Proceedings of Symposia in Pure Mathematics Volume 00, 19xx

Simple modules of Ariki–Koike algebras

Andrew Mathas

Abstract. In this note we classify the simple modules of the Ariki–Koike algebras when q = 1 and also describe the classification for those algebras considered in [2, 15], together with the underlying computation of the canonical bases of the affine quantum group U(ŝl_{e}).

1. Introduction

The Ariki–Koike algebras were introduced by Michel Broué and Gunter Malle [4] in their paper on cyclotomic Hecke algebras; in part, they were motivated by the conjectures (in the case of Lie type groups) which are the focus of Broué’s paper in this proceedings. In the semisimple case, the representation theory of these algebras was first studied by Ariki and Koike [3]. (Actually, this not completely accurate because the Iwahori–Hecke algebras of types A and B are special cases of the Ariki–Koike algebras and their generic representation theory was first investigated by Hoefsmit [10].)

In this paper we are concerned with the representation theory of the non– semisimple Ariki–Koike algebras. Dipper and James [5] were the first to construct and classify the irreducible modules of the Iwahori–Hecke algebras of type A. Dipper, James and Murphy [6] extended these results to the Iwahori–Hecke algebras of type B; however, they were unable to give a labelling of the simple modules. Using slightly different methods Graham and Lehrer [7] studied the general case (ie. r > 2); the techniques of [6] can also be used here [18, 15].

Dipper, James and Murphy [6] gave a conjecture for classifying the simple modules of the Iwahori–Hecke algebras of type B and they proved that this condition was sufficient; Graham and Lehrer [7] extended these results to the general case. In [15] we proved the Dipper–James–Murphy conjecture for all r, with some restrictions on the parameters defining the Ariki–Koike algebras. Our proof relies in a crucial way upon a deep result of Ariki [2] which allowed us to give an algorithm for computing the decomposition matrices of the Ariki–Koike algebras defined over fields of characteristic zero. This algorithm is very much in the spirit of the work of Lascoux, Leclerc and Thibon [13]; it first involves computing the canonical basis of a certain integral highest weight module of an affine quantum group.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

0082-0717/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. Primary 16G99; Secondary 17B37.

Supported in part by SERC grant GR/J37690.

In this note we classify the simple modules of the Ariki–Koike algebras when q = 1, a case not covered by the results of [15]. We also describe the results of [15] and prove several other new results along the way.

2. Ariki–Koike algebras

Throughout this paper we fix non–negative integers n ≥ 0 and r ≥ 1.

Let 𝔽 be a field, q an invertible element of 𝔽, and u_{1}, u_{2}, …, u_{r} arbitrary elements of 𝔽. The Ariki–Koike algebra ℋ = ℋ_{n,r} = ℋ_{𝔽,q;u_{1},... ,u_{r}} is the unital associative 𝔽–algebra with generators T_{0}, T_{1}, …, T_{n−1} and relations

(T_{0} − u_{1}) ⋯ (T_{0} − u_{r}) = 0,  T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},
(T_{i} + 1)(T_{i} − q) = 0,  T_{i+1}T_{i}T_{i+1} = T_{i}T_{i+1}T_{i},
T_{i}T_{j} = T_{j}T_{i} if |i − j| > 1,

where 1 ≤ i < n and 0 ≤ j < n.

Let 𝔖_{n} be the symmetric group on {1, 2, …, n} and let s_{i} = (i, i+1), 1 ≤ i < n, be the standard Coxeter generators of 𝔖_{n}. If w ∈ 𝔖_{n} then a word w = s_{i_{1}} … s_{i_{k}} for w is reduced if k is minimal; in this case we say that w has length k and write ℓ(w) = k. Given a reduced expression s_{i_{1}} … s_{i_{k}} for w ∈ 𝔖_{n} let T_{w} = T_{i_{1}} … T_{i_{k}}; the relations in ℋ ensure that T_{w} is independent of the choice of reduced expression. Thus, the span of {T_{w}}_{w∈𝔖_{n}} is a homomorphic image of the Iwahori–Hecke algebra ℋ (𝔖_{n}) of the symmetric group.

Define elements L_{1}, L_{2}, …, L_{n} by L_{m} = q^{1−m}T_{m−1} … T_{1}T_{0}T_{1} … T_{m−1}. Ariki and Koike [3] showed that

{ L_{1}^{a_{1}}L_{2}^{a_{2}} … L_{n}^{a_{r}}T_{w} : w ∈ 𝔖_{n}, and 0 ≤ a_{m} < r for m = 1, 2, …, n }

is a basis for ℋ . In particular, ℋ has dimension r^{n}n! over 𝔽 and ℋ (𝔖_{n}) is a subalgebra of ℋ .

2.1. Specht modules. A partition of n is a sequence σ = (σ_{1}, σ_{2}, …) of non–negative integers such that |σ| = ∑_{i≥1} σ_{i} = n and σ_{1} ≥ σ_{2} ≥ ⋯ ≥ 0. A multipartition of n (with r components) is an r–tuple of partitions λ = (λ^{(1)}, …, λ^{(r)}) such that ∑_{s=1}^{r} |λ^{(s)}| = n; we write λ ⊢ n. For each multipartition λ there exists a right ℋ –module S(λ), called a Specht module, which we now describe.

A node of a multipartition λ = (λ^{(1)}, …, λ^{(r)}) is a triple γ = (j, k)_{s} where 1 ≤ k ≤ λ_{j}^{(s)} and 1 ≤ s ≤ r; the diagram of λ is the set of all such nodes. Informally, the diagram of λ can be thought of as an r–tuple of boxes in the plane. For example, if λ = ((3, 2), (2^{2}, 1), (2, 1^{2})) then the diagram of λ is

(  ,  ,  ).

We make no distinction between a multipartition and its diagram.

A λ–tableau is a map from (the diagram of) λ into {1, 2, …, n}; so a λ–tableau corresponds to a labelled diagram.

Example 2.1. Let λ = ((3, 1), (2, 1)). Then two λ–tableaux are

𝔰 = ( 2 3 7 5 , 4 6 1 ) and 𝔱 = ( 2 3 6 4 , 1 7 5 ).

A λ–tableau 𝔱 is row standard if the entries in each row of 𝔱 are in increasing order; 𝔱 is standard if it is row standard and the entries in each column are also in increasing order. For example, in (2.1), 𝔰 is row standard, but not standard, and 𝔱 is standard. Let tab(λ) be the set of standard λ–tableaux.

The symmetric group 𝔖_{n} acts from the right on the set of bijective λ–tableaux. Let 𝔱^{λ} be the λ–tableau with the numbers 1, 2, …, n entered in order along the rows of λ^{(1)}, and then the rows of λ^{(2)} and so on. Then 𝔖_{λ} = 𝔖_{λ^{(1)}} × 𝔖_{λ^{(2)}} × ⋯ × 𝔖_{λ^{(r)}} is the row stabilizer of 𝔱^{λ} and the right cosets of 𝔖_{λ} in 𝔖_{n} correspond to the row standard λ–tableaux. More precisely, if 𝔰 is a row standard tableau let d(𝔰) be the unique element of 𝔖_{n} such that 𝔰 = 𝔱^{λ}d(𝔰). As in [5, Lemma 1.1] it is easy to see that d(𝔰) is a distinguished coset representative for 𝔖_{λ} in 𝔖_{n} (ie. d(𝔰) is the unique element of minimal length in its coset).

Example 2.2. Continuing Example 2.1 above, 𝔱^{λ} = ( 1 2 3 4 , 5 6 7 ),

d(𝔰) = (1, 2, 3, 7)(4, 5), and d(𝔱) = (1, 2, 3, 6, 7, 5).

Let |λ|_{s} = |λ^{(1)}| + ⋯ + |λ^{(s−1)}|, where 1 ≤ s ≤ r. The dominance ordering on the set of multipartitions is the partial order ⊵ where λ ⊵ μ if

|λ|_{s} + ∑_{j=1}^{k} λ_{j}^{(s)} ≥ |μ|_{s} + ∑_{j=k}^{k} μ_{j}^{(s)} for all 1 ≤ s ≤ r and k ≥ 1.

We write λ ▷ μ if λ ⊵ μ and λ ≠ μ. For example, when r = n = 2 we have ((2), (0)) ▷ ((1^{2}), (0)) ▷ ((1), (1)) ▷ ((0), (2)) ▷ ((0), (1^{2})).

Finally, define ∗: ℋ ⟶ℋ to be the linear anti–isomorphism of ℋ determined by T_{i}^{∗} = T_{i} for all 0 ≤ i < n. In particular, T_{w}^{∗} = T_{w^{−1}} for all w ∈ 𝔖_{n}.

Definition 2.3. Suppose that λ = (λ^{(1)}, …, λ^{(r)}) is a multipartition.

(i) S_{λ} = ( ∏_{s=2}^{r} ∏_{m=1}^{|λ|_{s}}(L_{m} − u_{s}))( ∑_{w∈𝔖_{λ}} T_{w});

(ii) If 𝔰 and 𝔱 are standard λ–tableaux let S_{𝔰𝔱}^{λ} = T_{d(𝔰)}^{∗}S_{λ}T_{d(𝔱)}.

(iii) Let ℋ_{λ} be the span of the S_{𝔲𝔳}^{μ} where μ ▷ λ and 𝔲, 𝔳 ∈ tab(μ).

The following theorem is proved in [17] and [6] when r = 1 and r = 2 respectfully (part (i) is only implicit). When r > 2 an equivalent result was first proved in [7]; the result below can be proved by extending the argument of [6, Theorem 4.14]. To prove the theorem it suffices to prove (i) and then apply the general theory of cellular algebras [7].

Theorem 2.4. [18, 15]

(i) The set { S_{𝔰𝔱}^{λ} : λ a multipartition of n; 𝔰, 𝔱 ∈ tab(λ) } is a (cellular) basis of ℋ . In particular, ℋ_{λ} is a two–sided ideal.

(ii) Let S(λ) = (ℋ_{λ} + S_{𝔲𝔳}^{λ})ℋ , for any standard tableaux 𝔲, 𝔳 ∈ tab(λ). Then { ℋ_{λ} +S_{𝔲𝔱}^{λ} : 𝔱 ∈ tab(λ) } is a basis of S(λ) and the isomorphism type of S(λ) depends only on λ.

(iii) Let D(μ) = S(μ)/ rad S(μ). Then D(μ) is absolutely irreducible or zero. Moreover, { D(μ) : D(μ) ≠ 0 } is a complete set of non–isomorphic irreducible ℋ –modules.

(iv) If D(μ) is non–zero, let d_{λμ} = [S(λ) : D(μ)] be multiplicity of D(μ) as a composition factor of S(λ). Then d_{μμ} = 1 and d_{λμ} = 0 unless λ ⊵ μ.

The module S(λ) is called a Specht module of ℋ ; in (iii), rad S(λ) is the radical of a bilinear form on S(λ); see section 3. Note that although (iii) tells us how to construct all of the simple ℋ –modules it does not tell us which of the D(μ) are non–zero. We turn to this question next.

3. The case q = 1

In this section we classify the simple modules of the Ariki–Koike algebras when r ≥ 2 and q = 1, generalizing the result of Dipper, James and Murphy [6, Theorem 7.3] when r = 2 and u_{1} ≠ 0. Throughout this section we work over an arbitrary field 𝔽.

Let λ = (λ^{(1)}, …, λ^{(r)}) be a multipartition of n. By Theorem 2.4, the Specht module S(λ) is isomorphic to the right ℋ –module (ℋ_{λ} + S_{λ})ℋ , and S(λ) has basis {S_{𝔱}^{λ} := ℋ_{λ} + S_{λ}T_{d(𝔱)}} where 𝔱 runs over the set of standard λ–tableaux (so we have taken 𝔳 = 𝔱^{λ} in Theorem 2.4(ii)).

The cellular algebra structure of ℋ , or the arguments of [6], show that there is an inner product on S(λ) where ⟨S_{𝔰}^{λ}, S_{𝔱}^{λ}⟩ is determined by

S_{λ}T_{d(𝔰)}T_{d(𝔱)}^{∗}S_{λ} = ⟨S_{𝔰}^{λ}, S_{𝔱}^{λ}⟩S_{λ} (mod ℋ_{λ}),

for 𝔰, 𝔱 ∈ tab(λ). Moreover, ⟨xh, y⟩ = ⟨x, yh^{∗}⟩ for all h ∈ ℋ and x, y ∈ S(λ). So the radical

rad S(λ) = { x ∈ S(λ) : ⟨x, y⟩ = 0 for all y ∈ S(λ) }.

of this bilinear form is an ℋ –module. Moreover, by Theorem 2.4(iii), D(λ) = S(λ)/ rad S(λ) is either zero or absolutely irreducible. Further, we see that D(λ) is non–zero if and only if there exists some w ∈ 𝔖_{n} such that S_{λ}T_{w}S_{λ} ∉ ℋ_{λ}.

Let T_{λ} = ∑_{w∈𝔖_{λ}} T_{w} and L_{λ} = ∏_{s=2}^{r} ∏_{m=1}^{|λ|_{s}}(L_{m} − u_{s}). Then S_{λ} = L_{λ}T_{λ}. Let |λ| be the composition (|λ^{(1)}|, |λ^{(2)}|, …, |λ^{(r)}|), and 𝔖_{|λ|} the obvious sub-

group of 𝔖_{n}. Then L_{λ} depends only on |λ| and if w ∈ 𝔖_{|λ|} then T_{w} and L_{λ} commute (this is a straightforward computation using the relations in ℋ ). In particular, S_{λ} = L_{λ}T_{λ} = T_{λ}L_{λ}.

Define e to be the smallest positive integer such that 1 + q + ⋯ + q^{e−1} = 0; if no such integer exists set e = ∞. Thus, either q is a primitive eth root of unity (q is transcendental if e = ∞), or q = 1 and 𝔽 is a field of characteristic e.

Definition 3.1. A partition σ = (σ_{1}, σ_{2}, …) is e–restricted if σ_{i} − σ_{i+1} < e for all i ≥ 1. In particular, all partitions are ∞–restricted.

Lemma 3.2 (cf. [6, Lemma 6.8]). Suppose that λ = (λ^{(1)}, …, λ^{(r)}) is a multipartition of n. Then T_{λ}T_{w}S_{λ} ≠ 0 (mod ℋ_{λ}) for some w ∈ 𝔖_{|λ|} if and only if each partition λ^{(s)} is e–restricted for all 1 ≤ s ≤ r.

Proof. Write 𝔖_{|λ|} = 𝔖_{|λ^{(1)}|} × 𝔖_{|λ^{(2)}|} × ⋯ × 𝔖_{|λ^{(r)}|}; then any element w of 𝔖_{|λ|} can be written as w = w_{1}w_{2} … w_{r} where all of these elements commute and w_{s} ∈ 𝔖_{|λ^{(s)}|} for 1 ≤ s ≤ r. By Theorem 2.4(i), or directly, T_{λ}T_{w}S_{λ} ∉ ℋ_{λ} if and only if T_{λ}T_{w}T_{λ} ∉ ℋ (𝔖_{n})T_{λ}ℋ (𝔖_{n}). Furthermore, using the obvious (abuse of) notation,

T_{λ}T_{w}T_{λ} = (T_{λ^{(1)}}T_{w_{1}}T_{λ^{(1)}})(T_{λ^{(2)}}T_{w_{2}}T_{λ^{(2)}}) … (T_{λ^{(r)}}T_{w_{r}}T_{λ^{(r)}}),

and all of these factors commute. Therefore, T_{λ}T_{w}S_{λ} ∉ ℋ_{λ} if and only if

T_{λ^{(s)}}T_{w_{s}}T_{λ^{(s)}} ∉ ℋ (𝔖_{|λ^{(s)}|})T_{λ^{(s)}}ℋ (𝔖_{|λ^{(s)}|}) for all 1 ≤ s ≤ r.

This completes the proof because Dipper and James [5, Theorem 7.7] showed that such a w_{s} ∈ 𝔖_{|λ^{(s)}|} exists if and only if λ^{(s)} is e–restricted.

Henceforth, we assume that q = 1. Consequently, T_{w}T_{v} = T_{wv} for any two elements w and v in 𝔖_{n}.

Lemma 3.3. Suppose that q = 1 and that m appears in the sth diagram of 𝔱^{λ} where 1 ≤ m ≤ n. Then S_{λ}L_{m} = u_{s}S_{λ} (mod ℋ_{λ}).

Proof. This is a special case of [15, Theorem 1.13] where S_{𝔰𝔱}^{λ}L_{m} is computed for arbitrary 𝔰, 𝔱 ∈ tab(λ). (Actually, this theorem is stated for the case q ≠ 1; however, as noted in [15], the argument applies more generally.)

Example 3.4. If λ = ((3, 1), (2, 1)), as in Example 2.2, then modulo ℋ_{λ}

S_{λ}L_{m} = \begin{cases}u_{1}S_{λ}, & if 1 ≤ m ≤ 4, \\ u_{2}S_{λ}, & if 5 ≤ m ≤ 7. \end{cases}

Applying Lemma 3.3 to the definition of L_{λ} yields.

Corollary 3.5. Suppose that q = 1 and that λ is a multipartition of n. Then

L_{λ}S_{λ} = ∏_{1≤s<t≤r}(u_{s} − u_{t})^{|λ^{(s)}|}S_{λ} (mod ℋ_{λ}).

Lemma 3.6. Suppose that q = 1 and that S_{λ}T_{w}S_{λ} ≠ 0 (mod ℋ_{λ}) for some w ∈ 𝔖_{n}. Then w ∈ 𝔖_{|λ|}.

Proof. Let w ∈ 𝔖_{n} and let L_{λ,s} = ∏_{m=1}^{a_{s}}(L_{m} − u_{s}) where a_{s} = |λ^{(s)}| and 2 ≤ s ≤ r. Then L_{λ} = L_{λ,2} … L_{λ,r} and L_{λ,s} commutes with T_{w} whenever w ∈ 𝔖_{(a_{s},n−a_{s})}. Because of Lemma 3.3, the argument of [6, Lemma 7.2] applies; consequently, if w ∈ 𝔖_{n} and S_{λ}T_{w}S_{λ} ≠ 0 (mod ℋ_{λ}) then w ∈ 𝔖_{(a_{s},n−a_{s})}. (Lemma 7.2 of [6] is essentially the case r = 2 of this lemma.) Therefore, if S_{λ}T_{w}S_{λ} ≠ 0 (mod ℋ_{λ}) then w ∈ ⋂_{s=2}^{r} 𝔖_{(a_{s},n−a_{s})} = 𝔖_{|λ|}.

Theorem 3.7. Let 𝔽 be an arbitrary field and suppose that q = 1 and let λ = (λ^{(1)}, …, λ^{(r)}) be a multipartition of n. Then the ℋ –module D(λ) is non– zero if and only if

(i) each partition λ^{(s)} is e–restricted for all 1 ≤ s ≤ r, and (ii) λ^{(s)} = ∅ whenever u_{s} = u_{t} and 1 ≤ s < t ≤ r.

Proof. By the previous lemma and the remarks above, D(λ) is non–zero if and only if there exists a w ∈ 𝔖_{|λ|} such that S_{λ}T_{w}S_{λ} ≠ 0 (mod ℋ_{λ}). If w ∈ 𝔖_{|λ|} then T_{w} and L_{λ} commute, so

S_{λ}T_{w}S_{λ} = T_{λ}T_{w}L_{λ}S_{λ} = ( ∏_{1≤s<t≤r}(u_{s} − u_{t})^{|λ^{(s)}|})T_{λ}T_{w}S_{λ} (mod ℋ_{λ}),

by Corollary 3.5. Now apply Lemma 3.2.

When r = 2 it is not immediately apparent that our result agrees with that of Dipper, James and Murphy [6, Theorem 7.3]; however, these results are reconciled by the observation that if u_{1} ≠ 0 then we may assume that u_{1} = −1 by replacing T_{0} with −u_{1}^{−1}T_{0} (and therefore u_{s} with −u_{s}u_{1}^{−1} for all 1 ≤ s ≤ r).

4. e–Restricted multipartitions

For the remainder of this paper we assume that q ≠ 1 and define e ≠ 1 to be the smallest non–negative integer such that q^{e} = 1; if no such integer exists we set e = ∞. We further assume that the parameters {u_{s}} defining ℋ are given by

(4.1) u_{s} = q^{e_{s}} where e_{1} ≤ e_{2} ≤ ⋯ ≤ e_{r} and e_{s} ∈ ℤ/eℤ for all 1 ≤ s ≤ r.

(Here we adopt the convention that ℤ/eℤ = ℤ when e = ∞.) We write e = (e; e_{1}, e_{2}, …, e_{r}) and let ℋ_{𝔽,e} = ℋ_{𝔽,q;q^{e_{1}} ,...,q^{er}}. Ariki [1] has shown that under the assumptions of (4.1) the Ariki–Koike algebra ℋ_{𝔽,e} is not semisimple.

Definition 4.2. [15] Let λ = (λ^{(1)}, …, λ^{(r)}) be a multipartition and let γ = (j, k)_{s} be a node in λ. The e–residue of γ is res_{e}(γ) = k − j + e_{s}. If res_{e}(γ) = i then we call γ an i–node.

Example 4.3. If e = (3; 0, 0, 1) and λ = ((2, 1), (1^{2}), (3, 2)) then the e–residues in λ are as follows

( 0 1 2 , 0 2 , \begin{matrix}1 & 2 & 0 \\ 0 & 1 \end{matrix} ).

So the e–residue of the (1, 1)_{s}th node is e_{s} and, modulo e, the entries increase by 1 along the rows and by e − 1 down the columns.

Remark 4.4. Let λ be a multipartition and 𝔰 and 𝔱 two λ–tableaux. Then, for m = 1, 2, …, n and working modulo bigger terms, S_{𝔰𝔱}^{λ}L_{m} = q^{res_{e}(𝔱^{−1}(m))}S_{𝔰𝔱}^{λ}; note that 𝔱^{−1}(m) is a node in λ. This is the origin of the definition of e–residue; see [15, Theorem 1.13] for more details.

If γ is an i–node then we say that

(i) γ is a removable i–node of λ if λ \ {γ} is a multipartition (and γ ∈ λ).

(ii) γ is an addable i–node of λ if λ ∪ {γ} is a multipartition (and γ ∉ λ). Here, and below, we implicitly assume that γ ∉ λ (resp. γ ∈ λ) whenever we write λ ∪ {γ} (resp. λ \ {γ}).

Given two nodes γ = (j, k)_{s} and γ′ = (l, m)_{t}, we say that γ is above γ′, or γ′ is below γ, if either s < t, or s = t and k < m. The picture to keep in mind is that λ^{(s)} is above λ^{(t)} whenever 1 ≤ s < t ≤ r.

Definition 4.5 ([15]; cf. [12, 13]). Let λ and μ be multipartitions and suppose that μ = λ ∪ {γ} where γ is a removable i–node in μ. Then

(i) N_{i}^{l}(γ) = #{γ′| γ′ an addable i–node of λ strictly below γ } − #{γ′| γ′ a removable i–node of λ strictly below γ }.

(ii) A removable i–node η of μ is normal if N_{i}^{l}(η) ≤ 0 and N_{i}^{l}(η) < N_{i}^{l}(γ) for all

removable i–nodes γ which are below η. (iii) A normal i–node η of μ is good if N_{i}^{l}(η) ≤ N_{i}^{l}(γ) for all normal i–nodes γ.

A good i–node, if it exists, is the highest i–node η such that the number of removable i–nodes strictly below η is greater than the number of addable i–nodes strictly below η. We give some examples after Definition 4.7.

We shall need the following simple property of normal nodes.

Lemma 4.6. Let λ be a multipartition and suppose that η is a normal i–node and that γ is an addable i–node in λ which is below η. Then there exists a removable i–node γ′ which is below η and above γ.

Proof. Suppose that no such γ′ exists. Then either η is the lowest removable i–node and N_{i}^{l}(η) ≥ 1, or N_{i}^{l}(η) > N_{i}^{l}(η′) where η′ is the highest removable i–node below η. Both alternatives contradict the assumption that η is normal.

Here, and below, we write ∅_{e} = ((0), …, (0)) for the multipartition of zero with r components.

Definition 4.7. [15] A multipartition μ is e–restricted if either μ = ∅_{e} or μ = λ ∪ {η} where λ is an e–restricted multipartition and η is a good node in μ.

If μ is e–restricted then, by removing good nodes, we get a chain of e–restricted multipartitions from μ to the empty multipartition ∅_{e}. For example, if e = (3; 0, 2),

((2, 1), (1^{2})) ⟶^{0} ((2, 1), (1)) ⟶^{2} ((2), (1)) ⟶^{2} ((2), (0)) ⟶^{1} ((1), (0)) ⟶^{0} ∅_{e};

here a superscript i on an arrow indicates that a good i–node has been removed from the previous multipartition. This chain corresponds to a path in the crystal graph of a certain highest weight module; one consequence of this is that there is at most one addable i–node η in λ such that η is a node good of λ ∪ {η} [15].

In [15, Prop. 4.1] several equivalent conditions are given for a multipartition to be e–restricted (one of which is used as the definition in that paper). In particular, we show that a multipartition μ is e–restricted if and only if it satisfies the condition conjectured in [6, 7] to be both necessary and sufficient to ensure that D(μ) is non– zero. One of the main results of [15] is the following:

Theorem 4.8 ([15, Theorem 1.17]). Suppose that 𝔽 is an arbitrary field and that q ≠ 1. Let e = (e; e_{1}, …, e_{r}) as above. Then

{ D(μ) : μ an e–restricted multipartition of n }

is a complete set of non–isomorphic irreducible ℋ_{𝔽,e}–modules.

Remark 4.9. When r = 1, if ℋ is a not semisimple then either q = 1 or ℋ is of the form ℋ_{𝔽,e} for some e [1]. When r ≥ 2 there exist non–semisimple Ariki–Koike algebras with q ≠ 1 which are not of the form ℋ_{𝔽,e}.

We will say something of the proof of Theorem 4.8 in the next section. For arbitrary ℋ , it can be shown using Lemma 3.2 (cf. [6, Theorem 6.9]), that any multipartition μ such that D(μ) ≠ 0 satisfies the conclusion of the next proposition.

Proposition 4.10. Suppose that μ = (μ^{(1)}, …, μ^{(r)}) is an e–restricted multipartition. Then each partition μ^{(s)} is e–restricted for 1 ≤ s ≤ r,

Proof. If μ = ∅_{e} there is nothing to prove, so suppose that μ = λ ∪ {η} where λ is e–restricted and η = (j, k)_{s} is good. By induction the only way in which the proposition can fail for μ is if μ^{(s)} is not e–restricted, in which case μ_{j}^{(s)} − μ_{j+1}^{(s)} = e. If this is true then the (j + 1)th row of μ^{(s)} ends in the addable node γ = (j + 1, k − e + 1)_{s}. However, η and γ have the same e–residue, which is impossible by Lemma 4.6.

When e = 2 there is a simple combinatorial criterion for a multipartition to be e–restricted.

Proposition 4.11. Suppose that e = 2 or, equivalently, q = −1. Then a multipartition μ = (μ^{(1)}, …, μ^{(r)}) is e–restricted if and only if

(i) each partition μ^{(s)} is 2–restricted for all 1 ≤ s ≤ r, and (ii) (μ^{(s)})′_{1} + e_{s} ≤ μ_{1}^{(s+1)} + e_{s+1} for 1 ≤ s < r; here (μ^{(s)})′ denotes the partition

which is conjugate to μ^{(s)}.

Proof. First suppose that μ is e–restricted. By the previous proposition μ satisfies (i). If μ = ∅_{e} then (ii) certainly holds so suppose that μ = λ∪{η} where η = (j, k)_{s} is good. By induction, (ii) is false only if (μ^{(s)})′_{1}+e_{s} = μ_{1}^{(s+1)}+e_{s+1}+1. Thus, s < r and η is at the bottom of the first column of μ^{(s)}; so k = 1 and j = (μ^{(s)})′_{1}. Therefore, the first row of μ^{(s+1)} ends in an addable node γ = (1, μ_{1}^{(s)} +1)_{s+1} which has the same e–residue as η; this contradicts Lemma 4.6.

Conversely, suppose that μ ▷ ∅_{e} satisfies both (i) and (ii). Let γ be the lowest removable node in μ. Because of (i), γ must be at the bottom of the first column of μ^{(s)}. Now if s = r then γ is normal. If s < r then μ^{(s+1)} = ∅ so by (ii), μ^{(s)} = (1) and 0 = e_{s} < e_{s+1} = 1; so γ = (1, 1)_{s} is again normal. Since μ has a normal node, it also has a good node η. Let λ = μ \ {η}; then λ satisfies both (i) and (ii) because otherwise there would be a higher normal node in μ. By induction λ is e–restricted, so μ is e–restricted as required.

Example 4.12. To highlight some of the difficulties of extending Proposition 4.11 to the case where e ≠ 2 let e = (3; 0, 0). Then the multipartitions

( 0 2 , 0 2 ), ( 0 1 2 , 0 1 2 2 ), and ( 0 2 , 0 1 2 1 )

are all e–restricted, whereas the following multipartitions

( 0 2 , 0 ), ( 0 1 2 , 0 2 ), and ( 0 2 , 0 1 2 )

are not.

5. Decomposition numbers in characteristic zero

As in (4.1) we let e = (e; e_{1}, …, e_{r}) where e ≠ 1 is a non–negative integer and e_{1} ≤ ⋯ ≤ e_{r} are elements of ℤ/eℤ. In this section we describe an algorithm for computing the decomposition matrices of the algebra ℋ_{ℂ,e}.

5.1. The affine quantum group U(ŝl_{e}). Let 𝔥 be the free ℤ–module with basis { h_{i}, d : i ∈ ℤ/eℤ } and let { Λ_{i}, δ : i ∈ ℤ/eℤ } be the dual basis of 𝔥^{∗}, with ⟨ , ⟩: 𝔥^{∗} × 𝔥⟶ℤ the canonical pairing (with ⟨δ, d⟩ = 1). Given i ∈ ℤ/eℤ, let

α_{i} = \begin{cases}2Λ_{0} − Λ_{e−1} − Λ_{1} + δ, & if i = 0, \\ 2Λ_{i} − Λ_{i−1} − Λ_{i+1}, & if i > 0. \end{cases}

(If e = ∞ then Λ_{e−1} is to be interpreted as Λ_{−1}.) The α_{i} are the simple roots of 𝔥^{∗}, the Λ_{i} the fundamental weights, and the matrix (⟨α_{i}, h_{j}⟩)_{i,j∈ℤ/eℤ} is the Cartan matrix of 𝔥.

Let v be an indeterminate over ℂ and let U(ŝl_{e}) be the affine quantum group of type A_{e}. This is the unital associative ℂ(v)–algebra generated by elements E_{i}, F_{i}, K_{h} which satisfy the quantum Serre relations (i ∈ ℤ/eℤ and h ∈ 𝔥). We shall assume that the reader is familiar with the basic representation theory and terminology of quantum groups; see [14, 15] for more details. In particular, the elements of X^{+} = ⊕_{i=0}^{e−1}ℕΛ_{i} are the dominant weights of U(ŝl_{e}) and for each dominant weight Λ ∈ X^{+} there is a unique integrable highest weight module L(Λ).

5.2. The Fock space. Recall the sequence e = (e; e_{1}, e_{2}, …, e_{r}) of integers from the beginning of this section and let Λ_{e} = Λ_{e_{1}} + ⋯ + Λ_{e_{r}}; then Λ_{e} is a dominant weight of U(ŝl_{e}).

The (generalized) Fock space ℱ_{e} is the free ℂ(v)–module with basis the set of all multipartitions of n ≥ 0 with r components; that is,

ℱ_{e} = ⨁_{n≥0} ⨁_{λ⊢n} ℂ(v)λ.

As a vector space ℱ_{e} does not depend upon e; however, the action of U(ŝl_{e}) on ℱ_{e} does. As we shall see, L(Λ_{e}) is a submodule of ℱ_{e}.

Recall that a node γ = (j, k)_{s} is above γ′ = (l, m)_{t} if either t > s or t = s and m < k; equivalently, γ′ is below γ. We defined the integers N_{i}^{l}(γ) in (4.5), we also need:

Definition 5.1 ([15]; cf. [13]). Let λ and μ be multipartitions and suppose that μ = λ ∪ {γ} where res_{e}(γ) = i. Then

(i) N_{i}(λ) = #{addable i–nodes of λ} − #{removable i–nodes of λ}.

(ii) N_{e}(λ) = #{γ = (j, k)_{s} ∈ λ | res_{e}(γ) = e_{s}}.

(iii) N_{i}^{r}(γ) = #{γ′| γ′ an addable i–node of λ strictly above γ } − #{γ′| γ′ a removable i–node of λ strictly above γ }

This allows us to define an action of U(ŝl_{e}) on ℱ_{e}. When r = 1 the next result was proved by Hayashi [9]. We do not know if Hayashi considered the general case as his paper is not easily obtainable; in any case, the proof is not difficult and can be found in [15].

Proposition 5.2. [15, Prop. 3.2] The ℂ(v)–module ℱ_{e} is an integrable U(ŝl_{e})– module with action determined by

K_{h_{i}}λ = v^{N_{i}(λ)}λ,  K_{d}λ = v^{−N_{e}(λ)}λ
E_{i}λ = ∑_{res_{e}(γ)=i^{λ=μ∪{γ}}} v^{−N_{i}^{r}(γ)}μ,  F_{i}λ = ∑_{res_{e}(γ)=i^{μ=λ∪{γ}}} v^{N_{i}^{l}(γ)}μ

where i ∈ ℤ/eℤ and λ is multipartition of n.

Example 5.3. Suppose that e = (3; 0, 1) and let λ = ((3, 1), (4, 3)). The e–residue diagram of λ is

( 0 1 2 2 , \begin{matrix}1 & 2 & 0 & 1 \\ 0 & 1 & 2 \end{matrix} );

so \begin{aligned}E_{2}λ & = & v^{−1}((3, 1), (4, 2)) + v^{−1}((3), (4, 3)) + ((2, 1), (4, 3)), and \\ F_{0}λ & = & ((3, 1), (4, 4)) + v((3, 2), (4, 3)) + v^{2}((4, 1), (4, 3)). \end{aligned}

Because of the natural embedding ℋ_{n,r} ↪ ℋ_{n+1,r}, it makes sense to talk of induction and restriction of ℋ_{n,r}–modules to ℋ_{n+1,r} and ℋ_{n−1,r}–modules respectively. When v = 1 the action of ∑_{i} F_{i} on λ corresponds to inducing the Specht module S(λ), and ∑_{i} E_{i} corresponds to restriction; see [3, Cor. 3.2]. Assuming Conjecture 5.15, the individual operators E_{i} and F_{i} compute the projections of these modules onto the blocks of ℋ_{n,r}.

By Proposition 5.2 the empty multipartition ∅_{e} = ((0), …, (0)) is a highest weight vector of weight Λ_{e} in ℱ_{e}; that is, K_{h}∅_{e} = v^{Λ_{e}(h)}∅_{e} and E_{i}∅_{e} = 0 for all h ∈ 𝔥 and all i ∈ ℤ/eℤ. Therefore, L(Λ_{e}) ≅ U(ŝl_{e})∅_{e} is a direct summand of ℱ_{e} since integrable modules are completely reducible [14, 6.2.2].

Let be the ℂ–linear involution on U(ŝl_{e}) determined by

\overline{v} = v^{−1},  \overline{K_{h}} = K_{−h},  \overline{E_{i}} = E_{i}, and \overline{F_{i}} = F_{i},

and extend to L(Λ_{e}) by specifying that \overline{u∅_{e}} = \overline{u}∅_{e} for all u ∈ U(ŝl_{e}).

Theorem 5.4. [15] For each e–restricted multipartition μ of n there exists a unique element B_{μ} ∈ L(Λ_{e}) such that

(i) \overline{B_{μ}} = B_{μ}.

(ii) B_{μ} = μ + ∑_{λ▷μ^{λ⊢n}} b_{λμ}(v)λ, for some polynomials b_{λμ}(v) ∈ vℤ[v].

Moreover, B(Λ_{e}) = { B_{μ} : μ an e–restricted multipartition of n } is a basis of L(Λ_{e}).

The basis B(Λ_{e}) is called the canonical basis of L(Λ_{e}). Extend the definition of the polynomials b_{λμ}(v) so that b_{μμ}(v) = 1 when μ is e–restricted and b_{λμ}(v) = 0 when μ is not e–restricted. Then, using Theorem 5.4, the main result of [2] can be reformulated as follows.

Theorem 5.5 (Ariki [2, Theorem 4.4]). Let λ and μ be multipartitions. Let S(λ) and D(μ) be the ℋ_{ℂ,e}–modules defined in Theorem 2.4. Then

d_{λμ} = [S(λ) : D(μ)] = b_{λμ}(1).

When r = 1, Theorem 5.4 was proved by Lascoux, Leclerc, and Thibon [13], and Theorem 5.5 by Grojnowski [8].

As the canonical basis elements are indexed by e–restricted multipartitions we see at once that the ℋ_{ℂ,e}–module D(μ) is non–zero if and only if μ is e–restricted, proving Theorem 4.8 in this case. For an arbitrary field 𝔽, Theorem 4.8 can be deduced from the characteristic zero case using some general nonsense together with the fact that D(μ) is non–zero when μ is e–restricted [6, 7, 15]; see [15] for details.

To prove Theorem 5.4 we first showed that the crystal graph of L(Λ_{e}) is the directed graph with vertices a set of e–restricted multipartitions and edges given by adjoining good nodes. Then, using Lusztig’s (and Kashiwara’s) result on the existence of the canonical basis, we gave an explicit combinatorial algorithm for computing the canonical basis elements B_{μ}. By virtue of Theorem 5.5 this gives an algorithm for computing the decomposition numbers of ℋ_{ℂ,e}; this we describe next.

5.3. The algorithm. ([15]; cf. [13]) We now describe how to calculate B_{μ} for an e–restricted multipartition μ.

Given an integer k ≥ 0 let [k] = (v^{k} −v^{−k})/(v −v^{−1}) and [k]! = [k][k −1] ⋯ [1]. For i ∈ ℤ/eℤ, let F_{i}^{(k)} = F_{i}^{k}/[k]! be the divided power operator; then F_{i}^{(k)} is

invariant under the bar involution; that is, \overline{F_{i}^{(k)}} = F_{i}^{(k)}.

When μ = ∅_{e}, B_{μ} = ∅_{e} and there is nothing to do. Suppose then that μ ▷ ∅_{e}. Let i be the e–residue of the lowest removable node in μ and suppose that γ_{1}, γ_{2}, …, γ_{k} are the k lowest removable i–nodes in μ where k is maximal with respect to the property that there is no addable i–node below γ_{k} (k > 0 since μ is e–restricted). Now let τ = μ \ {γ_{1}, …, γ_{k}}; by [15, Prop. 4.1], τ is e–restricted. Using induction we may assume that B_{τ} is known and let A_{μ} = F_{i}^{(k)}B_{τ}. Then \overline{A_{μ}} = A_{μ} and one can show that

B_{μ} = A_{μ} − ∑_{ν▷μ} α_{νμ}(v)B_{ν}, (5.6)

where the polynomials α_{νμ}(v) ∈ ℤ[v, v^{−1}] are completely determined by Theorem 5.4(ii) and the condition that \overline{α_{νμ}} = α_{νμ}. Since we may assume that B_{ν} is already known when ν ▷ μ this allows us to compute B_{μ}.

Example 5.7. We call the matrix B_{e} = (b_{λμ}(v)) the crystalized decomposition matrix of ℋ . When r = 3 some of these matrices are given in the appendix of [15]. When e = (2; 0, 1) the first four of these matrices are as follows.

\begin{aligned}n = 1 & 0 ; & 1 & 1 \\ & 1 ; & 0 & . & 1 \\ n = 2 & 0 ; & 1^{2} & 1 \\ & 0 ; & 2 & v \\ & 1 ; & 1 & v^{2} & 1 \\ & 1^{2} ; & 0 & . & v \\ & 2 ; & 0 & . & v^{2} \\ n = 3 & 0 ; & 1^{3} & 1 \\ & 0 ; & 2, 1 & . & 1 \\ & 0 ; & 3 & v & . \\ & 1 ; & 1^{2} & . & v & 1 \\ & 1 ; & 2 & . & v^{2} & v \\ & 1^{2} ; & 1 & v & . & . & 1 \\ & 2 ; & 1 & v^{2} & . & . & v \\ & 1^{3} ; & 0 & . & . & v & . \\ & 2, 1 ; & 0 & . & . & . & v^{2} \\ & 3 ; & 0 & . & . & v^{2} & . \end{aligned} n = 4 \begin{matrix}0 ; & 1^{4} & 1 \\ 0 ; & 2, 1^{2} & v & 1 \\ 0 ; & 2^{2} & . & v \\ 0 ; & 3, 1 & v & v^{2} \\ 0 ; & 4 & v^{2} & . \\ 1 ; & 1^{3} & v^{2} & v & 1 \\ 1 ; & 2, 1 & . & . & . & 1 \\ 1 ; & 3 & v^{3} & v^{2} & v & . \\ 1^{2} ; & 1^{2} & v & v^{2} & v & . & 1 \\ 1^{2} ; & 2 & v^{2} & v^{3} & v^{2} & . & v \\ 2 ; & 1^{2} & v^{2} & v^{3} & v^{2} & . & v \\ 2 ; & 2 & v^{3} & v^{4} & v^{3} & . & v^{2} \\ 1^{3} ; & 1 & v^{3} & . & v & . & v^{2} \\ 2, 1 ; & 1 & . & . & . & . & . & 1 \\ 3 ; & 1 & v^{4} & . & v^{2} & . & v^{3} & . \\ 1^{4} ; & 0 & . & . & v^{2} & . & . & . \\ 2, 1^{2} ; & 0 & . & . & v^{3} & . & v^{2} & . \\ 2^{2} ; & 0 & . & . & . & . & v^{3} & . \\ 3, 1 ; & 0 & . & . & v^{3} & . & v^{4} & . \\ 4 ; & 0 & . & . & v^{4} & . & . & . \end{matrix}

These matrices were calculated using Specht [16].

Example 5.8. Non–trivial examples when r > 1 involve many multipartitions so we consider the case e = (2; 0) and μ = (1^{5}). Then, in the notation above, τ = (1^{4}), i = 0 and k = 1. Now B_{(1^{4})} = (1^{4}) + v(2, 1^{2}) + v(3, 1) + v^{2}(4), so

A_{(1^{5})} = F_{0}B_{(1^{4})} = (1^{5}) + (2^{2}, 1) + 2v(3, 1^{2}) + v^{2}(3, 2) + v^{2}(5).

The coefficient of (2^{2}, 1) in A_{(1^{5})} is 1 so B_{(2^{2},1)} must split off. Since B_{(2^{2},1)} = (2^{2}, 1) + v(3, 1^{2}) + v^{2}(3, 2), this gives A_{(1^{5})} = (1^{5}) + v(3, 1^{2}) + v^{2}(5). Setting v = 1, we see that

d_{λ(1^{5})} = \begin{cases}1, & if λ = (1^{5}), (3, 1^{2}), or (5), \\ 0, & otherwise. \end{cases}

5.4. A reduction theorem. To conclude, we prove a result on the decomposition numbers of ℋ_{ℂ,e} using the canonical basis. It would be surprising if this result did not hold for all Ariki–Koike algebras.

Given any multipartition λ = (λ^{(1)}, …, λ^{(r)}) of n with r components, let λ^{#} be the multipartition (λ^{(2)}, …, λ^{(r)}) of n − |λ^{(1)}| with (r − 1) components.

If e = (e; e_{1}, …, e_{r}) we also let e^{#} = (e; e_{2}, …, e_{r}) and write d_{λμ} and d_{λ^{#}μ^{#}} for the decomposition numbers of ℋ_{ℂ,e} and ℋ_{ℂ,e^{#}} respectively (see Theorem 2.4(iv)).

Theorem 5.9. Suppose that λ and μ are multipartitions of n and that μ is e– restricted and λ^{(1)} = μ^{(1)}. Then μ^{#} is e^{#}–restricted and the decomposition numbers of the Ariki–Koike algebras ℋ_{ℂ,e} and ℋ_{ℂ,e^{#}} satisfy d_{λμ} = d_{λ^{#}μ^{#}}.

Proof. First we note that because μ is e–restricted we can find a sequence of multipartitions from ∅_{e} to μ where at each stage we add a good node to the previous multipartition. Because the nodes in μ^{(1)} are higher than all of the other nodes, by omitting the nodes in μ^{(1)} from this sequence we obtain a path from ∅_{e^{#}} to μ^{#} in which each multipartition again differs from the previous one by a good node. Hence μ^{#} is e^{#}–restricted.

Next recall that B_{μ} = ∑_{λ⊵μ} b_{λμ}(v)λ for certain polynomials b_{λμ}(v) ∈ ℤ[v]. By Theorem 5.5 it is sufficient to prove that b_{λμ}(v) = b_{λ^{#}μ^{#}}(v) whenever λ^{(1)} = μ^{(1)}. When μ = ∅_{e} this is clear, so suppose that μ ▷ ∅_{e}.

Let μ = τ ∪ {γ_{1}, …, γ_{k}} and A_{μ} = F_{i}^{(k)}B_{τ} as in section (5.3); we also assume that γ_{s} is below γ_{t} whenever s < t. Define polynomials a_{λμ}(v) ∈ ℤ[v, v^{−1}] by A_{μ} = ∑_{λ⊵μ} a_{λμ}(v)λ, and similarly for a_{λ^{#}μ^{#}}(v). We claim that a_{λμ}(v) = a_{λ^{#}μ^{#}}(v) whenever λ^{(1)} = μ^{(1)}. This, combined with (5.6), will complete the proof because if ν is any e–restricted multipartition such that ν ⊵ μ and ν^{(1)} ≠ μ^{(1)} then λ ⋭ ν and consequently b_{λν}(v) = 0.

To prove our claim we need the easy lemma (cf. Proposition 5.2).

Lemma 5.10. [15, Lemma 3.4] Suppose that i ∈ ℤ/eℤ and k > 0. For two multipartitions σ and λ such that λ = σ ∪{γ′_{1}, …, γ′_{k}}, where γ′_{1}, …, γ′_{k} are distinct addable i–nodes of σ. Let

N_{i}^{l}(σ, λ) = ∑_{γ′∈λ/σ} #{α| α an addable i–node of λ strictly below γ′ } − #{α| α a removable i–node of σ strictly below γ′ }.

Then F_{i}^{(k)}σ = ∑_{res_{e}(γ_{j})=i ∀j^{λ=σ∪{γ_{1},...,γ_{k}}}} v^{N_{i}^{l}(σ,λ)}λ.

Therefore the polynomials a_{λμ}(v) are given by

a_{λμ}(v) = ∑_{σ} v^{N_{i}^{l}(σ,λ)}b_{στ} (v) (5.11)

where as in the lemma the sum is over those multipartitions σ ⊵ τ such that λ = σ ∪ {γ′_{1}, …, γ′_{k}}. Note that σ^{(1)} ⊵ τ^{(1)} by the definition of dominance.

First suppose that σ^{(1)} ▷ τ^{(1)}. With respect to dominance μ is the smallest multipartition in F_{i}^{(k)}B_{τ} and it is obtained by placing {γ_{1}, …, γ_{k}} as low as possible in τ. Now σ ▷ τ so τ can be obtained by moving some of the nodes in σ to lower positions; in particular some of the nodes in σ^{(1)} must be lowered in order to reach τ. Because k was also chosen to be maximal, either γ_{k} is the highest removable i–node in μ or there exists an addable i–node above γ_{k}. Therefore, if λ appears in F_{i}^{(k)}σ then λ^{(1)} ▷ μ^{(1)}.

Consequently, if λ is a multipartition appearing in A_{μ} and λ^{(1)} = μ^{(1)} then all of the multipartitions σ in (5.11) have first component σ^{(1)} = τ^{(1)}. Again because the nodes {γ_{1}, …, γ_{k}} are placed as low as possible in τ in order to obtain μ, and because k is maximal, either τ^{(1)} = μ^{(1)} or λ = μ (since λ ⊵ μ). Either way, N_{i}^{l}(σ, λ) = N_{i}^{l}(σ^{#}, λ^{#}). By induction, b_{στ} (v) = b_{σ^{#}τ^{#}}(v) so (5.11) now establishes our claim and hence the theorem.

The theorem says that the decomposition matrix of ℋ_{ℂ,e} is almost completely determined by the decomposition matrices of the Iwahori–Hecke algebras ℋ (𝔖_{m}) for m ≤ n. We suspect that this statement holds without qualification; more precisely we make the following conjecture.

Conjecture 5.12. Suppose that λ and μ are multipartitions of n with r components such that μ is e–restricted. Then there exist polynomials b_{λμ}^{σ}(v) ∈ ℕ[v] such that

b_{λμ}(v) = ∑_{σ} b_{λμ}^{σ}(v)b_{λ^{#}σ}(v),

where σ runs over the e–restricted multipartitions of n − |λ^{(1)}| with (r − 1) components. Moreover, b_{λμ}^{σ}(v) ∉ vℕ[v] only if b_{λμ}^{σ}(v) = 1 and λ^{(1)} = μ^{(1)}.

Combining Theorem 5.9 with the Nakayama conjecture for the Hecke algebras ℋ (𝔖_{m}) (see [11, Theorem 4.29]) proves our next corollary when 𝔽 = ℂ. For an arbitrary field 𝔽, [15, Theorem 4.5] shows that there exists a square unitriangular adjustment matrix A such that D(ℋ_{𝔽,e}) = D(ℋ_{ℂ,e})A, where D(ℋ ) denotes the decomposition matrix of the Ariki–Koike algebra ℋ . This, together with the corollary for ℋ_{ℂ,e}, implies the result for ℋ_{𝔽,e}.

Corollary 5.13. Let 𝔽 be an arbitrary field and suppose that λ and μ are two multipartitions of n such that λ^{(s)} = μ^{(s)} for s = 1, 2, …, r − 1 and that λ^{(r)} and μ^{(r)} have the same e–core. Then the ℋ_{𝔽,e}–modules S(λ) and S(μ) belong to the same block.

This result is still far short of the conjectures of [6, 7] which describe the block structure of the Ariki–Koike algebras. To state this conjecture for the algebra ℋ_{𝔽,e} we need some more notation. Let res_{e}(λ) be the multiset of e–residues in λ; thus,

res_{e}(λ) = { res_{e}(γ) : γ ∈ λ },

where we allow repetitions.

Example 5.14. If e = (4; 0, 2, 3) and λ = ((4, 3), (2, 1), (3, 1)) then the e–residue diagram of λ is

( \begin{matrix}0 & 1 & 2 & 3 \\ 3 & 0 & 1 \end{matrix} , 2 3 1 , 3 0 1 2 )

so res_{e}(λ) = {0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3}.

It is easy to see that if b_{λμ}(v) ≠ 0, or if λ and μ satisfy the assumptions of Corollary 5.13, then res_{e}(λ) = res_{e}(μ).

Conjecture 5.15. [6, 7] Suppose that 𝔽 is an arbitrary field and let λ and μ be two multipartitions of n. Then the ℋ_{𝔽,e}–modules S(λ) and S(μ) belong to the same block if and only if res_{e}(λ) = res_{e}(μ).

As above, we note that by [15, Theorem 4.5] it is sufficient to prove this conjecture for the Ariki–Koike algebras ℋ_{ℂ,e}. It is known [6, 7] that if S(λ) and S(μ) belong to the same block then res_{e}(λ) = res_{e}(μ). As far as we know, Corollary 5.13 is the only result in the reverse direction.

References

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

2. , On the decomposition numbers of the Hecke algebra of G(m, 1, n), preprint (1996).

3. Susumu Ariki and Kazchiko Koike, A Hecke algebra of (ℤ/rℤ) ≀ 𝔖_{n} and construction of its irreducible representations, Adv. Math 106 (1994), no. 2, 216–243.

4. Michel Broué and Gunter Malle, Zcyklotomische Heckealgebren, Asterisque 212 (1993), 119– 189.

5. Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), 20–52.

6. Richard Dipper, Gordon James, and Eugene Murphy, Hecke algebras of type B_{n} at roots of unity, Proc. London Math. Soc. (3) 70 (1995), 505–528.

7. J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. 123 (1996), 1–34.

8. Ian Grojnowski, Representations of affine Hecke algebras (and affine quantum GL_{n}) at roots of unity, Math. Research Notes (1994), 215–217.

9. T. Hayashi, q–analogues of Clifford and Weyl algebras, Comm. Math. Phys. 127 (1990), 129–144.

10. Peter Norbert Hoefsmit, Representations of Hecke algebras of finite groups with BN–pairs of classical type, Ph.D. thesis, University of British Columbia, 1979.

11. Gordon D James and Andrew Mathas, A q–analogue of the Jantzen–Schaper theorem, Proc. London Math. Soc. (3) 74 (1997), 241–274.

12. Alexander Kleshchev, Branching rules for the modular representations of symmetric groups III: some corollaries and a problem of Mullineux, J. London Math. Soc. (to appear).

13. Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.

14. George Lusztig, Introduction to quantum groups, Progress in mathematics, vol. 110, Birkhäuser, Boston, 1993.

15. Andrew Mathas, Canonical bases and the decomposition matrices of Ariki–Koike algebras, preprint (1996).

16. , Decomposition matrices of Hecke algebras of type A, Gap: groups, algorithms and programming, 3.4.4 (M. Schönert et. al., ed.), RWTH Aachen, 3.4.4 ed., 1996.

17. G. E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1982), 492–513.

18. , Representations of Hecke algebras of C_{r} ≀ 𝔖_{n} at roots of unity, in preparation (1996).

Imperial College, Queen’s Gate, London SW7 2BZ

 E-mail address: a.mathas@ic.ac.uk