \atop α_{j} <α_{i}−h}}^{m}[h]^{−d((α_{1},... ,α_{i}−h,... ,α_{j}−1+h,... ,α_{m}),(β_{1},... ,β_{m}))}

= X_{λ},

Similarly, we can gather terms to obtain Y_{λ}.

The only remaining terms which come from Proposition 4.9 and induction are B_{1}, B_{2}, B_{3}, B_{4}, B_{5} and B_{6}, where

B_{1} = ∏_{h=1}^{∞} ∏_{k=1}^{m−1} ∏_{{i=1 \atop i≠k}}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{k}−1,... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{j}+h,... ,β_{m}))}

B_{2} = ∏_{h=1}^{∞} ∏_{k=1}^{m−1} ∏_{i=1}^{m} ∏_{{j=1 \atop j≠k}}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{k}−1,... ,β_{j}+h,... ,β_{m}))}

B_{3} = ∏_{h=1}^{∞} ∏_{i=1}^{m−1} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h−1} + 1)^{d((α_{1},... ,α_{i}−h−1,... ,α_{m}),(β_{1},... ,β_{j}+h,... ,β_{m}))}

B_{4} = ∏_{h=1}^{∞} ∏_{i=1}^{m} ∏_{j=1}^{m−1}(Qq^{α_{i}−β_{j}−h+1} + 1)^{d((α_{1},... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{j}+h−1,... ,β_{m}))}
B_{5} = ∏_{i=1}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}+1} + 1)^{d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}

B_{6} = ∏_{i=1}^{m} ∏_{j=1}^{m−1}(Qq^{α_{i}−β_{j}} + 1)^{−d((α_{1},... ,α_{i}−1,... ,α_{m}),(β_{1},... ,β_{m}))}.

We may allow the second product in B_{3} to run from i = 1 to i = m, since this merely introduces factors whose exponent is zero. Therefore

B_{3}B_{5} = ∏_{h=1}^{∞} ∏_{i=1}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{j}+h−1,... ,β_{m}))}.

Also

B_{4}B_{6} = ∏_{h=1}^{∞} ∏_{i=1}^{m} ∏_{j=1}^{m−1}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h−1,... ,α_{m}),(β_{1},... ,β_{j}+h,... ,β_{m}))}.

Here, we may allow the third product to run from j = 1 to j = m, by Lemma 5.1(2).

We now have B_{1}B_{2}B_{3}B_{4}B_{5}B_{6} = B, where

B = ∏_{h=1}^{∞} ∏_{k=1}^{m−1} ∏_{{i=1 \atop i≠k}}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{k}−1,... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{j}+h,... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{k=1}^{m−1} ∏_{i=1}^{m} ∏_{{j=1 \atop j≠k}}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{k}−1,... ,β_{j}+h,... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h,... ,α_{m}),(β_{1},... ,β_{j}+h−1,... ,β_{m}))}

× ∏_{h=1}^{∞} ∏_{i=1}^{m} ∏_{j=1}^{m}(Qq^{α_{i}−β_{j}−h} + 1)^{d((α_{1},... ,α_{i}−h−1,... ,α_{m}),(β_{1},... ,β_{j}+h,... ,β_{m}))}

= Z_{λ},

by (4.7). This completes the proof of Theorem 5.2. □

We remark that the terms X_{λ} and Y_{λ} in Theorem 5.2 can be deduced from the Branching Theorem for Determinants in a way similar to the calculations in [5, section 3]. However, to translate from [5], it is necessary to take the conjugates of all partitions, since the H(W_{n})-modules S̃^{λ} are generalisations of the duals of Specht modules.

We used β-numbers in order to state Theorem 5.2, and to facilitate the proof of the theorem. But the best way to understand the formula for det(λ) involves “unwrapping a skew hook from [λ] and wrapping it on somewhere else”, (compare the discussion in [5, section 1]). We shall now describe this process, examining X_{λ}, Y_{λ} and Z_{λ} of Theorem 5.2, in turn.

Let h_{ab} denote the length of the (a, b)-hook in [λ^{(1)}], the first of the two arrays of crosses which make up [λ]. Then X_{λ} can be expressed as follows (cf.[5, p.233 and p.224]). To obtain X_{λ} take the product over all a, b and c with b < c of the quotient [h_{ab}]/[h_{ac}] raised to an exponent which depends upon a, b and c. The exponent is obtained by removing from [λ^{(1)}] the skew (a, c)-hook, and wrapping it on again, so that its hand is now in column b; if this does not give a diagram then the exponent is zero; if the diagram [μ^{(1)}] is obtained, then the exponent is ± dim(μ^{(1)}, λ^{(2)}) the sign being + if and only if the sum of the leg lengths of the unwrapped skew hook and the wrapped on skew hook is even.

Example 5.3 Suppose that λ^{(1)} = (3^{2}, 1). Then the hook lengths in [λ^{(1)}] are given by

5 3 2 4 2 1 1

We get a contribution of ([5]/[3])^{− dim((2,1^{5}),λ^{(2)})} to X_{λ}, the exponent determined by the manoeuvre illustrated below.

\begin{aligned}× & × & & × \\ & & & | \\ × & × & − & × \\ × \end{aligned}  ⟶  \begin{matrix}× & × \\ × \\ × \\ × \\ | \\ × \\ | \\ × \end{matrix}

The answer for X_{λ} is

X_{λ} = (\frac{[5]}{[3]})^{− dim((2,1^{5}),λ^{(2)})} (\frac{[5]}{[2]})^{dim((2^{2},1^{3}),λ^{(2)})} (\frac{[3]}{[2]})^{0}

× (\frac{[4]}{[2]})^{− dim((3,1^{4}),λ^{(2)})} (\frac{[4]}{[1]})^{dim((3,2,1^{2}),λ^{(2)})} (\frac{[2]}{[1]})^{dim((3,2^{2}),λ^{(2)})} .

The term Y_{λ} in Theorem 5.2 is obtained in a similar way, working with [λ^{(2)}] in place of [λ^{(1)}].

Finally, we turn to Z_{λ}. Unwrap from [λ^{(1)}] a skew hook whose hand position has residue r(x), and wrap this skew hook on to [λ^{(2)}] and suppose that the wrapped on skew hook has r(y) as the residue of its hand position. If the new diagram is [μ^{(1)}, μ^{(2)}] then we get a contribution of

(r(x) − r(y))^{± dim(μ^{(1)},μ^{(2)})}

to Z_{λ}; the sign is + if and only if the sum of the leg lengths of the unwrapped and wrapped on skew hooks is even. The product of all terms obtained this way is equal to (a power of q) × Z_{λ}.

Example 5.4 Suppose that λ = ((3, 1), (2^{2})). The residues in [λ] are indicated by

(\begin{matrix}Q & Qq & Qq^{2} \\ Qq^{−1} \end{matrix}  ,  \begin{matrix}−1 & −q \\ −q^{−1} & −1 \end{matrix})

We get contributions of

(Qq^{2} + q^{5})^{− dim((0),(6,2))} and (Qq^{2} + q^{2})^{− dim((1^{2}),(3^{2}))}

to Z_{λ}, the exponents being determined by the manoeuvres illustrated below.

(\begin{matrix}× & − & × & − & × \\ | \\ × \end{matrix}  ,  \begin{matrix}× & × \\ × & × \end{matrix})

⟶ (\begin{matrix}. & . & . \\ . \end{matrix}  ,  \begin{matrix}× & × & × & − & × & − & × & − & × \\ × & × \end{matrix})

respectively

(\begin{matrix}× & × & − & × \\ × \end{matrix}  ,  \begin{matrix}× & × \\ × & × \end{matrix})

⟶ (\begin{matrix}× \\ × \end{matrix}  ,  \begin{matrix}× & × & × \\ & & | \\ × & × & × \end{matrix})

The answer for Z_{λ} is (a power of q) multiplied by (Qq^{2} + q^{2})^{dim((2,1),(3,2))}(Qq^{2} + q^{−2})^{dim((2,1),(2^{2},1))}

× (Qq^{2} + q^{3})^{dim((1^{2}),(4,2))}(Qq^{2} + q^{2})^{− dim((1^{2}),(3^{2}))}

× (Qq^{2} + q^{−1})^{dim((1^{2}),(2^{3}))}(Qq^{2} + q^{−2})^{− dim((1^{2}),(2^{2},1^{2}))}

× (Qq^{2} + q^{5})^{− dim((0),(6,2))}(Qq^{2} + q^{4})^{dim((0),(5,3))}

× (Qq^{2} + q^{−1})^{− dim((0),(2^{3},1^{2}))}(Qq^{2} + q^{−2})^{dim((0),(2^{2},1^{4}))}

× (Qq^{−1} + q^{2})^{dim((3),(3,2))}(Qq^{−1} + q^{−2})^{dim((3),(2^{2},1))} = (a power of q) × (Q + 1)^{560}(Qq^{4} + 1)^{560}(Q + q)^{252} × (Q + 1)^{−140}(Qq^{3} + 1)^{140}(Qq^{4} + 1)^{−256}(Q + q^{3})^{−20} × (Q + q^{2})^{28}(Qq^{3} + 1)^{−28}(Qq^{4} + 1)^{20}(Q + q^{3})^{280}(Qq + 1)^{280} = (a power of q) × (Qq^{4} + 1)^{324}(Qq^{3} + 1)^{92}(Qq + 1)^{280} × (Q + 1)^{420}(Q + q)^{252}(Q + q^{2})^{28}(Q + q^{3})^{280}.

In this example,

X_{λ} = (\frac{[4]}{[2]})^{− dim((1^{4}),(2^{2}))} (\frac{[4]}{[1]})^{dim((2,1^{2}),(2^{2}))} (\frac{[2]}{[1]})^{dim((2^{2}),(2,1))}

= [2]^{420}[4]^{280}

and

Y_{λ} = (\frac{[3]}{[2]})^{dim((3,1),(1^{4}))} (\frac{[2]}{[1]})^{dim((3,1),(2,1^{2}))}
= [2]^{420}[3]^{210}.

Therefore

det(λ) = (a power of q) × (1 + q)^{840}(1 + q + q^{2})^{210}(1 + q + q^{2} + q^{3})^{280}

× (Qq^{4} + 1)^{324}(Qq^{3} + 1)^{92}(Qq + 1)^{280}(Q + 1)^{420} × (Q + q^{5})^{252}(Q + q^{2})^{28}(Q + q^{3})^{280}.

The definition of det(λ) shows that it is a polynomial in the two indeterminates Q and q. Without that assurance, it would be amazing that our formula in Theorem 5.2 always yields a polynomial, as it did in the example above.

References

[1] R. Dipper, G.D. James Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math.Soc.(3), 54 (1987), 57–82.

GRAM DETERMINANTS 25

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

[3] 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.

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

[5] G.D. James, G.E. Murphy The determinant of the Gram matrix for a Specht module, J. Algebra 59 (1979), 222–235.

[6] G.E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992), 492–513.