MATRIX UNITS AND GENERIC DEGREES FOR THE ARIKI–KOIKE ALGEBRAS

ANDREW MATHAS

Abstract. We compute the generic degrees of the Ariki–Koike algebras by first constructing a basis of matrix units in the semisimple case. As a consequence, we also obtain an explicit isomorphism from any semisimple Ariki– Koike algebra to the group algebra of the corresponding complex reflection group.

1. Introduction

The cyclotomic Hecke algebras were introduced by Ariki and Koike [2, 4] and Broué and Malle [7]. It is conjectured [7] that these algebras play a rôle in the representation theory of reductive groups similar to (but more complicated than) that played by the Iwahori–Hecke algebras (see, for example, [8]). In particular, it should be possible to use these algebras to compute the degrees (and more generally characters) of certain representations of reductive groups; more precisely, we can attach a polynomial to each irreducible representation ℋ , called its generic degree, and appropriate specialisations of this polynomial should compute the dimensions of the corresponding irreducible representations of the finite groups of Lie type.

The purpose of this paper is to compute the generic degrees for the cyclotomic Hecke algebras of type G(r, 1, n); these polynomials have also been computed by Geck, Iancu and Malle [15]. Further results of Malle [19] and Malle and the author [21] give the generic degrees for all of the cyclotomic Hecke algebras corresponding to imprimitive complex reflection groups. Malle [20] has recently computed the generic degrees for cyclotomic algebras for the primitive complex reflection groups (modulo the assumption that the corresponding cyclotomic Hecke algebras are symmetric), so this completes the calculation of the generic degrees of the cyclotomic Hecke algebras associated with complex reflection groups.

Two important special cases of the Ariki–Koike algebras are the Iwahori–Hecke algebras of types A and B; the generic degrees of these algebras are well–known and were first computed by Hoefsmit [16]. Later Murphy [25] gave an easier derivation of Hoefsmit’s formulae for the generic degrees of the Iwahori–Hecke algebras of type A using different, but related, techniques.

This article is largely inspired by Murphy’s paper [25]; however, with hindsight we are able to take quite a few shortcuts. Along the way we give a new and quite elegant treatment of the representation theory of the semisimple Ariki–Koike algebras. In particular, we explicitly construct the primitive idempotents and the matrix units in the Wedderburn decomposition of ℋ . One of the nice features of our approach is that we use the modular theory (more accurately, the cellular theory) to understand the semisimple case. Another advantage of our approach is that our calculation of the generic degrees is almost entirely algebraic and it is only near of the end of the paper that we need to introduce combinatorial arguments (however, this is not to say that our approach is painless; in addition, showing that our result agrees with [15] is a long and unpleasant calculation).

1991 Mathematics Subject Classification. 20C08, 20G05, 33D80.

Key words and phrases. Ariki–Koike algebras, generic degrees and complex reflection groups.

2. The Ariki–Koike algebras

Let R be a commutative domain with 1 and fix elements q, Q_{1}, …, Q_{r} in R with q invertible. Let q = (q; Q_{1}, …, Q_{r}). The Ariki–Koike algebra ℋ = ℋ_{q}(n) is the unital associative algebra with generators T_{0}, T_{1}, …, T_{n−1} and relations

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

The Ariki–Koike algebra is a deformation of the group algebra of W_{r,n}, where W_{r,n} = (ℤ/rℤ) ≀ 𝔖_{n} is the wreath product of the cyclic group of order r and the symmetric group 𝔖_{n} of degree n.

Define elements L_{m} = q^{1−m}T_{m−1} … T_{1}T_{0}T_{1} … T_{m−1} for m = 1, 2, …, n; these are analogues of the q–Murphy operators of the Iwahori–Hecke algebras of type A [10, 25]. An easy calculation using the relations in ℋ (see [4, 3.3] and [10, (2.1), (2.2)]) shows that we have the following results. 2.1 Suppose that 1 ≤ i ≤ n − 1 and 1 ≤ m ≤ n. Then

(i) L_{i} and L_{m} commute.

(ii) T_{i} and L_{m} commute if i ≠ m − 1, m. (iii) T_{i} commutes with L_{i}L_{i+1} and L_{i} + L_{i+1}. (iv) If a ∈ R and i ≠ m then T_{i} commutes with (L_{1} − a)(L_{2} − a) … (L_{m} − a). Using the elements T_{w} and L_{m} defined above, Ariki and Koike proved the fol-

lowing result which gives a basis for ℋ . 2.2 (Ariki–Koike [4, (3.10)]) The algebra ℋ is free as an R–module with basis

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

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

We call this basis the Ariki–Koike basis of ℋ . Let ℋ (𝔖_{n}) be the subalgebra of ℋ generated by T_{1}, …, T_{n−1}. It follows from (2.2) that ℋ (𝔖_{n}) is isomorphic to the Iwahori–Hecke algebra of 𝔖_{n} and is free as an R–module with basis { T_{w} | w ∈ 𝔖_{n} }.

Let τ : ℋ ⟶R be the R–linear map determined by

τ(L_{1}^{c_{1}}L_{2}^{c_{2}} … L_{n}^{c_{n}}T_{w}) = \begin{cases}1, & if c_{1} = ⋯ = c_{n} = 0 and w = 1, \\ 0, & otherwise, \end{cases}

where w ∈ 𝔖_{n} and 0 ≤ c_{i} < r for i = 1, …, n.

The function τ was introduced by Bremke and Malle [5] who showed that τ is a trace form and that τ is essentially independent on the choice of basis of ℋ . It is not obvious from the definition above that τ coincides with the form introduced by Bremke and Malle; however, this was proved by Malle and the author in [21] where we also showed that τ is non–degenerate whenever Q_{1}, …, Q_{r} are invertible in R.

For future reference we note the following two important properties of τ; the first is Bremke and Malle’s result that τ is a trace form and the second follows easily from the definition and well–known properties of the trace form τ in the case r = 1 (see, for example, [22, Prop. 1.16]). 2.3 (i) Suppose h_{1}, h_{2} ∈ ℋ . Then τ(h_{1}h_{2}) = (h_{2}h_{1}).

(ii) Suppose that x, y ∈ 𝔖_{n} and that 0 ≤ c_{i} < r for i = 1, …, n. Then

τ(L_{1}^{c_{1}}L_{2}^{c_{2}} … L_{n}^{c_{n}}T_{x}T_{y}) = \begin{cases}q^{ℓ(x)}, & if c_{1} = ⋯ = c_{n} = 0 and x = y^{−1}, \\ 0, & otherwise. \end{cases}

In this paper we will mainly be concerned with the semisimple Ariki–Koike algebras; these were classified by Ariki [1] who showed that when R is a field ℋ is semisimple if and only if

(2.4)  P_{ℋ} (q) = ∏_{i=1}^{n}(1 + q + ⋯ + q^{i−1}) · ∏_{1≤i<j≤r} ∏_{−n<d<n}(q^{d}Q_{i} − Q_{j})

is a non–zero element of R. For most of what we do it will be enough to assume that R is a ring in which P_{ℋ} (q) is invertible.

A multipartition of n is a ordered r–tuple λ = (λ^{(1)}, …, λ^{(r)}) of partitions λ^{(s)} such that n = ∑_{s=1}^{r} |λ^{(s)}|; we write λ ⊢ n. In the semisimple case Ariki and Koike [4] constructed an irreducible ℋ –module S^{λ}, called a Specht module, for each multipartition λ of n. Further, they showed that { S^{λ} | λ ⊢ n } is a complete set of pairwise non–isomorphic irreducible ℋ –modules. Let χ^{λ} be the character of S^{λ}.

Assume that ℋ is semisimple. Then τ can be written as a linear combination of the irreducible characters of ℋ because τ is a trace form.

2.5. Definition. Suppose that R is a field and that P_{ℋ} (q) ≠ 0. The Schur elements of ℋ are the elements s_{λ}(q) ∈ R such that

τ = ∑_{λ} \frac{1}{s_{λ}(q)}χ^{λ},

where λ runs over the multipartitions of n.

The rational functions \frac{1}{s_{λ}(q)} are also called the weights of ℋ .

The generic degrees of W_{r,n} are certain “spetsial” specializations of the rational functions s_{η}(q)/s_{λ}(q), where η = ((n), (0), …, (0)) — so s_{η}(q) is the Schur element corresponding to the trivial representation of ℋ (s_{η}(q) = s_{η_{1}}(q) is computed in Example 2.7). The spetsial specializations of the rational functions s_{η}(q)/s_{λ}(q) are polynomials in q with rational coefficients; moreover, for these specializations s_{η}(q) is equal to the Poincaré polynomial of the coinvariant algebra of the reflection representation of W_{r,n}. These results are due to Malle and can be found in [19,20].

In the special case when r = 1, 2 the group W_{r,n} is a Weyl group (rather than just a complex reflection group), and here the generic degrees were first computed by Hoefsmit [16]; they can be found, for example, in [8].

One of the motivations for writing this paper was to compute the Schur elements and hence the generic degrees. Ostensibly the Schur elements depend in a non– uniform way upon the choice of q, Q_{1}, …, Q_{r}; however, we shall see that in fact they can be expressed as rational functions in q, Q_{1}, …, Q_{r} which depend only on λ. The expression we obtain is a generalization of the hook length formula of Frame, Robinson and Thrall [14] for the dimensions of the irreducible representations of the symmetric groups. This is not unexpected because ℋ ≅ RW_{r,n} when R = ℂ, q = 1 and Q_{s} = ζ^{s}, so the orthogonality relations for the characters of W_{r,n} imply that χ^{λ}(1) = |W_{r,n}|/s_{λ}(1; ζ^{s}). As the referee remarked, it is worth noting that in general P_{ℋ} (1; ζ^{s}) is only a scalar multiple of |W_{r,n}| = r^{n}n! since ∏_{1≤i<j≤r}(ζ^{i} −

ζ^{j}) = (−1)^{\frac{1}{2}}^{\binom{r−1}{2}}e^{\frac{e}{2}} by [19, 2.22].

We will compute the Schur elements of ℋ by explicitly constructing a set of primitive idempotents in ℋ and then applying the following Lemma (which is really a well–known fact about symmetric algebras).

2.6. Lemma. Assume that R is a field and that ℋ is semisimple. Let λ be a multipartition of n and suppose that e_{λ} is a primitive idempotent in ℋ such that S^{λ} ≅ e_{λ}ℋ . Then s_{λ}(q) = \frac{1}{τ(e_{λ})}.

Proof. Suppose first that R is a field of characteristic zero. Let E_{λ} be the primitive central idempotent corresponding to the irreducible module S^{λ}. By definition τ = ∑_{μ} \frac{1}{s_{μ}(q)}χ^{μ}, where μ runs over the multipartitions of n; therefore,

τ(E_{λ}) = ∑_{μ⊢n} \frac{1}{s_{μ}(q)}χ^{μ}(E_{λ}) = \frac{1}{s_{λ}(q)}χ^{λ}(E_{λ}) = \frac{1}{s_{λ}(q)}χ^{λ}(1).

Now E_{λ} = e_{1}+⋯+e_{N}, where e_{1}, …, e_{N} are primitive idempotents with e_{i}ℋ ≅ S^{λ}, for all i, and N = dim S^{λ} = χ^{λ}(1). The idempotents e_{1}, …, e_{N} belong to the same Wedderburn component of ℋ , so there exist invertible elements u_{i} ∈ ℋ such that e_{i} = u_{i}e_{1}u_{i}^{−1} for all i. Consequently, τ(e_{a}) = τ(u_{i}e_{1}u_{i}^{−1}) = τ(e_{1}) since τ is a trace form; hence, τ(E_{λ}) = χ^{λ}(1)τ(e_{1}). Without loss, e_{1} = e_{λ} so the Lemma follows.

The case where R is a field of positive characteristic now follows by a specialization argument (using, for example, Theorem 3.14) which we leave to the reader. □

2.7. Example. Fix t with 1 ≤ t ≤ r and let η_{t} = (η_{t}^{(1)}, …, η_{t}^{(r)}) be the multipartition of n with η_{t}^{(s)} = (n) if s = t and η_{t}^{(s)} = (0) otherwise. We will compute the Schur elements s_{η_{t}}(q). Let x_{η_{t}} = ∑_{w∈𝔖_{n}} T_{w} and u_{η_{t}} = ∏_{s≠t} ∏_{k=1}^{n}(L_{k} − Q_{s}) — in the product, 1 ≤ s ≤ r — and set m_{η_{t}} = u_{η_{t}}x_{η_{t}} = x_{η_{t}}u_{η_{t}} (cf. (3.1)). It follows from (2.1) that u_{η_{t}} is central in ℋ . Further, the relations imply that T_{0}u_{η_{t}} = Q_{t}u_{η_{t}} and T_{w}x_{η_{t}} = q^{ℓ(w)}x_{η_{t}} for w ∈ 𝔖_{n}; it follows that L_{k}m_{η_{t}} = q^{k−1}Q_{t}m_{η_{t}} for k = 1, …, n. Therefore, the module m_{η_{t}}ℋ is one dimensional and, in particular, irreducible; in fact, S^{η_{t}} ≅ m_{η_{t}}ℋ = Rm_{η_{t}} (for example, use (3.2)). Moreover, by what we have said

m_{η_{t}}^{2} = [n]_{q}^{!} ∏_{s≠t}^{r} ∏_{k=1}^{n}(q^{k−1}Q_{t} − Q_{s}) · m_{η_{t}},

where [n]_{q}^{!} = ∏_{k=1}^{n}(1 + q + ⋯ + q^{k−1}); so m_{η_{t}} is a scalar multiple of the primitive idempotent which generates S^{η_{t}}. Hence, by the Lemma,

s_{η_{t}}(q) = (τ(m_{η_{t}}))^{−1}[n]_{q}^{!} ∏_{s≠t} ∏_{k=0}^{n−1}(q^{k}Q_{t} − Q_{s})

= (−1)^{n(r−1)}[n]_{q}^{!} ∏_{s≠t} Q_{s}^{−n} · ∏_{s≠t} ∏_{k=0}^{n−1}(q^{k}Q_{t} − Q_{s}).

Similar arguments give the Schur elements for the multipartition which is conjugate to η_{t}; alternatively, they are given by Corollary 4.5 and the calculation above.

There is an action of 𝔖_{r} on the set of multipartitions of n (by permuting components) and also on the rational functions in Q_{1}, …, Q_{r} (by permuting parameters). When ℋ is semisimple the Specht modules are determined up to isomorphism by the action of L_{1}, …, L_{n}; as the relation ∏_{s=1}^{r}(T_{0} − Q_{s}) = 0 is invariant under the 𝔖_{r}–action it follows that s_{v·λ}(q) = v ·s_{λ}(q) for all multipartitions λ and all v ∈ 𝔖_{r}; this is also clear from Theorem 3.14(i). In the case where λ = η_{t} this symmetry is evident in the formulae above.

3. An orthogonal basis for ℋ

If R is a field and P_{ℋ} (q) is non–zero then ℋ is a split semisimple algebra; hence, ℋ has a basis which corresponds to the matrix units in its Wedderburn decomposition. In this section we explicitly construct a Wedderburn basis for ℋ . We begin by recalling the standard basis of ℋ from [12].

A multipartition of n (with r components) is an r–tuple λ = (λ^{(1)}, …, λ^{(r)}) of partitions such that |λ^{(1)}| + ⋯ + |λ^{(r)}| = n. Let Λ^{+} be the set of multipartitions of n; then Λ^{+} becomes a poset under dominance where λ ⊵ μ if for all 1 ≤ s ≤ r and all i ≥ 1 we have

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

We also write λ ▷ μ if λ ⊵ μ and λ ≠ μ.

The diagram of a multipartition λ is the set

[λ] = { (i, j, c) | 1 ≤ j ≤ λ_{i}^{(c)} and 1 ≤ c ≤ r } .

We will think of [λ] as being the r–tuple of diagrams of the partitions λ^{(c)}, for 1 ≤ c ≤ r. A λ–tableau is a bijection 𝔱: [λ] ⟶ {1, 2, …, n}. If 𝔱 is a λ–tableau write Shape(𝔱) = λ. As with diagrams, we will think of a tableau 𝔱 as an r–tuple of tableaux 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}), where 𝔱^{(c)} is a λ^{(c)}–tableau. The tableaux 𝔱^{(c)} are called the components of 𝔱. A tableau is standard if in each component the entries increase along the rows and down the columns; let Std(λ) be the set of standard λ–tableaux.

We identity a tableau 𝔱 with an r–tuple of labelled diagrams; for example

( 1 2 3 4 , 5 6 7 , 8 9 )  and  ( 1 3 6 2 , 4 9 7 , 5 8 )

are two ((3, 1), (2, 1), (1^{2}))–tableaux. Both of these tableaux are standard.

Given a multipartition λ let 𝔱^{λ} be the λ–tableau with the numbers 1, 2, …, n entered in order first along the rows of 𝔱^{λ^{(1)}} and then the rows of 𝔱^{λ^{(2)}} and so on. For example, the first of the ((3, 1), (2, 1), (1^{2}))–tableaux above is 𝔱^{((3,1),(2,1),(1^{2}))}.

The symmetric group 𝔖_{n} acts from the right on the set of λ–tableaux; let 𝔖_{λ} = 𝔖_{λ^{(1)}} × ⋯ × 𝔖_{λ^{(r)}} be the row stabilizer of 𝔱^{λ}. For any λ–tableau 𝔱 let d(𝔱) be the unique element of 𝔖_{n} such that 𝔱 = 𝔱^{λ}d(𝔱).

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

We can now define the cellular basis of ℋ constructed in [12]. Fix a multipartition λ and let 𝔞 = (𝔞_{1}, …, 𝔞_{r}), where 𝔞_{s} = |λ^{(1)}| + ⋯ + |λ^{(s−1)}| for 1 ≤ s ≤ r. Define m_{λ} = x_{λ}u_{λ}^{+}, where

x_{λ} = ∑_{w∈𝔖_{λ}} T_{w}  and  u_{λ}^{+} = ∏_{s=2}^{r} ∏_{k=1}^{𝔞_{s}}(L_{k} − Q_{s}).

It follows from (2.1) that x_{λ} and u_{λ}^{+} commute. Finally, given standard λ–tableaux 𝔰 and 𝔱 let m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}.

Whenever we write m_{𝔰𝔱} in what follows 𝔰 and 𝔱 will be standard tableaux of the same shape (and similarly, for f_{𝔰𝔱} etc.). 3.1 (Dipper–James–Mathas [12, Theorem 3.26]) The Ariki–Koike algebra ℋ is free as an R–module with cellular basis { m_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} }.

One consequence of this result is that the R–module ℋ̌ ^{λ} which has as basis the set of m_{𝔲𝔳} where Shape(𝔲) = Shape(𝔳) ▷ λ is a two–sided ideal of ℋ . It follows from (3.1) that m_{μ}ℋ ⊂ ℋ̌ ^{λ} whenever μ ▷ λ. The Specht module S^{λ} is the right ℋ –module m_{λ}ℋ /(m_{λ}ℋ ∩ ℋ̌ ^{λ} ), a submodule of ℋ /ℋ̌ ^{λ} . Thus, S^{λ} is free as an R–module with basis { m_{𝔱} | 𝔱 ∈ Std(λ) }, where m_{𝔱} = m_{𝔱^{λ}𝔱} + ℋ̌ ^{λ} . Further, there is a natural associative bilinear form ⟨ , ⟩ on S^{λ} which is determined by

⟨m_{𝔰}, m_{𝔱}⟩m_{λ} ≡ m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} mod ℋ̌ ^{λ} .

Before we can begin we need some further notation and a result from [17]. If 𝔱 is any tableau and k ≥ 0 is an integer let 𝔱↓k be the subtableau of 𝔱 which contains the integers 1, 2, …, k. Observe that 𝔱 is standard if and only if Shape(𝔱↓k) is a multipartition for all k. We extend the dominance order to the set of standard tableaux by defining 𝔰 ⊵ 𝔱 if Shape(𝔰↓k) ⊵ Shape(𝔱↓k) for k = 1, 2, …, n; again, we write 𝔰 ▷ 𝔱 if 𝔰 ⊵ 𝔱 and 𝔰 ≠ 𝔱.

Write res_{𝔱}(k) = q^{j−i}Q_{c} if k appears in row i and column j of 𝔱^{(c)}; then res_{𝔱}(k) is the residue of k in 𝔱. We can now state the result we need from [17]. 3.2 [17, Prop. 3.7] Let 𝔰 and 𝔱 be standard λ–tableaux and suppose that k is an integer with 1 ≤ k ≤ n. Then there exist a_{𝔳} ∈ R such that

m_{𝔰𝔱}L_{k} = res_{𝔱}(k)m_{𝔰𝔱} + ∑_{𝔳▷𝔱^{v∈Std(λ)}} a_{𝔳}m_{𝔰𝔳} mod ℋ̌ ^{λ} .

From our current point of view the importance of this result derives from the observation that m_{𝔰𝔱}(L_{k} − res_{𝔲}(k))/( res_{𝔱}(k) − res_{𝔲}(k)) = m_{𝔰𝔱} plus a linear combination of more dominant terms, providing that res_{𝔲}(k) ≠ res_{𝔱}(k); this motivates the next definition.

Let R(k) = { q^{d}Q_{s} | 1 ≤ s ≤ r, |d| < k and d ≠ 0 if r = 1 and k = 2, 3 }; be the set of possible residues res_{𝔱}(k) as 𝔱 runs over the standard tableaux.

3.3. Definition (cf. [17, Defn. 3.11]). Suppose that 𝔰 and 𝔱 are standard λ–tableaux.

(i) Let F_{𝔱} = ∏_{k=1}^{n} ∏_{\begin{matrix}c∈R(k) \\ c≠res_{𝔱}(k) \end{matrix}} \frac{L_{k} − c}{res_{𝔱}(k) − c}.

(ii) Let f_{𝔰𝔱} = F_{𝔰}m_{𝔰𝔱}F_{𝔱}.

There are some remarks worth making about the definition of F_{𝔱}. First, we do not need to specify an order for the product in the definition of F_{𝔱} since the L_{k} generate a commutative subalgebra of ℋ . Secondly, the definition of F_{𝔱} is very conservative in the sense that many of the factors of F_{𝔱} can be omitted without changing the element f_{𝔰𝔱}. Finally, historically this construction has been used to produce an orthogonal basis for the Specht modules; we are going to modify this procedure to give an orthogonal basis for the whole of ℋ .

Let 𝔰 an 𝔱 be two standard tableaux, not necessarily of the same shape. The proof of next result rests upon the easy fact [17, Lemma 3.12] that, because P_{ℋ} (q) is invertible, 𝔰 = 𝔱 if and only if res_{𝔰}(k) = res_{𝔱}(k) for k = 1, …, n.

3.4. Proposition. Suppose that P_{ℋ} is invertible in R and that 𝔰 and 𝔱 are standard λ–tableau and that k is an integer with 1 ≤ k ≤ n. Then

(i) f_{𝔰𝔱} = m_{𝔰𝔱} + ∑_{(𝔲,𝔳)▷(𝔰,𝔱)} a_{𝔲𝔳}m_{𝔲𝔳} for some a_{𝔲𝔳} ∈ R;

(ii) f_{𝔰𝔱}L_{k} = res_{𝔱}(k)f_{𝔰𝔱}; (iii) f_{𝔰𝔱}F_{𝔲} = δ_{𝔱𝔲}f_{𝔰𝔱}; and, (iv) F_{𝔲}f_{𝔰𝔱} = δ_{𝔰𝔲}f_{𝔰𝔱}.

Proof. Given (3.2), this is a variation on a well known argument; see, for example, [22, Prop. 3.35]. The proof in [22] can be copied out verbatim except that N should be replaced by ∑_{k=1}^{n} |R(k)|. □

In particular, part (i) together with (3.1) shows that

{ f_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ⊢ n }

is a basis of ℋ ; shortly we will see that it is an orthogonal basis of ℋ with respect to the trace form τ. As a first step we describe the action of ℋ (𝔖_{n}) on this basis; note that the action of the L_{k} (and, in particular, T_{0} = L_{1}), on this basis is given by Proposition 3.4(ii).

3.5. Proposition. Suppose that 𝔱 = 𝔰(i, i+1) where 𝔰 and 𝔲 are standard λ–tableaux and i is an integer with 1 ≤ i < n. If 𝔱 is standard then

f_{𝔲𝔰}T_{i} = \begin{cases}\frac{(q−1) res_{𝔱}(i)}{res_{𝔱}(i)−res_{𝔰}(i)}f_{𝔲𝔰} + f_{𝔲𝔱}, & if 𝔰 ▷ 𝔱, \\ \frac{(q−1) res_{𝔱}(i)}{res_{𝔱}(i)−res_{𝔰}(i)}f_{𝔲𝔰} + \frac{(q res_{𝔰}(i)−res_{𝔱}(i))(res_{𝔰}(i)−q res_{𝔱}(i))}{(res_{𝔱}(i)−res_{𝔰}(i))^{2}}f_{𝔲𝔱}, & if 𝔱 ▷ 𝔰. \end{cases}

If 𝔱 is not standard then

f_{𝔲𝔰}T_{i} = \begin{cases}qf_{𝔲𝔰}, & if i and i + 1 are in the same row of 𝔰, \\ −f_{𝔲𝔰}, & if i and i + 1 are in the same column of 𝔰. \end{cases}

Proof. By the above remarks, {f_{𝔲𝔰}} is a basis of ℋ , so f_{𝔲𝔰}T_{i} = ∑_{𝔞,𝔟} r_{𝔞𝔟}f_{𝔞𝔟} for some r_{𝔞𝔟} ∈ R. By Proposition 3.4(iv), F_{𝔲}f_{𝔞𝔟} = δ_{𝔞𝔲}f_{𝔲𝔟}. Therefore, multiplying the equation for f_{𝔲𝔰}T_{i} on the left by F_{𝔲} shows that r_{𝔞𝔟} = 0 whenever 𝔞 ≠ 𝔲; in particular, r_{𝔞𝔟} = 0 if Shape(𝔟) ≠ λ. Hence, f_{𝔲𝔰}T_{i} = ∑_{𝔟} a_{𝔟}f_{𝔲𝔟}, for some a_{𝔟} ∈ R, where 𝔟 runs over the set of standard λ–tableaux. The argument of [22, Theorem 3.36] can now be repeated, essentially word for word, to complete the proof. □

For each standard λ–tableau 𝔰 let f_{𝔰} = f_{𝔱^{λ}𝔰} + ℋ̌ ^{λ} . Then { f_{𝔰} | 𝔰 ∈ Std(λ) } is a basis of the Specht module S^{λ} by Proposition 3.4(i). Note that f_{𝔱^{λ}} = m_{𝔱^{λ}}; further, by Proposition 3.5, if 𝔰 and 𝔱 are standard λ–tableaux with 𝔰 = 𝔱(i, i + 1) ▷ 𝔱 then f_{𝔱} = f_{𝔰}(T_{i} − α), where α = (q − 1) res_{𝔱}(i)/( res_{𝔱}(i) − res_{𝔰}(i)); hence, by [17, Cor. 3.14], { f_{𝔰} | 𝔰 ∈ Std(λ) } is the orthogonal basis of S^{λ} which was constructed in [17, Theorem 3.13(vi)]. (In the notation of [17], f_{𝔰} = ψ_{𝔰}(m_{λ}).)

The inner products ⟨f_{𝔰}, f_{𝔱}⟩, for 𝔰, 𝔱 ∈ Std(λ) were computed explicitly (as rational functions) in [17]; to describe this we need some more notation. Recall that a node y ∉ [λ] is an addable node for λ if [λ] ∪ {y} is the diagram of a multipartition; similarly, y ∈ [λ] is a removable node for λ if [λ] \ {y} is the diagram of a multipartition. Given two nodes x = (i, j, k) and y = (a, b, c), write y < x if either c < k, or c = k and b > j.

Let 𝔰 be a λ–tableau. Then for each integer i there is a unique node x ∈ [λ] such that 𝔰(x) = i, 1 ≤ i ≤ n. Let 𝒜_{𝔰}(i) be the set of addable nodes for Shape(𝔰↓i) which are strictly less than x (with respect to <); similarly, let ℛ_{𝔰}(i) be the set of removable nodes strictly less than x for the multipartition Shape(𝔰↓i−1). Essentially as in [17, Defn. 3.15] let

(3.6)  γ_{𝔰} = q^{ℓ(d(𝔰))+α(λ)} ∏_{i=1}^{n} \frac{∏_{y∈𝒜_{𝔰}(i)} ( res_{𝔰}(i) − res(y))}{∏_{y∈ℛ_{𝔰}(i)} ( res_{𝔰}(i) − res(y))}

where α(λ) = \frac{1}{2} ∑_{s=1}^{r} ∑_{i≥1}(λ_{i}^{(s)} − 1)λ_{i}^{(s)}.

If k is an integer let [k]_{q} = \frac{q^{k}−1}{q−1} if q ≠ 1; more generally, set [k]_{q} = 1+q+⋯+q^{k−1} if k ≥ 0 and [k]_{q} = −q^{k}[−k]_{q} if k < 0. When k ≥ 0 we also set [k]_{q}^{!} = [1]_{q}[2]_{q} … [k]_{q}. Finally, if λ is a multipartition let [λ]_{q}^{!} = ∏_{s=1}^{r} ∏_{i≥1}[λ_{i}^{(s)}]_{q}^{!}. 3.7 (James–Mathas [17, (3.17)–(3.19)]) Suppose that P_{ℋ} (q) is invertible in R and let 𝔱 ∈ Std(λ). Then γ_{𝔱} is uniquely determined by the two conditions

(i) γ_{𝔱^{λ}} = [λ]_{q}^{!} ∏_{1≤s<t≤r} ∏_{(i,j)∈[λ^{(s)}]}(q^{j−i}Q_{s} − Q_{t}); and,

(ii) if 𝔰 = 𝔱(i, i + 1) ▷ 𝔱 then γ_{𝔱} = \frac{(q res_{𝔰}(i)−res_{𝔱}(i))(res_{𝔰}(i)−q res_{𝔱}(i))}{(res_{𝔰}(i)−res_{𝔱}(i))^{2}}γ_{𝔰}.

Furthermore, ⟨f_{𝔰}, f_{𝔱}⟩ = δ_{𝔰𝔱}γ_{𝔱} for all 𝔰, 𝔱 ∈ Std(λ).

3.8. Remarks. (i) In [17] the rational functions γ_{𝔱} were important only up to a power of q (which is a unit in R). In this paper we need to know the inner products ⟨f_{𝔰}, f_{𝔱}⟩ exactly; for this reason our definition of γ_{𝔰} differs from that in [17] by the factor q^{ℓ(d(𝔰))+α(λ)}. The argument in [17] computes ⟨f_{𝔰}, f_{𝔱}⟩ using only (3.7)(i) and (3.7)(ii); the inner product ⟨f_{𝔱^{λ}}, f_{𝔱^{λ}}⟩ is easily seen to be given by the formula on the right hand side of (3.7)(i). (ii) A factor of q was omitted from the formula for γ_{𝔱} in [17, Lemma 3.17)];

this is corrected in (3.7)(ii) — note that ℓ(d(𝔱)) = ℓ(d(𝔰)) + 1. (iii) The definition of γ_{𝔰} above is simpler than that given in [17] because 𝔰 is a standard tableau (rather than the more general semistandard tableau which were considered in [17] ).

Let ( , ) be the inner product on ℋ given by (h_{1}, h_{2}) = τ(h_{1}h_{2}^{∗}), for h_{1}, h_{2} ∈ ℋ . Then ( , ) is a symmetric associative bilinear form on ℋ .

3.9. Theorem. Suppose that P_{ℋ} (q) is invertible in R. Then

{ f_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} }

is a orthogonal basis of ℋ with respect to the trace form τ. In addition, if 𝔰, 𝔱, 𝔲 and 𝔳 are standard tableaux then f_{𝔰𝔱}f_{𝔲𝔳} = δ_{𝔲𝔱}γ_{𝔱}f_{𝔰𝔳}.

Proof. By (3.1) the set {m_{𝔰𝔱}} is a basis of ℋ ; therefore, {f_{𝔰𝔱}} is also a basis of ℋ by Proposition 3.4(i). Next we prove that f_{𝔰𝔱}f_{𝔲𝔳} = δ_{𝔲𝔱}γ_{𝔱}f_{𝔰𝔳}. First, if 𝔲 ≠ 𝔱 then f_{𝔰𝔱}f_{𝔲𝔳} = f_{𝔰𝔱}F_{𝔲}m_{𝔲𝔳}F_{𝔳} = 0 by Proposition 3.4(iii). Now consider f_{𝔰𝔱}f_{𝔱𝔳}; since {f_{𝔞𝔟}} is a basis we can write f_{𝔰𝔱}f_{𝔱𝔳} = ∑_{𝔞,𝔟} r_{𝔞𝔟}^{𝔱}f_{𝔞𝔟} for some r_{𝔞𝔟}^{𝔱} ∈ R. Applying Proposition 3.4(iii) and its left handed analogue shows that

f_{𝔰𝔱}f_{𝔱𝔳} = F_{𝔰}f_{𝔰𝔱}f_{𝔱𝔳}F_{𝔳} = ∑_{𝔞,𝔟} r_{𝔞𝔟}^{𝔱}F_{𝔰}f_{𝔞𝔟}F_{𝔳} = r_{𝔰𝔳}^{𝔱}f_{𝔰𝔳}.

Now, for any 𝔰, 𝔳 ∈ Std(λ) the definition of the inner product on the Specht module gives

⟨f_{𝔱}, f_{𝔱}⟩f_{𝔰𝔳} ≡ f_{𝔰𝔱}f_{𝔱𝔳} mod ℋ̌ ^{λ} .

Therefore, r_{𝔰𝔳}^{𝔱} = ⟨f_{𝔱}, f_{𝔱}⟩ = γ_{𝔱} by (3.7); so r_{𝔰𝔳}^{𝔱} depends only on 𝔱 and f_{𝔰𝔱}f_{𝔱𝔳} = γ_{𝔱}f_{𝔰𝔳} as claimed.

Finally, it remains to show that the basis {f_{𝔰𝔱}} is orthogonal with respect to the bilinear form ( , ). First, (f_{𝔰𝔱}, f_{𝔲𝔳}) = τ(f_{𝔰𝔱}f_{𝔲𝔳}^{∗}) = τ(f_{𝔰𝔱}f_{𝔳𝔲}) = δ_{𝔱𝔳}γ_{𝔱}τ(f_{𝔰𝔲}); in particular, (f_{𝔰𝔱}, f_{𝔲𝔳}) = 0 if 𝔱 ≠ 𝔳. On the other hand, τ is a trace form so (f_{𝔰𝔱}, f_{𝔲𝔳}) = τ(f_{𝔰𝔱}f_{𝔳𝔲}) = τ(f_{𝔳𝔲}f_{𝔰𝔱}) = δ_{𝔰𝔲}γ_{𝔰}τ(f_{𝔳𝔱}). Therefore, (f_{𝔰𝔱}, f_{𝔲𝔳}) = δ_{𝔰𝔲}δ_{𝔱𝔳}γ_{𝔱}τ(f_{𝔰𝔰}) = δ_{𝔰𝔲}δ_{𝔱𝔳}γ_{𝔰}τ(f_{𝔱𝔱}). Consequently, {f_{𝔰𝔱}} is an orthogonal basis of ℋ (and τ(f_{𝔱𝔱}) is non–zero for all 𝔱). □

The basis {f_{𝔰𝔱}} is cellular (with respect to the involution ∗); but this is not surprising as is was constructed from a cellular basis.

3.10. Remark. In [21] it was shown that if ℋ is defined over a ring R in which the parameters q, Q_{1}, …, Q_{r} are invertible then ℋ is a symmetric algebra with respect to the trace form τ; however, this was proved indirectly without constructing a pair of dual bases. The Theorem gives a self–dual basis of the semisimple Ariki–Koike algebras; no such basis is known in general.

As a first consequence, Theorem 3.9 identifies a submodule of ℋ which is isomorphic to the Specht module S^{λ}.

3.11. Corollary. Suppose that P_{ℋ} (q) is invertible in R and let 𝔰 and 𝔱 be standard λ–tableaux. Then S^{λ} ≅ f_{𝔰𝔱}ℋ = ∑_{𝔳∈Std(λ)} Rf_{𝔰𝔳}.

Proof. By the Theorem, f_{𝔰𝔱}ℋ has as basis the set { f_{𝔰𝔳} | 𝔳 ∈ Std(λ) }. The isomorphism is given by the linear map S^{λ} ⟶ f_{𝔰𝔱}ℋ determined by f_{𝔳} ⟼f_{𝔰𝔳} for all tableaux 𝔳 ∈ Std(λ). □

Set f̃_{𝔰𝔱} = γ_{𝔱}^{−1}f_{𝔰𝔱}. Then f̃_{𝔰𝔱}f̃_{𝔲𝔳} = δ_{𝔱𝔲}f̃_{𝔰𝔳} and {f̃_{𝔰𝔱}} is a basis of ℋ . Hence, we have the following.

3.12. Corollary. Suppose that P_{ℋ} (q) is invertible in R. Then

{ f̃_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} }

is a basis of matrix units in ℋ .

The last result yields an explicit isomorphism from ℋ to the group ring of W_{r,n} when P_{ℋ} (q) is invertible. Assume that R contains a primitive rth root of unity ζ; then, ℋ ≅ RW_{r,n} when q = 1 and Q_{s} = ζ^{s} for s = 1, 2, …, r. Write f̃_{𝔰𝔱}^{1} for the element of RW_{r,n} corresponding to f̃_{𝔰𝔱} ∈ ℋ under the canonical isomorphism ℋ ⟶ RW_{r,n}.

3.13. Corollary. Assume that R contains a primitive rth root of unity and that P_{ℋ} (q) is invertible in R. Then ℋ ≅ RW_{r,n} via the R–algebra homomorphism determined by f̃_{𝔰𝔱} ⟼f̃_{𝔰𝔱}^{1}.

By parts (i) and (iii) of Theorem 3.14 below, T_{i} = ∑_{𝔱} f̃_{𝔱𝔱}T_{i}, for 0 ≤ i < n; so, in principle, we can determine the image of the generators of ℋ under this isomorphism.

Lusztig [18] has shown that there exists a homomorphism Φ from the Hecke algebra ℋ (W) of any finite Weyl group W to the group ring RW and he shows that Φ induces an isomorphism when ℋ (W) is semisimple. Our map is not an analogue of Lusztig’s isomorphism; rather it is an explicit realization of the Tits deformation theorem in this setting. It would be good to find a generalization of Lusztig’s isomorphism theorem for the Ariki–Koike algebras.

We next construct the primitive (central) idempotents in ℋ .

3.14. Theorem. Suppose that R is a field and that P_{ℋ} (q) ≠ 0.

(i) Let 𝔱 be a standard λ–tableau. Then F_{𝔱} = \frac{1}{γ_{𝔱}} f_{𝔱𝔱} and F_{𝔱} is a primitive idempotent with S^{λ} ≅ F_{𝔱}ℋ .

(ii) For any multipartition λ let F_{λ} = ∑_{𝔱∈Std(λ)} F_{𝔱}. Then F_{λ} is a primitive

central idempotent. (iii) { F_{λ} | λ ⊢ n } is a complete set of primitive central idempotents; in partic-

ular,

1 = ∑_{λ⊢n} F_{λ} = ∑_{𝔱 standard} F_{𝔱}.

Proof. We may write F_{𝔱} = ∑_{𝔲,𝔳} a_{𝔲𝔳}f_{𝔲𝔳} for some a_{𝔲𝔳} ∈ R by Theorem 3.9. By Proposition 3.4(iii) and Theorem 3.9, f_{𝔰𝔱} = f_{𝔰𝔱}F_{𝔱} = ∑_{𝔲,𝔳} a_{𝔲𝔳}f_{𝔰𝔱}f_{𝔲𝔳} = ∑_{𝔳} γ_{𝔱}a_{𝔱𝔳}f_{𝔰𝔳}; equating coefficients on both sides shows that a_{𝔱𝔳} = 0 if 𝔳 ≠ 𝔱 and that 1 = a_{𝔱𝔱}γ_{t}. Since F_{𝔱}^{∗} = F_{t} we also have that a_{𝔳𝔱} = 0 if 𝔳 ≠ 𝔱. Hence, F_{𝔱} = \frac{1}{γ_{𝔱}} f_{𝔱𝔱} as claimed. By Theorem 3.9, \frac{1}{γ_{𝔱}} f_{𝔱𝔱} is idempotent; further, it is primitive because S^{λ} is irreducible and S^{λ} ≅ \frac{1}{γ_{𝔱}} f_{𝔱𝔱}ℋ by Corollary 3.11. Hence, (i) is proved.

Parts (ii) and (iii) now follow because ℋ = ⨁_{λ⊢n} ⨁_{𝔱∈Std(λ)} F_{𝔱}ℋ is a decomposition of ℋ into a direct sum of simple modules with each simple module F_{𝔱}ℋ ≅ S^{λ} appearing with multiplicity equal to its dimension (the sum is direct because {f_{𝔱𝔳}} is a basis of F_{𝔱}ℋ by Theorem 3.9 and the set of all f_{𝔲𝔳} is a basis of ℋ ). □

3.15. Corollary. Let 𝔱 be a standard tableau and let k be an integer with 1 ≤ k ≤ n. Then F_{𝔱}L_{k} = res_{𝔱}(k)F_{𝔱} and f_{𝔰𝔱}L_{k} = res_{𝔱}(k)f_{𝔰𝔱}.

Proof. Since F_{𝔱} = \frac{1}{γ_{𝔱}} f_{𝔱𝔱} by part (i), the formula for F_{𝔱}L_{k} follows from Proposition 3.4(ii); this also implies the second statement because f_{𝔰𝔱}L_{k} = f_{𝔰𝔱}F_{𝔱}L_{k}. □

In particular, the Corollary describes the action of T_{0} = L_{1} on the orthogonal basis {f_{𝔰𝔱}} of ℋ . Proposition 3.5 gives the action of the remaining generators T_{1}, …, T_{n−1} of ℋ on the basis {f_{𝔰𝔱}}.

3.16. Corollary. Suppose that 1 ≤ k ≤ n. Then ∏_{c∈R(k)}(L_{k} − c) = 0 and this is the minimum polynomial for L_{k} acting on ℋ .

Proof. By Theorem 3.14,

∏_{c∈R(k)}(L_{k} − c) = 1 · ∏_{c∈R(k)}(L_{k} − c) = ∑_{𝔱 standard} F_{𝔱} ∏_{c∈R(k)}(L_{k} − c) = 0,

since F_{𝔱}(L_{k} − res_{𝔱}(k)) = 0 by Corollary 3.15. Moreover, if we remove any factor (L_{k} − c) from the product ∏_{c∈R(k)}(L_{k} − c) then what remains is non–zero because it divides F_{𝔱} for some standard tableau 𝔱. Hence, ∏_{c∈R(k)}(L_{k} − c) is the minimum polynomial of L_{k}. □

3.17. Corollary. Suppose that 1 ≤ k ≤ n. Then L_{k} = ∑_{𝔱} res_{𝔱}(k)F_{𝔱}, where the sum is over the set of all standard tableaux (of arbitrary shape).

Proof. Combining part (iii) of Theorem 3.14 with Corollary 3.15 shows that

L_{k} = 1 · L_{k} = ∑_{𝔱 standard} F_{𝔱}L_{k} = ∑_{𝔱 standard} res_{𝔱}(k)F_{𝔱}

as required. □

We close this section with a description of the centre of ℋ in the semisimple case; this result is due to Ariki and Koike [4, Theorem 3.20].

3.18. Theorem (Ariki–Koike). Suppose that R is a field and that P_{ℋ} (q) ≠ 0. (i) The centre of ℋ is the set of symmetric polynomials in L_{1}, L_{2}, …, L_{n}. (ii) Let 𝕃 be the subalgebra of ℋ generated by L_{1}, L_{2}, …, L_{n}. Then 𝕃 is a

maximal abelian subalgebra of ℋ .

Proof. First consider (ii). By definition 𝕃 contains each of the primitive idempotents F_{𝔱}, for an arbitrary standard tableau 𝔱. On the other hand, by Corollary 3.17, 𝕃 is contained in the subalgebra of ℋ generated by the primitive idempotents F_{𝔱}; hence, 𝕃 is the subalgebra of ℋ generated by the idempotents F_{𝔱}. As the primitive idempotents generate a maximal abelian subalgebra of ℋ the result follows.

Now consider (i). By (2.1) every symmetric polynomial in L_{1}, …, L_{n} belongs to the centre of ℋ . Conversely, the centre of ℋ has as basis the set of idempotents F_{λ}, as λ runs over the multipartitions of n. So to prove (i) it is enough to show that each F_{λ} is symmetric in L_{1}, …, L_{n}.

Let R = ∪_{k=1}^{n}R(k) be the set of all possible residues for ℋ . By Corollary 3.15 if 𝔱 is a λ–tableau and 1 ≤ m ≤ n then F_{𝔱}(L_{k} − c) = (res_{𝔱}(k) − c)F_{𝔱}; therefore, multiplying F_{𝔱} by the appropriate extra factors we see that

F_{𝔱} =^{def} ∏_{k=1}^{n} ∏_{c∈R(k)\\{res_{𝔱}(k)\}} \frac{L_{k} − c}{res_{𝔱}(k) − c} = ∏_{k=1}^{n} ∏_{c∈R\\{res_{𝔱}(k)\}} \frac{L_{k} − c}{res_{𝔱}(k) − c}.

Notice that the denominator is equal to f_{λ} = ∏_{k=1}^{n} ∏_{c∈R\{res_{𝔱^{λ}} (k)}}(res_{𝔱^{λ}}(k) − c) and f_{λ} depends only on λ and not directly on 𝔱.

Next, following Murphy [25] say that a λ–tableau is regular if its entries increase from left to right along the nodes in [λ] of constant residue. Now, because P_{ℋ} (q) ≠ 0 two nodes in [λ] have the same residue if and only if they lie on the same diagonal { (i + d, j + d, s) ∈ [λ^{(s)}] | d ≥ 0 } of [λ]; thus, a tableau is regular if and only if its entries increase from left to right along each diagonal. (So, for example, every standard tableau is regular, but not conversely.) Extending the formula above for F_{𝔱}, for each regular tableau 𝔱 define

F_{𝔱} = ∏_{k=1}^{n} ∏_{c∈R\\{res_{𝔱}(k)\}} \frac{L_{k} − c}{res_{𝔱}(k) − c} = \frac{1}{f_{λ}} ∏_{k=1}^{n} ∏_{c∈R\\{res_{𝔱}(k)\}}(L_{k} − c).

Observe that if c ∈ R then L_{k} − c is a factor of F_{𝔱} if and only if c ≠ res_{𝔱}(k); therefore, F_{𝔱} determines res_{𝔱}(k), for k = 1, …, n. As remarked above, the residues on the different diagonals of [λ] are distinct; consequently, a regular tableau 𝔱 is uniquely determined by the sequence of residues ( res_{𝔱}(1), …, res_{𝔱}(n)) and hence by the ‘polynomial’ F_{𝔱}. It follows that if we permute L_{1}, …, L_{n} then F_{𝔱} is mapped to F_{𝔰} where 𝔰 is the regular tableau determined by the corresponding permutation of the residue sequence of 𝔱 — again, 𝔰 is necessarily a λ–tableau because the shape of 𝔰 is determined by the lengths of its diagonals which, in turn, are determined by the multiplicity of each residue in 𝔰 (or 𝔱).

Finally, notice that a regular tableau 𝔱 is not standard if and only if res_{𝔱}(k) ∉ R(k) for some k. Therefore, if 𝔱 is not standard then ∏_{c∈R(k)}(L_{k} − c) is a factor of F_{𝔱} and, consequently, F_{𝔱} = 0 by Corollary 3.16; hence,

F_{λ} = ∑_{𝔱 a standard λ–tableau} F_{𝔱} = ∑_{𝔱 a regular ; λ–tableau} F_{𝔱}.

By the last paragraph the right hand side is a symmetric polynomial in L_{1}, …, L_{n} so the theorem follows. □

For the Iwahori–Hecke algebras of type A (that is, when r = 1), it is conjectured that the centre of ℋ is always the set of symmetric polynomials in L_{1}, …, L_{n}, even when ℋ is not semisimple; see [10,23]. When r > 1 and ℋ is not semisimple there are cases where the centre of ℋ is larger than the set of symmetric polynomials in L_{1}, …, L_{n}; for an example see [3, p. 792].

4. Another construction of the Specht modules

In this section we give another two constructions of the Specht modules. The first is via a second cellular basis of ℋ which is, in a certain sense, dual to the basis described in the previous section. The second construction combines these two approaches to produce submodules of ℋ which are isomorphic to the Specht modules S^{λ}. Some of these results can be found in the work of Du and Rui [13]. In the next section we will use these results to compute the Schur elements of ℋ .

Let Z = ℤ[q̂, q̂^{−1}, Q̂_{1}, …, Q̂_{r}], where q̂, Q̂_{1}, …, Q̂_{r} are indeterminates over ℤ, and let ℋ_{Z} be the Ariki–Koike algebra with parameters q̂, Q̂_{1}, …, Q̂_{r}. Consider the ring R as a Z–module by letting q̂ act on R as multiplication by q and Q̂_{s} by multiplication by Q_{s}, for 1 ≤ s ≤ r. Then ℋ ≅ ℋ_{Z} ⊗_{Z} R, since ℋ is free as an R–module; we say that ℋ is a specialization of ℋ_{Z} and call the map which sends h ∈ ℋ_{Z} to h ⊗ 1 ∈ ℋ the specialization homomorphism.

Let ′: Z ⟶ Z be the ℤ–linear map given by q̂⟼ q̂^{−1} and Q̂_{s} ⟼ Q̂_{r−s+1} for 1 ≤ s ≤ r. Define T′_{0} = T_{0} and T′_{i} = −q̂^{−1}T_{i} for 1 ≤ i < n; using the relations of ℋ_{Z} it is easy to verify that ′ now extends to a ℤ–linear ring involution ′: ℋ_{Z} ⟶ ℋ_{Z} of ℋ_{Z}. Hereafter, we drop the distinction between q̂ and q, and Q̂_{s} and Q_{s}.

Suppose that h ∈ ℋ . Then there exists a (not necessarily unique) h_{Z} ∈ ℋ_{Z} such that h = h_{Z} ⊗ 1 under specialization; we sometimes abuse notation and write h′ = h′_{Z} ⊗ 1 ∈ ℋ . As the map ′ does not in general define a semilinear involution on R, this notation is not well–defined on elements of ℋ ; however, in the cases where we employ it there should be no ambiguity. For example, T′_{w} = (−q)^{−ℓ(w)}T_{w} and L′_{i} = L_{i} for all w ∈ 𝔖_{n} and 1 ≤ i ≤ n.

If λ is a multipartition of n let y_{λ} = ∑_{w∈𝔖_{λ}}(−q)^{−ℓ(w)}T_{w}; then y_{λ} = x′_{λ}. Similarly, we define n_{λ} = y_{λ}u_{λ}^{−} = m′_{λ} where

u_{λ}^{−} = ∏_{s=2}^{r} ∏_{k=1}^{𝔞_{s}}(L_{k} − Q_{r−s+1}) = ∏_{s=1}^{r−1} ∏_{k=1}^{𝔞_{r−s+1}}(L_{k} − Q_{s}).

Observe that u_{λ}^{−} = (u_{λ}^{+})′; here, as usual, 𝔞_{s} = |λ^{(1)}| + ⋯ + |λ^{(s−1)}| for all s. For standard tableaux 𝔰, 𝔱 ∈ Std(λ) set n_{𝔰𝔱} = T_{d(𝔰)}^{∗}′n_{λ}T_{d(𝔱)}′; then m′_{𝔰𝔱} ∈ ℋ_{Z} is mapped to n_{𝔰𝔱} under specialization. Hence, from (3.1) we obtain the following. 4.1 [13, (2.7)] The Ariki–Koike algebra ℋ is free as an R–module with cellular basis { n_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} }.

Let λ be a multipartition of n. Then (ℋ̌ ^{λ} )′ is a two–sided ideal of ℋ which is free as an R–module with basis { n_{𝔲𝔳} | 𝔲, 𝔳 ∈ Std(μ) for some μ ▷ λ }. Let S̃^{λ} be the Specht module (or cell module) corresponding to λ determined by the basis {n_{𝔰𝔱}}; then S̃^{λ} ≅ n_{λ}ℋ /(n_{λ}ℋ ∩ (ℋ̌ ^{λ} )′) and S̃^{λ} is free as an R–module with basis { n_{𝔱} | 𝔱 ∈ Std(λ) }, where n_{𝔱} = n_{𝔱^{λ}𝔱} + (ℋ̌ ^{λ} )′ = (m_{𝔱^{λ}𝔱} + ℋ̌ ^{λ} )′ for all 𝔱 ∈ Std(λ).

In order to compare the two modules S^{λ} and S̃^{λ} we need to introduce some more notation. Given a partition σ let σ′ = (σ′_{1}, σ′_{2}, …) be the partition which is conjugate to σ; thus, σ′_{i} is the number of nodes in column i of the diagram of σ. If λ = (λ^{(1)}, …, λ^{(r)}) is a multipartition then the conjugate λ′ = (λ′^{(1)}, …, λ′^{(r)}) of λ′ is the multipartition with λ′^{(s)} = (λ^{(r−s+1)})′ for 1 ≤ s ≤ r.

Now suppose that 𝔱 = (𝔱^{(1)}, …, 𝔱^{(r)}) is a standard λ–tableau. Then the conjugate of 𝔱 is the standard λ′–tableau 𝔱′ = (𝔱′^{(1)}, …, 𝔱′^{(r)}) where 𝔱′^{(s)} is the tableau obtained by interchanging the rows and columns of 𝔱^{(r−s+1)}.

With these definitions in place, we see that the following holds in Z. 4.2 Let 𝔱 be a standard λ–tableau. Then ( res_{𝔱}(k))′ = res_{𝔱′}(k) in Z, for 1 ≤ k ≤ n.

The expression res_{𝔱′}(k) is always well–defined; whereas (res_{𝔱}(k))′ is ambiguous for certain rings R. As a first consequence we determine how the L_{k} act on the n_{𝔰𝔱}–basis of ℋ .

4.3. Proposition. Let 𝔰 and 𝔱 be standard λ–tableaux and suppose that k is an integer with 1 ≤ m ≤ n. Then there exist a_{𝔳} ∈ R such that

n_{𝔰𝔱}L_{k} = res_{𝔱′}(k)n_{𝔰𝔱} + ∑_{𝔳▷𝔱^{𝔳∈Std(λ)}} a_{𝔳}n_{𝔰𝔳} mod (ℋ̌ ^{λ} )′.

Proof. First assume that R = Z. Then ′ is a ℤ–linear ring involution on ℋ_{Z} and L′_{k} = L_{k}; therefore, by (3.2),

n_{𝔰𝔱}L_{k} = (m_{𝔰𝔱}L_{k})′ = ( res_{𝔱}(k)m_{𝔰𝔱} + ∑_{𝔳▷𝔱^{𝔳∈Std(λ)}} a_{𝔳}m_{𝔰𝔳} mod ℋ̌ ^{λ} )′.

Using (4.2) this proves the Proposition for ℋ_{Z}. The general case now follows by specialization since ℋ ≅ ℋ_{Z} ⊗_{Z} R. □

Next consider the orthogonal basis {f_{𝔰𝔱}} of ℋ in the case where P_{ℋ} (q) is invertible. Let Z_{P} be the localization of Z at P_{ℋ} (q) and let ℋ_{Z_{P}} be the corresponding Ariki–Koike algebra. The involution ′ extends to ℋ_{Z_{P}} and ℋ is a specialization of ℋ_{Z_{P}} whenever P_{ℋ} (q) is invertible in R. (Note that Q_{1}, …, Q_{r} are indeterminates in Z_{P}.)

In general, f_{𝔰𝔱} ∉ ℋ_{Z}; however, if 𝔱 ≠ 𝔲 then res_{𝔱}(k)−res_{𝔲}(k) is a factor of P_{ℋ} (q) for all k, so f_{𝔰𝔱} ∈ ℋ_{Z_{P}} and we can speak of the elements F_{𝔱} and f_{𝔰𝔱} ∈ ℋ_{Z_{P}}. More generally, whenever P_{ℋ} (q) is invertible in R we have an element f′_{𝔰𝔱} ∈ ℋ via specialization because ℋ ≅ ℋ_{Z_{P}} ⊗_{Z_{P}} R.

4.4. Proposition. Suppose that 𝔱 is a standard tableau. Then F′_{𝔱} = F_{𝔱′} in ℋ_{Z_{P}}.

Proof. Applying the definitions together with (4.2) gives

F′_{𝔱} = ∏_{k=1}^{n} ∏_{\begin{matrix}c∈R(k) \\ c≠res_{𝔱}(k) \end{matrix}} (\frac{L_{k} − c}{res_{𝔱}(k) − c})′ = ∏_{k=1}^{n} ∏_{\begin{matrix}c∈R(k) \\ c≠res_{𝔱}(k) \end{matrix}} \frac{L_{k} − c′}{res_{𝔱′}(k) − c′}

= ∏_{k=1}^{n} ∏_{\begin{matrix}c∈R(k) \\ c≠res_{𝔱′} (k) \end{matrix}} \frac{L_{k} − c}{res_{𝔱′}(k) − c} = F_{𝔱′},

the last equality following because R(k) is invariant under ′. □

By Lemma 2.6 and Theorem 3.14 the Schur elements are given by s_{λ}(q) = τ(F_{𝔱^{λ}})^{−1}; consequently, the Schur elements have the following “palindromy” property.

4.5. Corollary. Suppose that λ is a multipartition of n. Then s_{λ′}(q) = (s_{λ}(q))′.

Returning to the general case, let g_{𝔰𝔱} = F_{𝔰′}n_{𝔰𝔱}F_{𝔱′}; then g_{𝔰𝔱} = F′_{𝔰}m′_{𝔰𝔱}F′_{𝔱} = f′_{𝔰𝔱} (in ℋ_{Z_{p}}). Applying ′ to {f_{𝔰𝔱}} and using Theorem 3.9 (and a specialization argument) shows that { g_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ^{+} } is a basis of ℋ . Consequently, as in Corollary 3.11, S̃^{λ} ≅ g_{𝔰𝔱}ℋ for any standard λ–tableaux 𝔰, 𝔱 ∈ Std(λ).

4.6. Remark. By the Proposition and Theorem 3.14(i),

g_{𝔱𝔱} = f′_{𝔱𝔱} = (γ_{𝔱}F_{𝔱})′ = γ′_{𝔱}F_{𝔱′} = \frac{γ′_{𝔱}}{γ_{𝔱′}} f_{𝔱′𝔱′}.

More generally, we can write g_{𝔰𝔱} = ∑_{𝔲,𝔳} a_{𝔲𝔳}f_{𝔲𝔳} for some a_{𝔲𝔳} ∈ R. By Proposition 3.4 and Proposition 4.4, F_{𝔰′}g_{𝔰𝔱}F_{𝔱′} = g_{𝔰𝔱}; so it follows that a_{𝔲𝔳} = 0 unless 𝔲 = 𝔰′ and 𝔳 = 𝔱′. Therefore, g_{𝔰𝔱} = α_{𝔰𝔱}f_{𝔰′𝔱′} for some α_{𝔰𝔱} ∈ R. Applying the ∗–involution shows that α_{𝔰𝔱} = α_{𝔱𝔰}. Finally, by looking at the product g_{𝔰𝔱}g_{𝔱𝔰} we see that α_{𝔰𝔱}^{2} = γ′_{𝔰}γ′_{𝔱}/γ_{𝔰′}γ_{𝔱′}. A priori, there is no reason why the square root of this element should belong to R; nor do I see a way to determine the sign of α_{𝔰𝔱}.

Combining Proposition 4.4 with Corollary 3.11 and the corresponding result for the g–basis shows that S^{λ} ≅ f_{𝔱𝔱}ℋ = g_{𝔱′𝔱′}ℋ ≅ S̃^{λ′}, for any 𝔱 ∈ Std(λ). Hence, we have the following.

4.7. Corollary. Suppose that P_{ℋ} (q) is invertible in R. Then S̃^{λ} ≅ S^{λ′}.

When R is field the assumption that P_{ℋ} (q) is invertible is equivalent to ℋ being semisimple. This assumption is necessary because, in general, S^{λ′} and S̃^{λ} are not isomorphic; rather, S^{λ′} is isomorphic to the dual of S̃^{λ} [24]. In the semisimple case both S^{λ} and S̃^{λ′} are irreducible, and hence self–dual, since they carry a non– degenerate bilinear form. Accordingly, we call the module S̃^{λ} a dual Specht module.

Here is another useful application of Proposition 4.4.

4.8. Corollary. Suppose that 𝔰 and 𝔱 are standard λ–tableaux and that 𝔲 and 𝔳 are standard μ–tableaux where λ and μ are multipartitions of n. Then f_{𝔰𝔱}g_{𝔲𝔳} = 0 if 𝔱 ≠ 𝔲′.

Proof. Applying the definitions, f_{𝔰𝔱}g_{𝔲𝔳} = F_{𝔰}m_{𝔰𝔱}F_{𝔱}F′_{𝔲}m_{𝔲𝔳}F′_{𝔳}; however, F′_{𝔲} = F_{𝔲′} by Proposition 4.4, so F_{𝔱}F′_{𝔲} = δ_{𝔱𝔲′}F_{𝔱} by Theorem 3.14(i), giving the result. (By Remark 4.6, f_{𝔰𝔱}g_{𝔱′𝔳} is a scalar multiple of f_{𝔰𝔳′}.) □

The Specht modules S^{λ} and the dual Specht modules S̃^{λ} are both constructed as quotient modules using the bases {m_{𝔰𝔱}} and {n_{𝔰𝔱}} respectively (see Corollary 3.11). In the cases where P_{ℋ} (q) is invertible in R we have also constructed these modules as submodules of ℋ . Next we produce submodules of an arbitrary Hecke algebra ℋ which are isomorphic to the Specht modules; these results will also play a rôle in computing the Schur elements in the next section.

Recall that 𝔱^{λ} is the λ–tableau which has the numbers 1, 2, …, n entered in order first along the rows of 𝔱^{λ}^{(1)} and then the rows of 𝔱^{λ}^{(2)} and so on. Let 𝔱_{λ} = (𝔱^{λ′})′; that is to say that 𝔱_{λ} is the λ–tableau with the numbers 1, 2, …, n entered in order first down the columns of 𝔱_{λ}^{(r)} and then the columns of 𝔱_{λ}^{(r−1)} etcetera. Observe that if 𝔱 is a standard λ–tableau then 𝔱^{λ} ⊵ 𝔱 ⊵ 𝔱_{λ}.

4.9. Proposition. Suppose that P_{ℋ} (q) is invertible and let λ be a multipartition of n. Then m_{λ}ℋ n_{λ′} = Rf_{𝔱^{λ}}_{𝔱_{λ}}.

Proof. By Proposition 3.4(i), m_{λ} = f_{𝔱^{λ}}_{𝔱^{λ}} + ∑_{𝔲,𝔳▷𝔱^{λ}} a_{𝔲𝔳}f_{𝔲𝔳} for some a_{𝔲𝔳} ∈ R. Therefore, interchanging the roles of λ and λ′ and applying the involution ′ (in ℋ_{Z_{p}} and then specializing) we see that there exist b_{𝔞𝔟} ∈ R such that

(4.10)  n_{λ′} = g_{𝔱^{λ′}}_{𝔱^{λ′}} + ∑_{𝔞,𝔟▷𝔱^{λ′}} b_{𝔞𝔟}g_{𝔞𝔟} = \frac{γ′_{𝔱^{λ′}}}{γ_{𝔱_{λ}}} f_{𝔱_{λ}𝔱_{λ}} + ∑_{𝔱_{λ}▷𝔞′,𝔟′} b_{𝔞𝔟}g_{𝔞𝔟}

where for the second equality we have used Remark 4.6 (note that (𝔱^{λ′})′ = 𝔱_{λ}) and the observation that 𝔞, 𝔟 ▷ 𝔱^{λ′} if and only if 𝔱_{λ} ▷ 𝔞′, 𝔟′. Now m_{λ}ℋ n_{λ′} is spanned by the elements m_{λ}f_{𝔰𝔱}n_{λ′}, where 𝔰 and 𝔱 range over all pairs of standard tableaux of the same shape. Now, (4.10) and Corollary 4.8 imply that

m_{λ}f_{𝔰𝔱}n_{λ′} = (f_{𝔱^{λ}}_{𝔱^{λ}} + ∑_{𝔲,𝔳▷𝔱^{λ}} a_{𝔲𝔳}f_{𝔲𝔳})f_{𝔰𝔱}(\frac{γ′_{𝔱^{λ′}}}{γ_{𝔱_{λ}}} f_{𝔱_{λ}𝔱_{λ}} + ∑_{𝔱_{λ}▷𝔞′,𝔟′} b_{𝔞𝔟}g_{𝔞𝔟})

= \frac{γ′_{𝔱^{λ′}}}{γ_{𝔱_{λ}}} f_{𝔱^{λ}}_{𝔱^{λ}}f_{𝔰𝔱}f_{𝔱_{λ}𝔱_{λ}} + ∑_{\begin{matrix}𝔱_{λ}▷𝔞′,𝔟′ \\ 𝔲,𝔳▷𝔱^{λ} \end{matrix}} a_{𝔲𝔳}b_{𝔞𝔟}f_{𝔲𝔳}f_{𝔰𝔱}g_{𝔞𝔟},

= \frac{γ′_{𝔱^{λ′}}}{γ_{𝔱_{λ}}} f_{𝔱^{λ}}_{𝔱^{λ}}f_{𝔰𝔱}f_{𝔱_{λ}𝔱_{λ}}

= \begin{cases}γ_{𝔱^{λ}}γ′_{𝔱^{λ′}} f_{𝔱^{λ}}_{𝔱_{λ}}, & if 𝔰 = 𝔱^{λ} and 𝔱 = 𝔱_{λ}, \\ 0, & otherwise, \end{cases}

with the last equality following from Theorem 3.9. Therefore, m_{λ}ℋ n_{λ′} = Rf_{𝔱^{λ}}_{𝔱_{λ}} as required. □

Let w_{λ} = d(𝔱_{λ}); thus, w_{λ} is the unique element of 𝔖_{n} such that 𝔱_{λ} = 𝔱^{λ}w_{λ}.

4.11. Example. Let λ = ((2, 1^{2}), (2, 1), (2)). Then

𝔱^{λ} = ( 1 2 3 4 , 5 6 7 , ^{8 9 )} and 𝔱_{λ} = ( 6 9 7 8 , 3 5 4 , 1 2 )

and w_{λ} = (1, 6, 5, 3, 7, 4, 8)(2, 9).

In order to compute the Schur elements we will need to know quite a few properties of the permutations w_{λ}; these permutations enter the story through the following definition and Corollary.

4.12. Definition. Suppose that λ is a multipartition of n. Let z_{λ} = m_{λ}T_{w_{λ}}n_{λ′}.

The element z_{λ} and the following result are crucial to our computation of the Schur elements.

4.13. Corollary. Suppose that P_{ℋ} (q) is invertible in R. Then z_{λ} = γ′_{𝔱^{λ′}} f_{𝔱^{λ}}_{𝔱_{λ}}. In particular, m_{λ}ℋ n_{λ′} = Rz_{λ}.

Proof. Consulting the definitions, z_{λ} = m_{λ}T_{w_{λ}}n_{λ′} = m_{𝔱^{λ}}_{𝔱_{λ}}n_{λ′}. Furthermore, by Proposition 3.4(i), there exist c_{𝔲𝔳} ∈ R such that m_{𝔱^{λ}}_{𝔱_{λ}} = f_{𝔱^{λ}}_{𝔱_{λ}} + ∑ c_{𝔲𝔳}f_{𝔲𝔳} where the sum is over the pairs (𝔲, 𝔳) of standard tableaux which strictly dominate 𝔱_{λ}. Therefore, by (4.10),

z_{λ} = (f_{𝔱^{λ}}_{𝔱_{λ}} + ∑_{𝔲,𝔳▷𝔱_{λ}} c_{𝔲𝔳}f_{𝔲𝔳})(\frac{γ′_{𝔱^{λ′}}}{γ_{𝔱_{λ}}} f_{𝔱_{λ}𝔱_{λ}} + ∑_{𝔱_{λ}▷𝔞′,𝔟′^{𝔞,𝔟}} b_{𝔞𝔟}g_{𝔞𝔟}).

By Theorem 3.9, f_{𝔱^{λ}}_{𝔱_{λ}}f_{𝔱_{λ}𝔱_{λ}} = γ_{𝔱_{λ}}f_{𝔱^{λ}}_{𝔱_{λ}} and this is the only non–zero term in this product by Corollary 4.8. Hence, z_{λ} = γ′_{𝔱^{λ′}} f_{𝔱^{λ}}_{𝔱_{λ}} as required. □

Now, z_{λ} is an element of ℋ_{Z}, so γ′_{𝔱^{λ′}} f_{𝔱^{λ}}_{𝔱_{λ}} ∈ ℋ_{Z}. By definition, z_{λ}ℋ is a submodule of m_{λ}ℋ and a quotient module of n_{λ′}ℋ . Over an arbitrary ring R, Du and Rui [13, Remark 2.5] showed that S^{λ} ≅ z_{λ}^{∗}ℋ and S̃^{λ′} ≅ z_{λ}ℋ as ℋ –modules, the isomorphisms being given by the natural quotient maps m_{λ}ℋ ⟶ z_{λ}^{∗}ℋ and n_{λ′}ℋ ⟶ z_{λ}ℋ . Note that S^{λ} ≅ S̃^{λ′} when ℋ is semisimple by Corollary 3.11.

5. The Schur elements

Using the results of the previous sections we are now ready to compute the Schur elements of ℋ . By Lemma 2.6, s_{λ}(q) = \frac{1}{τ(F_{𝔱^{λ}})}, so it is enough to calculate τ(F_{𝔱^{λ}}). Our basic strategy, which is inspired by Murphy [25], is to write F_{𝔱^{λ}} as a product of two terms and, in effect, to evaluate τ on each of these factors separately.

5.1. Proposition. Suppose that 𝔱 is a standard λ–tableau. Then there exist elements Φ_{𝔱} and Ψ_{𝔱} in ℋ (𝔖_{n}) such that

(i) Ψ_{𝔱}F_{𝔱} = F_{𝔱^{λ}}Φ_{𝔱};

(ii) Ψ_{𝔱} = T_{d(𝔱)} + ∑_{w<d(𝔱)} p_{𝔱w}T_{w}, for some p_{𝔱w} ∈ R; and,

(iii) γ_{𝔱^{λ}}Φ_{𝔱} Ψ_{𝔱}^{∗} = γ_{𝔱}.

Proof. We prove all three statements by induction on 𝔱. When 𝔱 = 𝔱^{λ} there is nothing to prove as we may take Φ_{𝔱^{λ}} = Ψ_{𝔱^{λ}} = 1. Suppose then that 𝔱 ≠ 𝔱^{λ}. Then there exists an integer i, with 1 ≤ i < n, such that 𝔰 = 𝔱(i, i + 1) ▷ 𝔱. Let α = \frac{(q−1) res_{𝔱}(i)}{res_{𝔱}(i)−res_{𝔰}(i)} and β = \frac{(q−1) res_{𝔰}(i)}{res_{𝔰}(i)−res_{𝔱}(i)}. Then f_{𝔰𝔰}(T_{i} − α) = f_{𝔰𝔱} by Proposition 3.5. Similarly, by the left hand analogue of Proposition 3.5 (interchanging the roles of 𝔰 and 𝔱), together with (3.7)(ii), (T_{i} − β)f_{𝔱𝔱} = (γ_{𝔱}/γ_{𝔰})f_{𝔰𝔱}. Therefore,

(T_{i} − β)F_{𝔱} = \frac{1}{γ_{𝔱}} (T_{i} − β)f_{𝔱𝔱} = \frac{1}{γ_{𝔰}} f_{𝔰𝔱} = \frac{1}{γ_{𝔰}} f_{𝔰𝔰}(T_{i} − α) = F_{𝔰}(T_{i} − α).

By induction, there exist elements Φ_{𝔰} and Ψ_{𝔰} which satisfy properties (i)–(iii). Define Ψ_{𝔱} = Ψ_{𝔰}(T_{i} − β) and Φ_{𝔱} = Φ_{𝔰}(T_{i} − α); then, by induction and the last equation,

Ψ_{𝔱}F_{𝔱} = Ψ_{𝔰}(T_{i} − β)F_{𝔱} = Ψ_{𝔰}F_{𝔰}(T_{i} − α) = F_{𝔱^{λ}}Φ_{𝔰}(T_{i} − α) = F_{𝔱^{λ}}Φ_{𝔱}.

Hence, (i) holds. Next, again by induction we have

Ψ_{𝔱} = Ψ_{𝔰}(T_{i} − β) = (T_{d(𝔰)} + ∑_{v<d(𝔰)} p_{𝔰v}T_{v})(T_{i} − β) = T_{d(𝔱)} + ∑_{w<d(𝔱)} p_{𝔱w}T_{w},

by standard properties of the Bruhat order since d(𝔱) = d(𝔰)(i, i + 1) > d(𝔰). This proves (ii). Finally, using induction once more (and a quick calculation for the second equality),

γ_{𝔱^{λ}}Φ_{𝔱} Ψ_{𝔱}^{∗} = γ_{𝔱^{λ}}Φ_{𝔰}(T_{i} − α)(T_{i} − β)Ψ_{𝔰}^{∗}

= \frac{(q res_{𝔰}(i) − res_{𝔱}(i))(res_{𝔰}(i) − q res_{𝔱}(i))}{(res_{𝔰}(i) − res_{𝔱}(i))^{2}}γ_{𝔱^{λ}}Φ_{𝔰}Ψ_{𝔰}^{∗}

= \frac{(q res_{𝔰}(i) − res_{𝔱}(i))(res_{𝔰}(i) − q res_{𝔱}(i))}{(res_{𝔰}(i) − res_{𝔱}(i))^{2}}γ_{𝔰} = γ_{𝔱},

the last equality coming from (3.7)(ii). This proves (iii) and so completes the proof. □

We are not claiming that the elements Φ_{𝔱} and Ψ_{𝔱} are uniquely determined by the conditions of the Proposition; ostensibly, these elements depend upon the choice of reduced expression for d(𝔱). In what follows we only need to know that elements with these properties exist.

5.2. Corollary. Suppose that 𝔱 is a standard λ–tableau. Then (i) F_{𝔱^{λ}} = \frac{γ_{𝔱^{λ}}}{γ_{𝔱}} Ψ_{𝔱}F_{𝔱}Ψ_{𝔱}^{∗}; and, (ii) F_{𝔱} = \frac{γ_{𝔱^{λ}}}{γ_{𝔱}} Φ_{𝔱}^{∗}F_{𝔱^{λ}}Φ_{𝔱}.

Proof. Using parts (iii) and (i) of the Proposition, respectively, shows that

F_{𝔱^{λ}} = \frac{γ_{𝔱^{λ}}}{γ_{𝔱}} F_{𝔱^{λ}}Φ_{𝔱}Ψ_{𝔱}^{∗} = \frac{γ_{𝔱^{λ}}}{γ_{𝔱}} Ψ_{𝔱}F_{𝔱}Ψ_{𝔱}^{∗};

this proves (i). Part (ii) follows from (i) by ‘conjugating’ (i) by Φ_{𝔱}. □

The main reason why we are interested in Ψ_{𝔱} and Φ_{𝔱} is the following.

5.3. Proposition. Suppose that 𝔰 and 𝔱 are standard λ–tableaux. Then

f_{𝔰𝔱} = Φ_{𝔰}^{∗}f_{𝔱^{λ}}_{𝔱^{λ}}Φ_{𝔱}.

Proof. By the definition of f_{𝔰𝔱} and Proposition 5.1(ii) we have

f_{𝔰𝔱} = F_{𝔰}T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}F_{𝔱} = F_{𝔰}Ψ_{𝔰}^{∗}m_{λ}Ψ_{𝔱}F_{𝔱} − ∑_{(v,w)<(d(𝔰),d(𝔱))} p_{𝔰v}p_{𝔱w}F_{𝔰}T_{v}^{∗}m_{λ}T_{w}F_{𝔱}.

Now if (v, w) < (d(𝔰), d(𝔱)) then T_{v}^{∗}m_{λ}T_{w} belongs to the span of the m_{𝔲𝔳} where (𝔲, 𝔳) ▷ (𝔰, 𝔱). Therefore, by Proposition 3.4(i), T_{v}^{∗}m_{λ}T_{w} belongs to the span of the f_{𝔲𝔳} where either 𝔲 and 𝔳 are standard λ–tableaux and (𝔲, 𝔳) ▷ (𝔰, 𝔱), or Shape(𝔲) = Shape(𝔳) ▷ λ; consequently, F_{𝔰}T_{v}^{∗}m_{λ}T_{w}F_{𝔱} = 0 by Proposition 3.4(iii). Hence, by Theorem 3.14(i) and Proposition 5.1(i),

f_{𝔰𝔱} = F_{𝔰}Ψ_{𝔰}^{∗}m_{λ}Ψ_{𝔱}F_{𝔱} = Φ_{𝔰}^{∗}F_{𝔱^{λ}}m_{λ}F_{𝔱^{λ}}Φ_{𝔱} = Φ_{𝔰}^{∗}f_{𝔱^{λ}}_{𝔱^{λ}}Φ_{𝔱}

as required. □

Applying this result to f_{𝔱^{λ}}_{𝔱_{λ}} shows that f_{𝔱^{λ}}_{𝔱_{λ}} = Φ_{𝔱^{λ}}^{∗}f_{𝔱^{λ}}_{𝔱^{λ}}Φ_{𝔱_{λ}} = γ_{𝔱^{λ}}F_{𝔱^{λ}}Φ_{𝔱_{λ}}, the last equality following because Φ_{𝔱^{λ}} = 1. Using Proposition 5.1(iii) to multiply this equation on the right by Ψ_{𝔱_{λ}}^{∗}, and recalling Corollary 4.13, now yields the following.

5.4. Corollary. Suppose that λ is a multipartition of n. Then

F_{𝔱^{λ}} = \frac{1}{γ_{𝔱_{λ}}} f_{𝔱^{λ}}_{𝔱_{λ}}Ψ_{𝔱_{λ}}^{∗} = \frac{1}{γ_{𝔱_{λ}}γ′_{𝔱^{λ′}}} z_{λ}Ψ_{𝔱_{λ}}^{∗}.

By Lemma 2.6 in order to compute the Schur elements it suffices to calculate τ(F_{𝔱^{λ}}) for each λ; so we are reduced to finding τ(z_{λ}Ψ_{𝔱_{λ}}^{∗}). To do this we rewrite z_{λ} with respect to the Ariki–Koike basis. In types A and B (that is, r = 1 or r = 2) this is reasonably straightforward; in general we have to work much harder.

Until further notice, fix a multipartition λ and let λ̄ = (λ̄^{(1)}, …, λ̄^{(r)}) be the multipartition with λ̄^{(s)} = (|λ^{(s)}|) for 1 ≤ s ≤ r. Then wλ̄ = d(𝔱λ̄) is the permutation

( \begin{matrix}1 & ⋯ & 𝔞_{2} & 𝔞_{2} + 1 & ⋯ & 𝔞_{3} & ⋯ & 𝔞_{r} + 1 & ⋯ & n \\ 𝔟_{1} + 1 & ⋯ & n & 𝔟_{2} + 1 & ⋯ & 𝔟_{1} & ⋯ & 1 & ⋯ & 𝔟_{r−1} \end{matrix} ),

where 𝔞_{s} = ∑_{t=1}^{s−1} |λ^{(t)}| and 𝔟_{s} = ∑_{t=s+1}^{r} |λ^{(t)}| for 1 ≤ s ≤ r.

Now, 𝔱λ̄wλ̄ is a standard tableau; therefore, wλ̄ is a distinguished right coset representative for 𝔖λ̄ in 𝔖_{n} (for example, by [22, Prop. 3.3]). Equivalently, wλ̄ is a distinguished left coset representative for 𝔖\overline{λ′} in 𝔖_{n} since 𝔖λ̄wλ̄ = wλ̄𝔖_{λ′}. Consequently, there exists w_{λ/λ̄} ∈ 𝔖\overline{λ′} such that w_{λ} = wλ̄w_{λ/λ̄} and ℓ(w_{λ}) = ℓ(wλ̄)+ ℓ(w_{λ/λ̄}). To proceed we need to factorize wλ̄.

Suppose that a and b are non–negative integers let w_{a,b} = w_{((a),(b))}; so, 𝔱_{((a),(b))} = 𝔱^{((a),(b))}w_{a,b}. More concretely, w_{a,0} = 1 = w_{0,b} and if a > 0 and b > 0 then

w_{a,b} = ( \begin{matrix}1 & 2 & ⋯ & a & a + 1 & a + 2 & ⋯ & a + b \\ b + 1 & b + 2 & ⋯ & a + b & 1 & 2 & ⋯ & b \end{matrix} ).

If i ≤ j we also set s_{i,j} = s_{i}s_{i+1} … s_{j} and s_{j,i} = 1; so s_{i,j} = (j + 1, j, …, i). For convenience let T_{i,j} = T_{s_{i,j}}. It is not hard to see that w_{a,b} = (s_{a+b−1,1})^{b}.

The permutations w_{a,b} were studied by Dipper and James [11]; in particular, they observed that w_{a,b} = s_{a,a+b−1}w_{a−1,b}, with the lengths adding, which implies the following. (Note that our s_{i,j} is Dipper and James’ s_{i,j−1} when i < j; also if i ≥ j then Dipper and James set s_{i,j} = (s_{j,i−1})^{−1} whereas we have s_{i,j} = 1.) We let 𝔖_{(b,a)} = 𝔖_{b} × 𝔖_{a} ↪ 𝔖_{n} (natural embedding).

5.5. Lemma. Suppose that a and b are positive integers with a + b ≤ n. Then w_{a,b} = s_{a,a+b−1} … s_{1,b} and ℓ(w_{a,b}) = ℓ(s_{a,a+b−1}) + ⋯ + ℓ(s_{1,b}). Moreover, w_{a,b} is a distinguished left coset representative for 𝔖_{(b,a)} in 𝔖_{n}; that is, ℓ(w_{a,b}v) = ℓ(w_{a,b}) + ℓ(v) for all v ∈ 𝔖_{(b,a)}.

For s = 1, …, r define w_{λ̄,s} = w_{n_{s},𝔟_{s}}, where n_{s} = |λ^{(s)}| and 𝔟_{s} = ∑_{t=s+1}^{r} |λ^{(t)}|; in particular, 𝔟_{0} = n and w_{λ̄,r} = 1. Now n_{s} + 𝔟_{s} = 𝔟_{s−1} so w_{λ̄,s} ∈ 𝔖_{𝔟_{s−1}} for each s; therefore, w_{λ̄,s} … w_{λ̄,r−1} is an element of 𝔖_{𝔟_{s−1}} and, consequently,

ℓ(w_{λ̄,1} … w_{λ̄,r−1}) = ℓ(w_{λ̄,1}) + ℓ(w_{λ̄,2} … w_{λ̄,r−1}) = ⋯ = ℓ(w_{λ̄,1}) + ⋯ + ℓ(w_{λ̄,r−1})

by Lemma 5.5. Noting that (k)wλ̄ = (k)w_{λ̄,1} … w_{λ̄,s} for k = 1, 2, …, 𝔞_{s+1} we have shown the following.

5.6 Let λ be a partition of n. Then w_{λ} = wλ̄w_{λ/λ̄} = w_{λ̄,1} … w_{λ̄,r−1}w_{λ/λ̄}. Moreover, ℓ(w_{λ}) = ℓ(wλ̄) + ℓ(w_{λ/λ̄}) = ℓ(w_{λ̄,1}) + ⋯ + ℓ(w_{λ̄,r−1}) + ℓ(w_{λ/λ̄}).

The point of the lengths adding is that T_{w_{λ}} = T_{w_{λ̄,1}} … T_{w_{λ̄,r−1}}T_{w_{λ/λ̄}}. We will use this below without further comment.

Although we won’t need it notice that ℓ(w_{a,n−a}) = a(n − a) by Lemma 5.5 (or directly); so (5.6) implies that ℓ(wλ̄) = ∑_{1≤s<t≤r} |λ^{(s)}||λ^{(t)}|.

5.7. Example. Let λ = ((2, 1^{2}), (2, 1), (2)). Then λ̄ = ((4), (3), (2)) and

w_{λ} = ( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 6 & 9 & 7 & 8 & 3 & 5 & 4 & 1 & 2 \end{matrix} ) = (1, 6, 5, 3, 7, 4, 8)(2, 9),
wλ̄ = ( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 6 & 7 & 8 & 9 & 3 & 4 & 5 & 1 & 2 \end{matrix} ) = (1, 6, 4, 9, 2, 7, 5, 3, 8);

therefore, w_{λ/λ̄} = (7, 9, 8)(4, 5). Further, 𝔟 = (5, 2, 0) so, using Lemma 5.5,

w_{λ̄,1} = w_{4,5} = ( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 6 & 7 & 8 & 9 & 1 & 2 & 3 & 4 & 5 \end{matrix} )
= (1, 6, 2, 7, 3, 8, 4, 9, 5) = s_{4,8}s_{3,7}s_{2,6}s_{1,5}
w_{λ̄,2} = w_{3,2} = ( \begin{matrix}1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 1 & 2 \end{matrix} ) = (1, 3, 5, 2, 4) = s_{3,4}s_{2,3}s_{1,2}

and w_{λ̄,3} = 1. It is evident that w_{λ} = w_{λ̄,1}w_{λ̄,2}w_{λ/λ̄}.

Given i ≤ j and Q ∈ R let L_{i,j}(Q) = (L_{i}−Q) … (L_{j}−Q); if i > j set L_{i,j}(Q) = 1. In particular, u_{λ′}^{−} = L_{1,𝔟_{1}}(Q_{1}) … L_{1,𝔟_{r−1}}(Q_{r−1}) where 𝔟_{s} = |λ^{(s+1)}| + ⋯ + |λ^{(r)}|, as above.

As a final piece of notation, given 0 ≤ l ≤ m < n let H_{k,m} be the R–submodule of ℋ spanned by the elements { L_{1}^{c_{1}} … L_{k}^{c_{k}}T_{w} | 0 ≤ c_{i} < r and w ∈ 𝔖_{m} }. In general H_{k,m} is neither a subalgebra nor a submodule of ℋ ; however, H_{k,m} is a right ℋ (𝔖_{m})–module.

One of the difficulties in working with ℋ is that, in general, the powers of the L_{k} are not necessarily linear combinations of products of the L_{i}^{s}, for 1 ≤ i ≤ k and 1 ≤ s < r; however, it is always true that L_{k}^{s} ∈ H_{k,m} for all s ≥ 0 provided that m ≥ k (since in this case ⟨T_{0}, …, T_{k−1}⟩ ⊆ H_{k,m} by (2.2)).

5.8. Lemma. Suppose that h ∈ H_{k−1,k} for some k with 1 ≤ k ≤ n.

(i) If 1 ≤ i < n then

T_{i}h ∈ \begin{cases}H_{k−1,k}, & if 1 ≤ i < k − 1, \\ H_{k,k}, & if i = k − 1, \\ H_{k−1,i+1}, & if k ≤ i < n. \end{cases}

(ii) If 1 ≤ s < r then hL_{k}^{s} = ∑_{t=0}^{s} L_{k}^{t}h_{t} for some h_{t} ∈ H_{k−1,k}.

Proof. By definition h is a linear combination of terms of the form L_{k−1}^{c_{k−1}} … L_{1}^{c_{1}}T_{w}, for some 0 ≤ c_{i} < r and w ∈ 𝔖_{k}. By (2.1)(ii), if i ≠ m − 1, m then T_{i}L_{m} = L_{m}T_{i}; whereas T_{m}L_{m} = L_{m+1}(T_{m} − q + 1) and T_{m−1}L_{m} = L_{m−1}T_{m−1} + (q − 1)L_{m}. Combining these facts proves (i).

For part (ii), observe that if w ∈ 𝔖_{k} then T_{w}L_{k} = L_{k}T_{w} unless s_{k−1} appears in a reduced expression of w; however, T_{k−1}L_{k} = L_{k−1}T_{k−1} + (q − 1)L_{k}. Part (ii) now follows by induction on s using the remarks of the last paragraph. □

5.9. Lemma. Fix Q ∈ R and let i, j, c and d be positive integers with i ≤ c ≤ d ≤ j < n. Then

T_{i,j}L_{c,d}(Q) = {L_{c+1,d+1}(Q) + ∑_{k=c+1}^{d+1} L_{k}h_{k}}T_{i,j}

for some elements h_{k} ∈ H_{k−1,k} for c + 1 ≤ k ≤ d + 1.

Proof. If d = j then by convention T_{d+1,j} = 1. Therefore, by (2.1)(iv) we have

T_{i,j}L_{c,d}(Q) = T_{i,c−1}T_{c,d}(L_{c} − Q) … (L_{d} − Q)T_{d+1,j}
= T_{i,c−1}T_{c}(L_{c} − Q) … T_{d}(L_{d} − Q)T_{d+1,j}.

Now, T_{k}L_{k} = qL_{k+1}T_{k}^{−1} = L_{k+1}T_{k} − (q − 1)L_{k+1}, so

T_{i,j}L_{c,d}(Q) = T_{i,c−1} ∏_{k=c+1}^{d+1} {(L_{k} − Q)T_{k−1} − (q − 1)L_{k}} · T_{d+1,j},

where we read the terms from left to right with increasing values of k — the order of the factors is important here. Now T_{k−1} commutes with L_{m} if k < m; therefore, each T_{k−1} in this product commutes with all of L_{m}’s which appear to its right. A straightforward induction on (d − c) using (2.1) shows that there exist h′_{k} ∈ H_{k−1,k} such that

T_{i,j}L_{c,d}(Q) = T_{i,c−1}{L_{c+1,d+1}(Q)T_{c,d} + ∑_{k=c+1}^{d+1} L_{k}h′_{k}T_{k,d}}T_{d+1,j}.

(Recall that our convention is that T_{d+1,d} = 1.) If c + 1 ≤ k ≤ d + 1 then by (2.1)

T_{i,c−1}L_{k}h′_{k}T_{k,d}T_{d+1,j} = L_{k}T_{i,c−1}h′_{k}T_{k,j} = L_{k}T_{i,c−1}h′_{k}T_{i,k−1}^{−1}T_{i,j}.

Also, using (2.1)(ii), T_{i,c−1}L_{c+1,d+1}(Q) = L_{c+1,d+1}(Q)T_{i,c−1}. Therefore,

T_{i,j}L_{c,d}(Q) = L_{c+1,d+1}(Q)T_{i,j} + ∑_{k=c+1}^{d+1} L_{k}h_{k}T_{i,j},

where h_{k} = T_{i,c−1}h′_{k}T_{i,k−1}^{−1} ∈ H_{k−1,k} by Lemma 5.8, for c + 1 ≤ k ≤ d + 1. □

This brings us to the key technical lemma.

5.10. Lemma. Fix Q ∈ R and let a, b, c and d be non–negative integers such that a + b ≤ n and 1 ≤ c ≤ d ≤ b. Then there exist elements h_{k} ∈ H_{k−1,k} such that

T_{w_{a,b}}L_{c,d}(Q) = {L_{a+c,a+d}(Q) + ∑_{k=a+c}^{a+d} L_{k}h_{k}}T_{w_{a,b}}.

Proof. We argue by induction on a. If a = 0 then w_{a,b} = 1 and there is nothing to prove; so suppose that a ≥ 1. Then w_{a,b} = s_{a,a+b−1}w_{a−1,b} by Lemma 5.5. Therefore, by induction and Lemma 5.9, respectively, there exist elements h′_{k}, h′′_{k} ∈ H_{k−1,k} such that

T_{w_{a,b}}L_{c,d}(Q) = T_{a,a+b−1}T_{w_{a−1,b}}L_{c,d}(Q)

= T_{a,a+b−1}{L_{a+c−1,a+d−1}(Q) + ∑_{k=a+c−1}^{a+d−1} L_{k}h′_{k}}T_{w_{a−1,b}}

= {L_{a+c,a+d}(Q) + ∑_{k=a+c}^{a+d} L_{k}h′′_{k}}T_{a,a+b−1}T_{w_{a−1,b}}

+ ∑_{k=a+c−1}^{a+d−1} T_{a,a+b−1}L_{k}h′_{k}T_{w_{a−1,b}}.

If a + c − 1 ≤ k ≤ a + d − 1 then by (2.1)

T_{a,a+b−1}L_{k}h′_{k}T_{w_{a−1,b}} = T_{a,k−1}T_{k}L_{k}T_{k+1,a+b−1}h′_{k}T_{w_{a−1,b}}
= qT_{a,k−1}L_{k+1}T_{k}^{−1}T_{k+1,a+b−1}h′_{k}T_{w_{a−1,b}}
= qL_{k+1}T_{a,k−1}T_{k}^{−1}h′_{k}T_{k+1,a+b−1}T_{w_{a−1,b}}
= qL_{k+1}T_{a,k−1}T_{k}^{−1}h′_{k}T_{a,k}^{−1}T_{a,a+b−1}T_{w_{a−1,b}}
= L_{k+1}T_{a,k−1}(T_{k} − q + 1)h′_{k}T_{a,k}^{−1}T_{w_{a,b}},

where the third equality follows by (2.1) because h′_{k} ∈ H_{k−1,k}. By Lemma 5.8 the element T_{a,k−1}(T_{k} − q + 1)h′_{k}T_{a,k}^{−1} belongs to H_{k,k+1}; so, combining these equations proves the Lemma. □

Lemma 5.10 is enough to compute the Schur elements when r = 2; in order to cover the general case we delicately apply Lemma 5.10 several times.

5.11. Corollary. Suppose that a, b, c and s are non–negative integers with a + b ≤ n, 1 ≤ c ≤ b and 1 ≤ s < r. Then T_{w_{a,b}}L_{c}^{s} = ∑_{t=1}^{s} L_{a+c}^{t}h_{t}T_{w_{a,b}} some h_{t} ∈ H_{a+c−1,a+c}.

Proof. If s = 1 this is the special case of Lemma 5.10 corresponding to the choices d = c and Q = 0; in particular, this implies that the exponent t is always at least 1. If s > 1 then the result follows by induction using Lemma 5.8(ii). □

For each integer k let c_{λ}(k) = s if k appears in component s of 𝔱^{λ}. Observe that for all k we have c_{λ}(k) = min { 1 ≤ s ≤ r | k ≤ |λ^{(1)}| + ⋯ + |λ^{(s)}| }. For the next Lemma recall that 𝔞_{s} = |λ^{(1)}| + ⋯ + |λ^{(s−1)}| and 𝔟_{s} = |λ^{(s+1)}| + ⋯ + |λ^{(r)}| for 1 ≤ s ≤ r.

5.12. Lemma. Suppose that λ is a multipartition of n. Then

T_{w_{λ}}u_{λ′}^{−} = L_{𝔞_{2}+1,n}(Q_{1}) … L_{𝔞_{r}+1,n}(Q_{r−1})T_{w_{λ}} + ∑_{k=1}^{n} ∑_{c=1}^{c_{λ}(k)−1} L_{k}^{c}h_{kc}

for some h_{kc} ∈ H_{k−1,n}.

Proof. First note that if r = 1 then u_{λ′}^{−} = 1 and there is nothing to prove; so we assume that r > 1. Next, T_{w_{λ}}u_{λ′}^{−} = T_{wλ̄}T_{w_{λ/λ̄}}u_{λ′}^{−} = T_{wλ̄}u_{λ′}^{−}T_{w_{λ/λ̄}} by part (iv) of (2.1) since w_{λ/λ̄} ∈ 𝔖\overline{λ′}. Consequently, it is enough to consider T_{wλ̄}u_{λ′}^{−}; equivalently, we may assume that λ = λ̄ (for clarity we will continue to write λ̄).

By (5.6), wλ̄ = w_{λ̄,1} … w_{λ̄,r−1} where w_{λ̄,s} ∈ 𝔖_{𝔟_{s−1}}. Therefore, by (2.1)(iv),

T_{wλ̄}u_{λ′}^{−} = T_{w_{λ̄,1}} … T_{w_{λ̄,r−1}}L_{1,𝔟_{1}}(Q_{1}) … L_{1,𝔟_{r−1}}(Q_{r−1})
= T_{w_{λ̄,1}}L_{1,𝔟_{1}}(Q_{1}) … T_{w_{λ̄,r−1}}L_{1,𝔟_{r−1}}(Q_{r−1}).

We also let n_{s} = |λ^{(s)}|. Suppose that 1 ≤ s ≤ r − 1. For k = 1, …, 𝔟_{s−1} set

c_{λ,s}(k) = min { 0 ≤ t ≤ r − s | k ≤ |λ^{(s)}| + ⋯ + |λ^{(s+t)}| }

and let H^{(s)} be the R–submodule of ℋ spanned by elements of the form L_{k}^{c}h_{kc} where n_{s} < k ≤ 𝔟_{s−1}, 1 ≤ c ≤ c_{λ,s}(k) and h_{kc} ∈ H_{k−1,𝔟_{s−1}}. We claim that

T_{w_{λ̄,s}}L_{1,𝔟_{s}}(Q_{s}) … T_{w_{λ̄,r−1}}L_{1,𝔟_{r−1}}(Q_{r−1}) ≡ P^{(s)} (mod H^{(s)}),

where P^{(s)} = ∏_{t=s}^{r−1} L_{n_{s}+···+n_{t}+1,𝔟_{s−1}}(Q_{t})·T_{w_{λ̄,s}} … T_{w_{λ̄,r−1}}. Taking s = 1 will prove the Lemma because 𝔟_{0} = n and c_{λ}(k) = c_{λ,1}(k) + 1 by the remarks before the Lemma. Note that P^{(s)} has degree c_{λ,s}(k) when we consider it as a polynomial in L_{k}.

To prove the claim we argue by downwards induction on s. If s ≥ c_{λ}(n) then 𝔟_{s} = 0 so that T_{w_{λ̄,s}} = L_{1,𝔟_{s}}(Q_{s}) = P^{(s)} = 1 and there is nothing to prove. Suppose then that s < c_{λ}(n). Then, by induction,

T_{w_{λ̄,s}}L_{1,𝔟_{s}}(Q_{s}) … T_{w_{λ̄,r−1}}L_{1,𝔟_{r−1}}(Q_{r−1}) = T_{w_{λ̄,s}}L_{1,𝔟_{s}}(Q_{s}){P^{(s+1)} + h^{(s+1)}}

for some h^{(s+1)} ∈ H^{(s+1)}. Now 𝔟_{s} > 0 since s < c_{λ}(n), so 𝔟_{s−1} = 𝔟_{s} + n_{s} ≥ 𝔟_{s} ≥ 1. Hence, by Lemma 5.10,

\begin{matrix}T_{w_{λ̄,s}}L_{1,𝔟_{s}}(Q_{s}) & \{P^{(s+1)} + h^{(s+1)}\} = T_{w_{ns,𝔟s}} L_{1,𝔟_{s}}(Q_{s})\{P^{(s+1)} + h^{(s+1)}\} \\ = & \{L_{n_{s}+1,𝔟_{s−1}}(Q_{s})T_{w_{ns,𝔟s}} + ∑_{k=n_{s}+1}^{𝔟_{s−1}} L_{k}h_{k}T_{w_{ns,𝔟s}} \}\{P^{(s+1)} + h^{(s+1)}\}, \end{matrix}

for some h_{k} ∈ H_{k−1,k}. We will move T_{w_{λ̄,s}} past P^{(s+1)} and h^{(s+1)} using Lemma 5.10.

Viewing P^{(s+1)} as a polynomial in the L_{m}’s, if n_{s+1} < m ≤ 𝔟_{s} then the degree of L_{m} in P^{(s+1)} is at most c_{λ,s+1}(m); further, L_{m} does not appear in P^{(s+1)} if either m ≤ n_{s+1} or m > 𝔟_{s}. Let p^{(s+1)} = ∏_{t=s+1}^{r−1} L_{n_{s+1}+···+n_{t}+1,𝔟_{s}}(Q_{t}). Then by Lemma 5.10 and Corollary 5.11 there exist h_{k,1}, h_{ck,2} ∈ H_{k−1,k} such that

T_{w_{ns,𝔟s}} p^{(s+1)} = {L_{n_{s}+n_{s+1}+1,𝔟_{s−1}}(Q_{s+1}) + ∑_{k=n_{s}+n_{s+1}+1}^{b_{s−1}} L_{k}h_{k,1}}T_{w_{ns,bs}} p^{(s+2)}

= {L_{n_{s}+n_{s+1}+1,𝔟_{s−1}}(Q_{s+1})L_{n_{s}+n_{s+1}+n_{s+2}+1,𝔟_{s−1}}(Q_{s+2})

+ ∑_{k=n_{s}+n_{s+1}+1}^{𝔟_{s−1}} ∑_{c=1}^{c_{2}(k)} L_{k}^{c}h_{ck,2}}T_{w_{ns,bs}} p^{(s+3)},

where c_{2}(k) = 1 if n_{s} + n_{s+1} < k ≤ n_{s} + n_{s+1} + n_{s+2} and c_{2}(k) = 2 if n_{s} + n_{s+1} + n_{s+2} < k ≤ 𝔟_{s−1}. For the second equality we have used Lemma 5.10, Lemma 5.8 and Corollary 5.11. Looking at the definition of c_{λ,s}(m), if n_{s+1} < m ≤ 𝔟_{s} then c_{λ,s+1}(m) = c_{λ,s}(m + n_{s}) − 1; therefore, continuing in this way we see that

T_{w_{ns,𝔟s}} p^{(s+1)} ≡ ∏_{t=s+1}^{r−1} L_{n_{s}+n_{s+1}+···+n_{t}+1,𝔟_{s−1}}(Q_{t}) · T_{w_{ns,𝔟s}} (mod H^{(s)}).

In obtaining this equation notice that if m > k then L_{k}h_{k}L_{m}h_{k} = L_{m}(L_{k}h_{k}h_{m}) and L_{k}h_{k}h_{m} ∈ H_{m−1,m}, for any h_{k} ∈ H_{k−1,k} and h_{m} ∈ H_{m−1,m}; similarly, if m < k then h_{k}L_{m}h_{m} ∈ H_{k−1,k} by Lemma 5.8. It follows that if L_{k}^{c}h appears in this expansion, for some h ∈ H_{k−1,k}, then n_{s} + n_{s+1} < k ≤ 𝔟_{s−1} and so 1 ≤ c ≤ c_{λ,s+1}(k − n_{s}) = c_{λ,s}(k) − 1 by the remarks above; hence, L_{k}h_{k}L_{m}h_{m} ∈ H^{(s)} for all k and m.

Now consider a term from the inductive step of the form L_{k}h_{k}T_{w_{ns,𝔟s}} P^{(s+1)}, where n_{s} < k ≤ 𝔟_{s−1} and h_{k} ∈ H_{k−1,k}. What we have just shown combined with Lemma 5.8(ii) shows that L_{k}h_{k}T_{w_{ns,𝔟s}} P^{(s+1)} is equal to a linear combination of terms of the form L_{m}^{c}h_{md}, where n_{s} < m ≤ 𝔟_{s−1}, 1 ≤ c ≤ c_{λ,s}(m) and h_{md} ∈ H_{m−1,m}. Moreover, c ≤ c_{λ,s}(m) with equality only if k = m > n_{s} + n_{s+1} and c = 1 = c_{λ,s}(m) if n_{s} < m ≤ n_{s} + n_{s+1}. Hence, L_{k}h_{k}T_{w_{ns,𝔟s}} P^{(s+1)} ∈ H^{(s)}.

Therefore, combining the last two paragraphs we have shown that

T_{w_{ns,𝔟s}} L_{1,𝔟_{s}}(Q_{s})P^{(s+1)} ≡ P^{(s)} (mod H^{(s)}).

By similar arguments, the terms L_{n_{s}+1,𝔟_{s−1}}(Q_{s})T_{w_{ns,𝔟s}} h^{(s+1)} and L_{k}h_{k}T_{w_{ns,𝔟s}} h^{(s+1)} from the inductive step also belong to H^{(s)}. We leave the details to the reader. □

We are now basically done. The next result essentially computes τ(z_{λ}Ψ_{𝔱_{λ}}^{∗}); we record it separately because it is also the key to showing that M^{λ} is a self–dual ℋ –module and that S̃^{λ′} is isomorphic to the dual of S^{λ}; see [24].

5.13. Proposition. Suppose that λ is a multipartition of n. Then

τ(z_{λ}T_{w_{λ}}^{∗}) = (−1)^{n(r−1)}q^{ℓ(w_{λ})} ∏_{s=1}^{r} Q_{s}^{n−|λ^{(s)}|}.

Proof. Let ũ_{λ′}^{−} = L_{𝔞_{2}+1,n}(Q_{1}) … L_{𝔞_{r}+1,n}(Q_{r−1}). Using (2.3)(i) and Lemma 5.12 shows that

τ(z_{λ}T_{w_{λ}}^{∗}) = τ(m_{λ}T_{w_{λ}}n_{λ′}T_{w_{λ}}^{∗}) = τ(x_{λ}u_{λ}^{+}T_{w_{λ}}u_{λ}^{−}y_{λ′}T_{w_{λ}}^{∗})
= τ(u_{λ}^{+}T_{w_{λ}}u_{λ}^{−}y_{λ′}T_{w_{λ}}^{∗}x_{λ})
= τ(u_{λ}^{+}ũ_{λ′}^{−}T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ}) + τ(u_{λ}^{+}hy_{λ′}T_{w_{λ}}^{∗}x_{λ}),

where h = ∑_{k=1}^{n} ∑_{c=1}^{c_{λ}(k)−1} L_{k}^{c}h_{kc} for some h_{kc} ∈ H_{k−1,n}. Consider u_{λ}^{+} as a polynomial in L_{k}. Then the degree of L_{k} in u_{λ}^{+} is r − c_{λ}(k). Therefore, if h_{kc} ≠ 0 then u_{λ}^{+}L_{k}^{c}h_{kc} is a polynomial in L_{k} which is left divisible by L_{k} and has degree at most r − 1. If m > k then L_{m} appears in u_{λ}^{+}L_{k}^{c}h_{kc} with exponent at most r − c_{λ}(m) < r. (If m < k then L_{m}^{d} can appear in u_{λ}^{+}L_{k}^{c}h_{kc} for d ≥ r; however, this does not matter because such terms can be written as a linear combination of Ariki–Koike basis elements in H_{k−1,k}.) Therefore, τ(u_{λ}^{+}L_{k}^{c}h_{kc}y_{λ′}T_{w_{λ}}^{∗}x_{λ}) = 0 by (2.3)(ii) and, consequently, τ(u_{λ}^{+}hy_{λ′}T_{w_{λ}}^{∗}x_{λ}) = 0; hence,

τ(z_{λ}T_{w_{λ}}^{∗}) = τ(u_{λ}^{+}ũ_{λ′}^{−}T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ}).

Considered as polynomials in L_{k}, u_{λ}^{+} has degree r − c_{λ}(k) and ũ_{λ′}^{−} has degree c_{λ}(k) − 1; consequently, each L_{k} has degree r − 1 in u_{λ}^{+}ũ_{λ′}^{−}. Therefore, by (2.3) again,

τ(z_{λ}T_{w_{λ}}^{∗}) = τ(u_{λ}^{+}ũ_{λ′}^{−}) · τ(T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ})

= ∏_{s=2}^{r}(−Q_{s})^{𝔞_{s}} · ∏_{s=1}^{r−1}(−Q_{s})^{𝔟_{s}} · τ(T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ})

= (−1)^{n(r−1)} ∏_{s=1}^{r} Q_{s}^{n−|λ^{(s)}|} · τ(T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ}),

since 𝔞_{s} + 𝔟_{s} = n − |λ^{(s)}| for all s (and 𝔞_{1} = 0 = 𝔟_{r}). To complete the proof recall that 𝔖_{λ} ∩ ^{w_{λ}}𝔖_{λ′} = {1} and that w_{λ} is a distinguished (𝔖_{λ}, 𝔖_{λ′})–double coset representative (consider the tableaux 𝔱^{λ} and 𝔱_{λ}), so

T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ} = ∑_{\begin{matrix}u∈𝔖_{λ′} \\ v∈𝔖_{λ} \end{matrix}}(−q)^{−ℓ(u)}T_{w_{λ}}T_{u}T_{w_{λ}}^{∗}T_{v} = ∑_{\begin{matrix}u∈𝔖_{λ′} \\ v∈𝔖_{λ} \end{matrix}}(−q)^{−ℓ(u)}T_{w_{λ}}T_{uw_{λ}^{−1}v}.

Therefore, τ(T_{w_{λ}}y_{λ′}T_{w_{λ}}^{∗}x_{λ}) = q^{ℓ(w_{λ})} by (2.3)(ii) (corresponding to u = v = 1), so the Proposition follows. □

We can now give our first formula for the Schur elements s_{λ}(q) of ℋ .

5.14. Corollary. Suppose that λ is a multipartition of n. Then

s_{λ}(q) = (−1)^{n(r−1)}q^{−ℓ(w_{λ})}γ_{𝔱_{λ}}γ′_{𝔱^{λ′}} ∏_{s=1}^{r} Q_{s}^{|λ^{(s)}|−n}.

Proof. By Lemma 2.6, s_{λ}(q) = 1/τ(F_{𝔱_{λ}}) and F_{𝔱^{λ}} = _{\frac{1}{γ_{𝔱_{λ}} γ′_{𝔱^{λ′}}}} z_{λ}Ψ_{𝔱_{λ}}^{∗} by Corollary 5.2;

therefore, s_{λ}(q) = γ_{𝔱_{λ}}γ′_{𝔱^{λ′}} /τ(z_{λ}Ψ_{𝔱_{λ}}^{∗}). Now, τ is a trace form by (2.3)(i), so

τ(z_{λ}Ψ_{𝔱_{λ}}^{∗}) = τ(x_{λ}u_{λ}^{+}T_{w_{λ}}u_{λ′}^{−}y_{λ′}Ψ_{𝔱_{λ}}^{∗}) = τ(u_{λ}^{+}T_{w_{λ}}u_{λ′}^{−}y_{λ′}Ψ_{𝔱_{λ}}^{∗}x_{λ}).

It is well–known and easy to check that the permutation w_{λ} has the “trivial intersection property”; that is, 𝔖_{λ} ∩^{w}𝔖_{λ′} ≠ {1} if and only if 𝔖_{λ}w𝔖_{λ′} = 𝔖_{λ}w_{λ}𝔖_{λ′}. Therefore, y_{λ′}T_{w}^{∗}x_{λ} ≠ 0 if and only if w ∈ 𝔖_{λ}w_{λ}𝔖_{λ′} (see, for example, [9, (4.9)]). Now, Ψ_{𝔱_{λ}}^{∗} = T_{w_{λ}}^{∗} + ∑_{w<w_{λ}} p_{𝔱_{λ}w}T_{w}^{∗} by Proposition 5.1(ii); so y_{λ′}Ψ_{𝔱_{λ}}^{∗}x_{λ} = y_{λ′}T_{w_{λ}}^{∗}x_{λ} since w_{λ} is the unique element of minimal length in 𝔖_{λ}w_{λ}𝔖_{λ′}. Therefore,

τ(z_{λ}Ψ_{𝔱_{λ}}^{∗}) = τ(u_{λ}^{+}T_{w_{λ}}u_{λ′}^{−}y_{λ′}T_{w_{λ}}^{∗}x_{λ′}) = τ(x_{λ}u_{λ}^{+}T_{w_{λ}}u_{λ′}^{−}y_{λ′}T_{w_{λ}}^{∗}) = τ(z_{λ}T_{w_{λ}}^{∗}).

The result now follows from Proposition 5.13. □

A closed formula for γ′_{𝔱^{λ′}} is given by (3.7)(i); therefore, in order to find an explicit formula for s_{λ}(q) we need only compute γ_{𝔱_{λ}}. Although the formula below looks formidable, its proof follows readily enough from the definition of γ_{𝔱_{λ}}.

Recall that the ijth hook in the diagram [λ^{(s)}] is the collection of nodes to the right of and below the node (i, j, s), including the node (i, j, s) itself. The ijth hook length h_{ij}^{λ^{(s)}} = λ_{i}^{(s)} + λ_{j}^{(s)′} − i − j + 1 is the number of nodes in the ijth hook and

the leg length, ℓ_{ij}^{λ^{(s)}} = λ_{j}^{(s)′} − j + 1, is the number of nodes in the “leg” of this hook. Observe that if (a, b, c) and (i, j, c) are two removable nodes in [λ^{(c)}] with a ≤ i and j ≤ b then h_{aj}^{λ^{(c)}} = b − a − j + i + 1.

5.15. Lemma. Suppose that λ is a multipartition of n. Then

γ_{𝔱_{λ}} = q^{ℓ(w_{λ})} ∏_{(i,j,s)∈[λ]}\frac{[h_{ij}^{λ^{(s)}}]_{q}}{[ℓ_{ij}^{λ^{(s)}}]_{q}} ∏_{t=s+1}^{r}(q^{j−i}Q_{s} − q^{λ_{1}^{(t)}}Q_{t}) ∏_{k=1}^{λ_{1}^{(t)}} \frac{(q^{j−i}Q_{s} − q^{k−1−λ_{k}^{(t)′}}Q_{t})}{(q^{j−i}Q_{s} − q^{k−λ_{k}^{(t)′}}Q_{t})}.

(Note that λ_{k}^{(t)′} is the length of the k^{th} column of [λ^{(t)}].)

Proof. We argue by induction on n. If n = 0, both sides are 1 and there is nothing to prove (by convention, empty products are 1). Suppose that n > 0. Let μ = Shape (𝔱_{λ}↓(n − 1)); then μ is a multipartition of n − 1. Recall that α(λ) = \frac{1}{2} ∑_{s=1}^{r} ∑_{i≥1}(λ_{i}^{(s)} − 1)λ_{i}^{(s)}. Applying the definitions (see (3.6)),

\frac{γ_{𝔱_{λ}}}{γ_{𝔱_{μ}}} = q^{ℓ(w_{λ})+α(λ)−ℓ(w_{μ})−α(μ)} \frac{∏_{x∈𝒜_{𝔱_{λ}} (n)} ( res_{𝔱_{λ}}(n) − res(x))}{∏_{y∈ℛ_{𝔱_{λ}} (n)} ( res_{𝔱_{λ}}(n) − res(y))}.

Assume that n appears in row a and column b of 𝔱_{λ}^{(c)}. First consider the contribution that the addable and removable nodes in [λ^{(c)}] make to γ_{𝔱_{λ}}. Looking at the definitions above (3.6), these nodes occur in pairs (x, y) where y < (a, b, c) is a removable node in row i and x is an addable node in row i + 1 for some i ≥ a. If x is in column d of [λ^{(c)}] and y is in column d′ then d ≤ d′ < b and

\frac{res_{𝔱_{λ}}(n) − res_{𝔱_{λ}}(x)}{res_{𝔱_{λ}}(n) − res_{𝔱_{λ}}(y)} = \frac{q^{b−a}Q_{c} − q^{d−(i+1)}Q_{c}}{q^{b−a}Q_{c} − q^{d′−i}Q_{c}} = ∏_{j=d}^{d′} \frac{q^{b−a} − q^{j−(i+1)}}{q^{b−a} − q^{j−i}}

= ∏_{j=d}^{d′} \frac{q^{j−i−1}(q^{b−a−j+i+1} − 1)}{q^{j−i}(q^{b−a−j+i} − 1)} = ∏_{j=d}^{d′} \frac{q^{−1}(q^{h_{aj}^{λ^{(c)}}} − 1)}{(q^{h_{aj}^{λ^{(c)}} −1} − 1)}

= ∏_{j=d}^{d′} \frac{q^{−1}[h_{aj}^{λ^{(c)}}]_{q}}{[h_{aj}^{λ^{(c)}} − 1]_{q}} .

Therefore,

\frac{∏_{(i,j,c)∈𝒜_{𝔱_{λ}} (n)} ( res_{𝔱_{λ}}(n) − res_{𝔱_{λ}}(i, j, c))}{∏_{(i,j,c)∈ℛ_{𝔱_{λ}} (n)} ( res_{𝔱_{λ}}(n) − res_{𝔱_{λ}}(i, j, c))} = ∏_{j=1}^{b−1} \frac{q^{−1}[h_{aj}^{λ^{(c)}}]_{q}}{[h_{aj}^{λ^{(c)}} − 1]_{q}} = q^{1−b} ∏_{j=1}^{b−1} \frac{[h_{aj}^{λ^{(c)}}]_{q}}{[h_{aj}^{μ^{(c)}}]_{q}} .

Note that α(λ) = α(μ) + b − 1; hence, by induction, this accounts for the factor q^{ℓ(w_{λ})} in Lemma 5.15.

Now γ_{𝔱_{μ}} is known by induction and it contains as a factor the left hand term

in the product below. Further, ℓ_{ij}^{μ^{(s)}} = ℓ_{ij}^{λ^{(s)}} and h_{ij}^{μ^{(s)}} = h_{ij}^{λ^{(s)}} if (i, s) ≠ (a, c), and

ℓ_{aj}^{μ^{(c)}} = ℓ_{aj}^{λ^{(c)}} for 1 ≤ j < b, so

( ∏_{(i,j,s)∈[μ]} \frac{[h_{ij}^{μ^{(s)}}]_{q}}{[ℓ_{ij}^{μ^{(s)}}]_{q}} )( ∏_{j=1}^{b−1} \frac{[h_{aj}^{λ^{(c)}}]_{q}}{[h_{aj}^{μ^{(c)}}]_{q}} ) = ( ∏_{\begin{matrix}(i,j,s)∈[λ] \\ (i,s)≠(a,c) \end{matrix}} \frac{[h_{ij}^{λ^{(s)}}]_{q}}{[ℓ_{ij}^{λ^{(s)}}]_{q}} )( ∏_{j=1}^{b−1} \frac{[h_{aj}^{λ^{(c)}}]_{q}}{[ℓ_{aj}^{λ^{(c)}}]_{q}} )

= ∏_{(i,j,s)∈[λ]} \frac{[h_{ij}^{λ^{(s)}}]_{q}}{[ℓ_{ij}^{λ^{(s)}}]_{q}} ,

since [h_{ab}^{λ^{(c)}}]_{q} = 1 = [ℓ_{ab}^{λ^{(c)}}]_{q}. This accounts for the left hand factor in the expression for γ_{𝔱_{λ}} given in the statement of the Lemma.

Finally, consider the nodes in 𝒜_{𝔱_{λ}}(n) and ℛ_{𝔱_{λ}}(n) which are in component t for some t > c (there are no such nodes for t < c). Again, almost all of the addable and removable nodes in component t occur in pairs placed in consecutive rows; however, this time there is also an additional addable node at the end of the first row of λ^{(t)}. As above, it is easier to insert extra factors which cancel out and so take a product over all of the columns of λ^{(t)}. An argument similar to that above shows that the nodes in 𝒜_{𝔱_{λ}}(n) and ℛ_{𝔱_{λ}}(n) which do not belong to component c contribute the factor

∏_{t=c+1}^{r}(q^{b−a}Q_{c} − q^{λ_{1}^{(t)}}Q_{t}) ∏_{k=1}^{λ_{1}^{(t)}} \frac{(q^{b−a}Q_{c} − q^{k−1−λ_{k}^{(t)′}}Q_{t})}{(q^{b−a}Q_{c} − q^{k−λ_{k}^{(t)′}}Q_{t})}

to γ_{𝔱_{λ}}. Using induction to combine the formulae above proves the Lemma. □

We can now give a closed formula for the Schur elements.

5.16. Corollary. Suppose that λ is a multipartition of n and for 1 ≤ s < t ≤ r let

X_{st}^{λ} = ∏_{(i,j)∈[λ^{(t)}]}(q^{j−i}Q_{t} − Q_{s}) · ∏_{(i,j)∈[λ^{(s)}]}(q^{j−i}Q_{s} − q^{λ_{1}^{(t)}}) ∏_{k=1}^{λ_{1}^{(t)}} \frac{(q^{j−i}Q_{s} − q^{k−1−λ_{k}^{(t)′}} Q_{t})}{(q^{j−i}Q_{s} − q^{k−λ_{k}^{(t)′}} Q_{t})}.

Then

s_{λ}(q) = (−1)^{n(r−1)}(Q_{1} … Q_{r})^{−n}q^{−α(λ′)} ∏_{s=1}^{r} ∏_{(i,j)=[λ^{(s)}]} Q_{s}[h_{ij}^{λ^{(s)}}]_{q} · ∏_{1≤s<t≤r} X_{st}^{λ}.

Proof. Looking at the definitions, [k]′_{q} = 1 + q^{−1} + ⋯ + q^{1−k} = q^{1−k}[k]_{q}; therefore, ([k]_{q}^{!})′ = q^{− \frac{1}{2} k(k−1)}[k]_{q}^{!}; consequently, ([λ′]_{q}^{!})′ = q^{−α(λ′)}[λ′]_{q}^{!}. Next observe that [λ′]_{q}^{!} = ∏_{(i,j,s)∈[λ]}[ℓ_{ij}^{λ^{(s)}}]_{q}. Further, (i, j) ∈ [λ^{(s)}] if and only if (j, i) ∈ [λ′^{(r−s+1)}].

Therefore, applying ′ to (3.7)(i) (and swapping the roles of s and t in the right hand factor),

γ′_{𝔱^{λ′}} = q^{−α(λ′)}( ∏_{(i,j,s)∈[λ]}[ℓ_{ij}^{λ^{(s)}}]_{q})( ∏_{1≤s<t≤r} ∏_{(i,j)∈[λ^{(t)}]}(q^{j−i}Q_{t} − Q_{s})).

By Corollary 5.14, s_{λ}(q) = (−1)^{n(r−1)}q^{−ℓ(w_{λ})}γ_{𝔱_{λ}}γ′_{𝔱^{λ′}} ∏_{s=1}^{r} Q_{s}^{|λ^{(s)}|−n}; so the result now follows by Lemma 5.15. □

5.17. Example. It is straightforward to check that Corollary 5.16 gives the same rational functions for the Schur elements s_{η_{t}}(q) as were obtained in Example 2.7.

It is not at all obvious that the formula in Corollary 5.16 for the Schur elements agrees with that conjectured by Malle, so we now show that this is the case. This takes quite a lot of work; however, it also results in a more symmetrical formula for s_{λ}(q). To do this we need to rewrite the Schur elements as functions of beta numbers (or first column hook lengths).

Define the length of a partition σ to be the smallest integer ℓ(σ) such that σ_{i} = 0 for all i > ℓ(σ). The length of a multipartition λ is ℓ(λ) = max { ℓ(λ^{(s)}) | 1 ≤ s ≤ r }.

Fix an integer L such that L ≥ ℓ(λ). The L–beta numbers for λ^{(s)} are the integers β_{i}^{(s)} = λ_{i}^{(s)} + L − i for i = 1, …, L; note that β_{1}^{(s)} > ⋯ > β_{L}^{(s)} ≥ 0. Malle calls the r × L matrix B = (β_{i}^{(s)})_{s,i} the L–symbol of λ. Actually, this is not quite one of Malle’s symbols; it is what Broué and Kim [6] call B an ordinary symbol. We also let B_{s} = {β_{1}^{(s)}, …, β_{L}^{(s)}} for s = 1, …, r.

If we change L to L + 1 the beta set B_{s} is shifted to {β_{1}^{(s)} + 1, …, β_{L}^{(s)} + 1, 0}. A function of beta numbers is invariant under beta shifts if it is unchanged by such transformations; equivalently, the function is independent of L provided that L is large enough. For example, the formula for s_{λ}(q) below is invariant under beta shifts since s_{λ}(q) does not depend on L.

5.18. Theorem. Suppose that λ is a multipartition of n with L–symbol B = (β_{i}^{(s)})_{s,i} such that L ≥ ℓ(λ). Then

s_{λ}(q) = (−1)^{a_{rL}}q^{b_{rL}} \frac{∏_{1≤s<t≤r}(Q_{s} − Q_{t})^{L} · ∏_{1≤s,t≤r} ∏_{α_{s}∈B_{s}} ∏_{1≤k≤α_{s}}(q^{k}Q_{s} − Q_{t})}{(q − 1)^{n}(Q_{1} … Q_{r})^{n} ∏_{1≤s≤t≤r} ∏_{\begin{matrix}(α_{s},α_{t})∈B_{s}×B_{t} \\ α_{s}>α_{t} if s=t \end{matrix}}(q^{α_{s}}Q_{s} − q^{α_{t}}Q_{t})},

where a_{rL} = n(r − 1) + \binom{r}{2}\binom{L}{2} and b_{rL} = \frac{rL(L−1)(2rL−r−3)}{12}.

Proof. Adopting the notation of Corollary 5.16,

s_{λ}(q) = (−1)^{n(r−1)}(Q_{1} … Q_{r})^{−n}q^{−α(λ′)} ∏_{s=1}^{r} ∏_{(i,j)=[λ^{(s)}]} Q_{s}[h_{ij}^{λ^{(s)}}]_{q} · ∏_{1≤s<t≤r} X_{st}^{λ}.

We consider each of these factors separately.

First, let σ be a partition of m with beta numbers (β_{1}, …, β_{L}), where L ≥ ℓ(σ). Then

∏_{(i,j)∈[σ]}[h_{ij}^{σ}]_{q} = (q − 1)^{−m} ∏_{(i,j)∈[σ]}(q^{h_{ij}^{σ}} − 1) = (q − 1)^{−m} \frac{∏_{i=1}^{L} ∏_{k=1}^{β_{i}}(q^{k} − 1)}{∏_{1≤i<j≤L}(q^{β_{i}−β_{j}} − 1)}.

This is quite well–known and is easily proved by first observing that the right hand side is invariant under beta shifts and then by arguing by induction on the number of columns.

Ignoring the leading term (q − 1)^{m} on the right hand side, the number of factors in the numerator is m + \binom{L}{2}, whereas the number of factors in the denominator is \binom{L}{2}. Consequently, multiplying the left hand side by Q^{m}, say, is the same as multiplying each factor, top and bottom, on the right hand side by Q. Therefore, returning to the multipartition λ,

q^{−α(λ′)} ∏_{s=1}^{r} ∏_{(i,j)∈[λ^{(s)}]} Q_{s}[h_{ij}^{λ^{(s)}}]_{q} = q^{N−α(λ′)}(q−1)^{−n} ∏_{s=1}^{r} \frac{∏_{i=1}^{L} ∏_{k=1}^{β_{i}^{(s)}}(q^{k}Q_{s} − Q_{s})}{∏_{1≤i<j≤L}(q^{β_{i}^{(s)}}Q_{s} − q^{β_{j}^{(s)}}Q_{s})},

where N = ∑_{s=1}^{r} ∑_{j=1}^{L}(j − 1)β_{j}^{(s)}. It is well–known (in the right circles) and easy

enough to check that α(λ′) = ∑_{s=1}^{r} ∑_{j=1}^{L}(j − 1)λ_{j}^{(s)}; therefore, N − α(λ′) = r\binom{L}{3}.

Fix s and t with 1 ≤ s < t ≤ r; there are \binom{r}{2} such choices. A quick calculation shows that b_{rL} = r\binom{L}{3} + \binom{r}{2}σ(L), where σ(L) = (2L − 1)L(L − 1)/6 = ∑_{j=1}^{L−1} j^{2}. Therefore, in order to complete the proof it is enough to show that X_{st}^{λ} = Y_{st}^{λ} where

Y_{𝔰𝔱}^{λ} = (−1)^{\binom{L}{2}}q^{σ(L)} \frac{(Q_{s} − Q_{t})^{L} ∏_{α_{s}∈B_{s}} ∏_{1≤k≤α_{s}}(q^{k}Q_{s} − Q_{t}) · ∏_{α_{t}∈B_{t}} ∏_{1≤k≤α_{t}}(q^{k}Q_{t} − Q_{s})}{∏_{(α_{s},α_{t})∈B_{s}×B_{t}}(q^{α_{s}}Q_{s} − q^{α_{t}}Q_{t})},

It is not hard to see that Y_{st}^{λ} is invariant under beta shifts so we may change L arbitrarily, provided that L ≥ max {ℓ(λ^{(s)}), ℓ(λ^{(t)})}. We prove our claim that X_{st}^{λ} = Y_{st}^{λ} by induction in three incremental steps. We start the induction by observing that both products are equal to 1 when λ^{(s)} = λ^{(t)} = (0) — to see this it is easiest to take L = 0.

Next consider the case where λ^{(t)} = (0). Since Y_{st}^{λ} is invariant under beta shifts we may assume that ℓ(λ^{(s)}) = L. Assume by way of induction that we have proved the claim for (λ_{1}^{(s)}, …, λ_{L−1}^{(s)}). Adding a non–empty L^{th} row to λ^{(s)} changes X_{st}^{λ} by the factor

∏_{k=1−L}^{λ_{L}^{(s)}−L}(q^{k}Q_{s} − Q_{t}) = ∏_{k=1}^{L−1} q^{−k}(Q_{s} − q^{k}Q_{t}) · (Q_{s} − Q_{t}) · ∏_{k=1}^{λ_{L}^{(s)}−L}(q^{k}Q_{s} − Q_{t})

= (−1)^{L−1} \frac{∏_{1≤k≤L−1} (q^{k}Q_{t} − Q_{s}) · (Q_{s} − Q_{t}) · ∏_{1≤k≤λ_{L}^{(s)}} (q^{k}Q_{s} − Q_{t})}{q^{\binom{L}{2}} ∏_{λ_{L}^{(s)}−L+1≤k≤λ_{L}^{(s)}}(q^{k}Q_{s} − Q_{t})}

= (−1)^{L−1} \frac{∏_{1≤k≤β_{L}^{(t)}} (q^{k}Q_{t} − Q_{s}) · (Q_{s} − Q_{t}) · ∏_{1≤k≤β_{L}^{(s)}} (q^{k}Q_{s} − Q_{t})}{∏_{α_{t}∈B_{t}}(q^{β_{L}^{(s)}}Q_{s} − q^{α_{t}}Q_{t})},

the last equality following because β_{L}^{(s)} = λ_{L}^{(s)} and β_{i}^{(t)} = L − i for i = 1, …, L. Notice that the (L − 1)–beta numbers for (λ_{1}^{(s)}, …, λ_{L−1}^{(s)}) all increase by 1 when we add the extra row λ_{L}^{(s)} to λ^{(s)}. Let B′_{s} be the set of (L − 1)–beta numbers for (λ_{1}^{(s)}, …, λ_{L−1}^{(s)}) and let B′_{t} = {0, 1, …, L−2} be the set of (L−1)–beta numbers for λ^{(t)}. Then

\begin{matrix}∏_{(α_{s},α_{t})∈B_{s}×B_{t}}(q^{α_{s}}Q_{s} − Q^{α_{t}}Q_{t}) = ∏_{1≤i≤L} ∏_{0≤k<L}(q^{β_{i}^{(s)}}Q_{s} − q^{k}Q_{t}) \\ = ∏_{α_{t}∈B_{t}}(q^{β_{L}^{(s)}}Q_{s} − q^{α_{t}}Q_{t}) · ∏_{1≤i<L}(q^{β_{i}^{(s)}}Q_{s} − Q_{t}) · ∏_{(α_{s},α_{t})∈B′_{s}×B′_{t}}q(q^{α_{s}}Q_{s} − q^{α_{t}}Q_{t}). \end{matrix}

This equation allows us to rewrite the denominator of the preceding equation and so see that the change in X_{st}^{λ} is the same as the change in Y_{st}^{λ} (in particular, the change of the scalar is (−1)^{L−1}q^{(L−1)^{2}} in both cases). This proves our claim when λ^{(t)} = (0).

The next step is to fix λ^{(s)} and assume that λ^{(t)} = (a) for some a ≥ 0. If λ^{(s)} = (0) in this case then it is straightforward to check the claim (or to modify the argument below), so assume that λ^{(s)} ≠ (0) and let L = ℓ(λ^{(s)}) ≥ 1 ≥ ℓ(λ^{(t)}). The case a = 0 we already understand. Next, changing λ^{(t)} from (a − 1) to (a) changes X_{st}^{λ} by the factor

(q^{a−1}Q_{t} − Q_{s}) ∏_{(i,j)∈[λ^{(s)}]} \frac{(q^{j−i}Q_{s} − q^{a}Q_{t})}{(q^{j−i}Q_{s} − q^{a−1}Q_{t})} \frac{(q^{j−i}Q_{s} − q^{a−2}Q_{t})}{(q^{j−i}Q_{s} − q^{a−1}Q_{t})}

= (q^{a−1}Q_{t} − Q_{s}) ∏_{(i,j)∈[λ^{(s)}]} \frac{(q^{a−1}Q_{t} − q^{j−i−1}Q_{s})(q^{j−i+1}Q_{s} − q^{a−1}Q_{t})}{(q^{j−i}Q_{s} − q^{a−1}Q_{t})(q^{a−1}Q_{t} − q^{j−i}Q_{s})}

= (q^{a−1}Q_{t} − Q_{s}) ∏_{i=1}^{L} \frac{(q^{a−1}Q_{t} − q^{−i}Q_{s})}{(q^{a−1}Q_{t} − q^{1−i}Q_{s})} \frac{(q^{λ_{i}^{(s)}−i+1}Q_{s} − q^{a−1}Q_{t})}{(q^{λ_{i}^{(s)}−i}Q_{s} − q^{a−1}Q_{t})}

= (q^{a−1}Q_{t} − q^{−L}Q_{s}) ∏_{i=1}^{L} \frac{(q^{β_{i}^{(s)}+1}Q_{s} − q^{β_{1}^{(t)}}Q_{t})}{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{1}^{(t)}}Q_{t})}

= (q^{β_{1}^{(t)}}Q_{t} − Q_{s}) ∏_{i=1}^{L} \frac{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{1}^{(t)}−1}Q_{t})}{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{1}^{(t)}}Q_{t})}.

This last product is exactly the change in Y_{st}^{λ} so we now know that X_{st}^{λ} = Y_{st}^{λ} when λ^{(t)} = (a).

Finally, suppose that λ^{(t)} has more than one row. For convenience, let l = ℓ(λ^{(s)}), m = ℓ(λ^{(t)}) > 1 and b = λ_{m}^{(t)} and assume that L ≥ l, m. When we add row m to λ^{(t)}, X_{st}^{λ} changes by the factor

∏_{j=1}^{b}(q^{j−m}Q_{t} − Q_{s}) · ∏_{(i,j)∈[λ^{(s)}]} ∏_{k=1}^{b} \frac{(q^{j−i}Q_{s} − q^{k−m−1}Q_{t})}{(q^{j−i}Q_{s} − q^{k−m}Q_{t})} \frac{(q^{j−i}Q_{s} − q^{k−m+1}Q_{t})}{(q^{j−i}Q_{s} − q^{k−m}Q_{t})}

= ∏_{j=1}^{b}(q^{j−m}Q_{t} − Q_{s}) · ∏_{(i,j)∈[λ^{(s)}]} ∏_{k=1}^{b} \frac{(q^{j−i+1}Q_{s} − q^{k−m}Q_{t})}{(q^{j−i}Q_{s} − q^{k−m}Q_{t})} \frac{(q^{k−m}Q_{t} − q^{j−i−1}Q_{s})}{(q^{k−m}Q_{t} − q^{j−i}Q_{s})}

= ∏_{j=1}^{b}(q^{j−m}Q_{t} − Q_{s}) · ∏_{i=1}^{l} ∏_{k=1}^{b} \frac{q(q^{λ_{i}^{(s)}−i}Q_{s} − q^{k−m−1}Q_{t})}{(q^{λ_{i}^{(s)}−i}Q_{s} − q^{k−m}Q_{t})} \frac{(q^{k−m}Q_{t} − q^{−i}Q_{s})}{(q^{k−m}Q_{t} − q^{1−i}Q_{s})}

= ∏_{j=1}^{b}(q^{j−m}Q_{t} − Q_{s}) · q^{lb} ∏_{i=1}^{l} \frac{(q^{λ_{i}^{(s)}−i}Q_{s} − q^{−m}Q_{t})}{(q^{λ_{i}^{(s)}−i}Q_{s} − q^{b−m}Q_{t})} · ∏_{k=1}^{b} \frac{(q^{k−m}Q_{t} − q^{−l}Q_{s})}{(q^{k−m}Q_{t} − Q_{s})}

= ∏_{k=1}^{b}(q^{k−m+l}Q_{t} − Q_{s}) · ∏_{i=1}^{l} \frac{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{m}^{(t)}−b}Q_{t})}{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{m}^{(t)}}Q_{t})}

= ∏_{k=L−m+1}^{β_{m}^{(t)}}(q^{k}Q_{t} − Q_{s}) · ∏_{i=1}^{L} \frac{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{m}^{(t)}−b}Q_{t})}{(q^{β_{i}^{(s)}}Q_{s} − q^{β_{m}^{(t)}}Q_{t})},

where the last line follows by a short calculation since β_{m}^{(t)} = b+L−m (and l ≤ L). As β_{m}^{(t)} is the only beta number that has changed, this factor is precisely the change in Y_{st}^{λ} when an extra row is added to λ^{(t)}.

We have now shown that X_{st}^{λ} = Y_{st}^{λ} in all cases, so the theorem is proved. □

Finally, comparing Theorem 5.18 with [6, Prop 3.17] we see that our formula for the Schur elements agrees with Malle’s — as it must because Malle’s conjecture has already been proved by Geck, Iancu and Malle [15]. Actually, there is still a small amount of work to be done in reconciling the two formulas because Broué and Kim [6] write the exponent of q as a sum of binomial coefficients; however, their expression simplifies to give b_{rL}.

Acknowledgements

I would like to thank the referee for many useful comments and corrections and Jean Michel for writing some Chevie code for computing the Schur elements.

References

[1] S. Ariki, On the semi–simplicity of the Hecke algebra of (ℤ/rℤ)≀𝔖_{n}, J. Algebra, 169 (1994), 216–225.

[2] , Representation theory of a Hecke algebra of G(r, p, n), J. Algebra, 177 (1995), 164– 185.

[3] , On the decomposition numbers of the Hecke algebra of G(m, 1, n), J. Math. Kyoto Univ., 36 (1996), 789–808.

[4] S. Ariki and K. Koike, A Hecke algebra of (Z/rZ) ≀ 𝔖_{n} and construction of its irreducible representations, Adv. Math., 106 (1994), 216–243.

[5] K. Bremke and G. Malle, Reduced words and a length function for G(e, 1, n), Indag. Math., 8 (1997), 453–469.

[6] M. Broué and S. Kim, Sur les familles de caractres des algbres de hecke cyclotomiques, preprint 2001.

[7] M. Broué and G. Malle, Zyklotomische Heckealgebren, Asterisque, 212 (1993), 119–189.

[8] R. W. Carter, Finite Groups of Lie Type, John Wiley, New York, 1985.

[9] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. L.M.S. (3), 52 (1986), 20–52.

[10] , Blocks and idempotents of Hecke algebras of general linear groups, Proc. L.M.S. (3), 54 (1987), 57–82.

[11] , Representations of Hecke algebras of type B_{n}, J. Algebra, 146 (1992), 454–481.

[12] R. Dipper, G. James, and A. Mathas, Cyclotomic q–Schur algebras, Math. Z., 229 (1999), 385–416.

[13] J. Du and H. Rui, Specht modules for Ariki–Koike algebras, Comm. Alg., 29 (2001), 4710– 4719.

[14] J. S. Frame, G. de. B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group, Canad. J. Math., 6 (1954), 316–324.

[15] M. Geck, L. Iancu, and G. Malle, Weights of Markov traces and generic degrees, Indag. Math., 11 (2000), 379–397.

[16] P. N. Hoefsmit, Representations of Hecke algebras of finite groups with BN–pairs of classical type, PhD thesis, University of British Columbia, 1979.

[17] G. D. James and A. Mathas, The Jantzen sum formula for cyclotomic q–Schur algebras, Trans. A.M.S., 352 (2000), 5381–5404.

[18] G. Lusztig, On a theorem of Benson and Curtis, J. Algebra, 71 (1981), 490–498.

[19] G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra, 177 (1995), 768–826.

[20] , On the generic degrees of cyclotomic algebras, Represent. Theory, 4 (2000), 342–369 (electronic).

[21] G. Malle and A. Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra, 205 (1998), 275–293.

[22] A. Mathas, Hecke algebras and Schur algebras of the symmetric group, Univ. Lecture Notes, 15, A.M.S., 1999.

[23] , Murphy operators and the centre of the Iwahori–Hecke algebras of type A, J. Alg. Comb., 9 (1999), 295–313.

[24] , Tilting modules for cyclotomic Schur algebras, in preparation 2001.

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

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