Séminaire Lotharingien de Combinatoire 69 (2013), Article B69a

ON THE REGULARIZATION PROCESS FOR ARIKI–KOIKE ALGEBRAS

N. JACON

Abstract. The aim of this note is to study a generalization of theorems by James and Fayers on the modular representations of the symmetric group and its Hecke algebra to the case of the complex reflection groups of type G(l, 1, n) and the associated Ariki– Koike algebra.

1. Introduction

One of the main and still open problems in the representation theory of finite groups is the explicit determination of the irreducible representations for the symmetric groups 𝔖_{n}, for n ∈ ℕ, over a field of characteristic p > 0. The main information on these representations is contained in a fundamental object: the decomposition matrix. By the work of James, the problem of computing this matrix may be attacked using the representation theory of Hecke algebras. Indeed, James’ conjecture predicts that, in the case where p^{2} > n, the decomposition matrix of the symmetric group corresponds to the decomposition matrix of a non-semisimple deformation of the group algebra ℂ𝔖_{n}: the Hecke algebra. Both matrices have their rows labelled by the partitions of rank n (which also label the simple modules of ℂ𝔖_{n}) whereas their columns are labelled by a certain subset of partitions, namely the p-regular partitions (which also label the simple modules of 𝔽𝔖_{n} where char(𝔽) = p).

An algorithm for the computation of the decomposition matrices for Hecke algebras over ℂ is available. This algorithm comes from a conjecture given by Lascoux, Leclerc and Thibon [11] and proved by Ariki [1]. It asserts that the decomposition matrix of the Hecke algebra over ℂ is given by the evaluation at v = 1 of the matrix of the canonical basis for the basic representation of the quantum algebra U_{v}(𝔰̂𝔩_{e}). Recently, several authors have shown that the matrix of the canonical basis itself (and not only its specialization at v = 1) admits an interpretation in terms of “graded representation theory” of Hecke algebras (see [10] and the references therein). This matrix can thus be called the v-decomposition matrix of the Hecke algebra.

A nice general property of these matrices has been revealed by James [9] (for the decomposition matrix of the symmetric group) and Fayers [4] (for the v-decomposition matrix of the Hecke algebra). One can explicitly associate to each partition λ of n a certain p-regular partition, the regularization of λ, and computes the associated decomposition number. This result uses a very important partial order on the set of partitions which appears in many ways in the representation theory of the symmetric group: the dominance order.

2010 Mathematics Subject Classification. 17B37, 20C30.

The author thanks the referee for useful comments and suggestions. The author is supported by ANR ACORT 12-JS01-0003-01.

This note is concerned with a generalization of the Hecke algebra of 𝔖_{n}: the Ariki– Koike algebra. We present a result analogous to the ones by James and Fayers for the decomposition matrices of these algebras and for the matrices of the canonical bases for the irreducible highest weight U_{v}(𝔰𝔩_{∞})-modules. The concept of partitions is here replaced by the concept of “multipartitions”, the set of p-regular partitions by the set of so called cylindric multipartitions and the dominance order on partitions by the dominance order on multipartitions. The main results, Theorem 5.1 and Theorem 6.3, give the desired analogues of James and Fayers’ Theorems. The paper will be organized as follows. In the first section, we introduce the main combinatorial objects we will use in this paper, namely multipartitions and symbols, and present some useful properties which they satisfy. Then we define the notion of regularization of multipartitions. The third part is devoted to a brief exposition of the representation theory of U_{v}(𝔰𝔩_{∞}) using the theory of Fock spaces. All these notions are then used in the two last parts to obtain our main results.

2. Multipartitions and symbols

In this part, we give the combinatorial definitions which are needed to present our main results.

2.1. Let l ∈ ℕ_{>0}. We write

S^{l} := {(s_{0}, …, s_{l−1}) ∈ ℤ^{l} | s_{i+1} ≥ s_{i} for i = 0, …, l − 2} .

Recall that a partition λ = (λ_{1}, …, λ_{r}) of n ∈ ℕ is an ordered sequence of weakly decreasing nonnegative integers such that |λ| := ∑_{1≤i≤r} λ_{i} = n. The integer h_{λ} := max(i ≥ 0 | λ_{i} ≠ 0) is called the height of λ with the convention that h_{λ} = 0 if λ is the empty partition, denoted by ∅. A multipartition or l-partition of n is an l-tuple of partitions λ = (λ^{0}, …, λ^{l−1}) such that, for each i ∈ {0, 1, …, l − 1}, λ^{i} is a partition of rank n_{i} ∈ ℕ and ∑_{0≤i≤l−1} n_{i} = n. If λ is a multipartition of n, we write λ ⊢_{l} n. The height of λ is the nonnegative integer

h_{λ} := max(h_{λ^{0}}, …, h_{λ^{l−1}}),

and we have h_{λ} = 0 if and only if λ is the empty multipartition, which is denoted by ∅. The dominance order on multipartitions is defined as follows. Let λ := (λ^{0}, …, λ^{l−1}) and μ := (μ^{0}, …, μ^{l−1}) be two multipartitions of n then we write

λ ⊵ μ if and only if ∑_{0≤i<c} |λ^{i}| + ∑_{1≤j≤k} λ_{j}^{c} ≥ ∑_{0≤i<c} |μ^{i}| + ∑_{1≤j≤k} μ_{j}^{c}

for c = 0, …, l − 1 and k ∈ ℕ,

where the partitions are considered with an infinite number of empty parts.

2.2. Let s = (s_{0}, …, s_{l−1}) ∈ S^{l}. We now define the notion of shifted symbol. Following [5, § 5.5.5], let β = (β_{1}, …, β_{k}) be a sequence of integers where k ∈ ℕ and let t be a positive integer. We set

β(t) := (0, 1, …, t − 1, β_{1} + t, …, β_{k} + t).

It is a sequence of exactly k + t integers. Let λ := (λ^{0}, …, λ^{l−1}) ⊢_{l} n. For i = 0, 1, …, l − 1, let h^{i} be the height of the partitions λ^{i}. We consider the following sequence of strictly increasing integers

β^{i} = (λ_{h^{i}}^{i} − h^{i} + h^{i}, …, λ_{j}^{i} − j + h^{i}, …, λ_{1}^{i} − 1 + h^{i}).

For i = 0, 1, …, l − 1, we put

hc^{i} := h^{i} − s_{i} and hc^{λ} := max(hc^{0}, …, hc^{l−1}).

Let h be an integer such that h ≥ hc^{λ} + 1. The shifted s-symbol of λ of size h is the family of sequences

𝔅_{(s,h)}(λ) = (𝔅^{0}, …, 𝔅^{l−1}),

such that for j = 0, …, l − 1, we have

𝔅^{j} = (β^{j}(h − hc^{j})).

Each sequence 𝔅^{j} contains exactly h + s_{j} elements (𝔅_{h+s_{j}}^{j}, …, 𝔅_{1}^{j}).

2.3. A shifted symbol is usually represented as (and identified with) an l-row tableau where the c-th row (starting from bottom) is 𝔅^{c} (see [5, § 5.5.5]). It is written as follows:

𝔅_{(s,h)}(λ) = ( \begin{matrix}𝔅_{h+s_{l−1}}^{l−1} & … & … & 𝔅_{2}^{l−1} & 𝔅_{1}^{l−1} \\ 𝔅_{h+s_{l−2}}^{l−2} & … & … & 𝔅_{1}^{l−2} \\ … & … & … \\ 𝔅_{h+s_{0}}^{0} & … & 𝔅_{1}^{0} \end{matrix} ) .

With our convention, note that the leftmost column of the symbol contains only entries 0. Also, note that the i^{th} column (starting from the right) of 𝔅_{(s,h)}(λ) contains exactly l − c(i) elements where c(i) = min(k ∈ {0, 1, …, l − 1} | i + s_{k} − s_{l−1} > 0) and this column is given by

( \begin{matrix}𝔅_{i}^{l−1} \\ 𝔅_{i+s_{l−2}−s_{l−1}}^{l−2} \\ … \\ 𝔅_{i+s_{c(i)}−s_{l−1}}^{c(i)} \end{matrix} ) .

It is easy to recover the multipartition λ from a given shifted symbol. Similarly, one can also easily recover s ∈ ℤ^{l} modulo a translation by an element (x, …, x) ∈ ℤ^{l}.

Example 2.4. With λ = (3, 2.2.2, 2.1) and s = (0, 0, 2), if we take h = 5 we obtain

𝔅_{((0,0,2),5)}(λ) = ( \begin{matrix}0 & 1 & 2 & 3 & 4 & 6 & 8 \\ 0 & 1 & 4 & 5 & 6 \\ 0 & 1 & 2 & 3 & 7 \end{matrix} ) .

With s = (0, 1, 2, 3) and λ = (3.1, 1.1, 2.1.1, 3) and h = 3, the shifted symbol is

𝔅_{((0,1,2,3),3)}(λ) = ( \begin{matrix}0 & 1 & 2 & 3 & 4 & 8 \\ 0 & 1 & 3 & 4 & 6 \\ 0 & 1 & 3 & 4 \\ 0 & 2 & 5 \end{matrix} ) .

3. Regularization of multipartitions

In this section, after having fixed an element in S^{l}, we associate to each multipartition another one which belongs to a certain class of multipartitions: the cylindric multipartitions.

Definition 3.1. Assume that s ∈ S^{l}. A shifted symbol 𝔅_{(s,h)}(λ) is called semistandard^{1} if and only if in each column of 𝔅_{(s,h)}(λ), the numbers weakly decrease from top to bottom.

Example 3.2. The first symbol in Example 2.4 is not semistandard whereas the second is. Let now s = (0, 1, 2) and λ = (3.1, 2.2.1, 2.1). If we take h = 5 we obtain

𝔅_{((0,1,2),5)}(λ) = ( \begin{matrix}0 & 1 & 2 & 3 & 4 & 6 & 8 \\ 0 & 1 & 2 & 4 & 6 & 7 \\ 0 & 1 & 2 & 4 & 7 \end{matrix} ) .

This symbol is semistandard.

Definition 3.3. Assume that s = (s_{0}, …, s_{l−1}) ∈ S^{l} then the λ = (λ^{0}, …, λ^{l−1}) ⊢_{l} n is called cylindric if for every c = 0, …, l − 2 and i ≥ 1, we have λ_{i}^{c} ≥ λ_{i+s_{c+1}−s_{c}}^{c+1} (the partitions are taken with an infinite number of empty parts). We denote by Φ_{s} the set of cylindric multipartitions associated to s ∈ S^{l} and by Φ_{s}(n) the set of cylindric multipartitions of rank n.

Proposition 3.4. Let s ∈ S^{l}, λ ⊢_{l} n and let 𝔅_{(s,h)}(λ) be an associated shifted symbol. Then λ is cylindric if and only if 𝔅_{(s,h)}(λ) is semistandard.

Proof. For all relevant i ∈ ℕ and c ∈ {0, 1, …, l − 2}, we have

λ_{i}^{c} − i + s_{c} ≥ λ_{i+s_{c+1}−s_{c}}^{c+1} − (i + s_{c+1} − s_{c}) + s_{c+1} if and only if λ_{i}^{c} ≥ λ_{i+s_{c+1}−s_{c}}^{c+1},

whence

𝔅_{i}^{c} ≥ 𝔅_{i+s_{c+1}−s_{c}}^{c+1} if and only if λ_{i}^{c} ≥ λ_{i+s_{c+1}−s_{c}}^{c+1},

which is exactly what is needed to prove the assertion. □

3.5. We now explain the process of regularization for multipartitions. Let λ ⊢_{l} n and let s = (s_{0}, …, s_{l−1}) ∈ S^{l}. Let 𝔅_{(s,h)}(λ) = (𝔅^{0}, …, 𝔅^{l−1}) be a shifted symbol associated to λ. Then for each column of the symbol, we reorder the elements so that it is weakly decreasing from top to bottom and we denote by 𝔅_{(s,h)}(λ)^{R} this new object. For example, the symbol in Example 2.4

𝔅_{((0,1,1),4)}(λ) = ( \begin{matrix}0 & 1 & 3 & 9 \\ 0 & 4 & 5 & 7 \\ 0 & 2 & 3 \end{matrix} )

becomes

( \begin{matrix}0 & 1 & 3 & 7 \\ 0 & 2 & 3 & 9 \\ 0 & 4 & 5 \end{matrix} ) .

1Such symbols are called standard in [12].

We claim that the resulting object is a well-defined semistandard symbol of a multipartition 𝔅_{(s,h)}(μ). Indeed, denote the symbol by B = (B^{0}, …, B^{l−1}) (in an obvious way) and assume that there exist c ∈ {0, 1, …, l − 1} and j ∈ ℕ such that

B_{j+1}^{c} ≥ B_{j}^{c}.

Note that the elements in each column of 𝔅_{(s,h)}(λ) are the same as the elements in the corresponding columns in B. Let A_{1} be the multiset of elements in the column containing the entry B_{j}^{c} which are less or equal than B_{j}^{c} in 𝔅_{(s,h)}(λ). Let A_{2} be the multiset of elements in the column containing the entry B_{j+1}^{c} which are greater or equal than B_{j}^{c} in 𝔅_{(s,h)}(λ). Assume that the column containing B_{j+1}^{c} contains m elements. We know that in the columns of B, the numbers are weakly decreasing from top to bottom. Thus, by the construction of B and the above assumption, we have ♯A_{1} ≥ l − c and ♯A_{2} ≥ m−l+c+1. Now 𝔅_{(s,h)}(λ) is a well-defined symbol so each row contains elements which are strictly increasing from left to right. This implies that the rows containing the elements of A_{1} in the column of 𝔅_{j}^{c} and the rows containing the elements of A_{2} in the column of 𝔅_{j+1}^{c} in 𝔅_{(s,h)}(λ) must be disjoint. So the sum ♯A_{1} +♯A_{2} must be less or equal to m. This is not the case, so the result follows. In addition, μ is an l-partition of n.

3.6. Let λ ⊢_{l} n and let 𝔅_{(s,h)}(λ) = (𝔅^{0}, …, 𝔅^{l−1}) be an associated shifted symbol. For all c_{1} ∈ {0, 1, …, l − 1} and j_{1} ∈ {1, 2, …, h + s_{c_{1}}}, we set

R(λ)_{(j_{1},c_{1})} = ♯ {c ∈ {0, 1, …, l − 1} | c > c_{1}, 𝔅_{j_{1}}^{c_{1}} < 𝔅_{j_{1}+s_{c}−s_{c_{1}}}^{c}, 𝔅_{j_{1}}^{c_{1}} ∉ 𝔅^{c}} ,

and

R(λ) = ∑_{0≤c≤l−1} ∑_{1≤j_{1}≤h+s_{c}} R(λ)_{(j_{1},c_{1})}.

Clearly, this number does not depend of the choice of h (and thus on the choice of the shifted symbol).

Definition 3.7. Let λ be a multipartition of n and let s = (s_{0}, …, s_{l−1}) ∈ S^{l}. Let 𝔅_{(s,h)}(λ) be the associated shifted symbol. By the discussion in §3.5, there exists a cylindric multipartition λ^{R} such that 𝔅_{(s,h)}(λ^{R}) = 𝔅_{(s,h)}(λ)^{R}. This multipartition is called the regularization of λ.

By construction, the regularization of a multipartition is a cylindric multipartition by Proposition 3.4. It is also clear that the regularization stabilizes the set of cylindric multipartitions.

Example 3.8. Consider the 3-partition λ := (∅, ∅, 6.2). We set s = (0, 1, 1), h = 3. Then we have

𝔅_{((0,1,1),3)}(λ) = ( \begin{matrix}0 & 1 & 4 & 9 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 2 \end{matrix} ) .

We obtain

𝔅_{((0,1,1),3)}(λ) = ( \begin{matrix}0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 9 \\ 0 & 1 & 4 \end{matrix} ) .

We have

R(λ)_{(j,c)} = \{ \begin{matrix}0 & if (j, c) ∉ \{(1, 0), (2, 1), (1, 1)\} \\ 1 & if (j, c) ∈ \{(1, 0), (2, 1), (1, 1)\} \end{matrix}

and so we have R(λ) = 3. We can check that λ^{R} = (2, 6, ∅).

Example 3.9. Consider the 3-partition λ := (5, ∅, 2.1). We set s = (0, 1, 1), h = 3. Then we have

𝔅_{((0,1,1),3)}(λ) = ( \begin{matrix}0 & 1 & 3 & 5 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 7 \end{matrix} ) .

We obtain

𝔅_{((0,1,1),3)}(λ) = ( \begin{matrix}0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 5 \\ 0 & 1 & 7 \end{matrix} ) .

and we have R(λ) = R(λ)_{(2,1)} = 1. We can check that λ^{R} = (5, 2.1, ∅).

4. Action of U_{v}(𝔰𝔩_{∞}) on the Fock space

4.1. Let v be an indeterminate and let U_{v}(𝔰𝔩_{∞}) be the quantized enveloping algebra of 𝔰𝔩_{∞} with Chevalley generators e_{j}, f_{j} and t_{j} (j ∈ ℤ), see for example [5, § 6.1]. The simple roots and fundamental weights are denoted by α_{k} and Λ_{k} for k ∈ ℤ respectively. Let now s ∈ S^{l} and let F^{s} be the associated Fock space. This is the ℚ(v)-vector space defined as follows:

F_{s} = ⨁_{n∈ℤ_{≥0}} ⨁_{λ⊢_{l}n} ℚ(v)λ.

One can define an action of U_{v}(𝔰𝔩_{∞}) which turns F_{s} into an integrable U_{v}(𝔰𝔩_{∞})-module.

4.2. Let β := (β_{1}, …, β_{m}) be a sequence of strictly increasing positive numbers. We write j ∈ β if the number j appears in the sequence β and j ∉ β otherwise. In this case, we write β ∪ {j} for the sequence of strictly positive numbers obtained by inserting the number j in β. Similarly, if j ∈ β then β \ {j} is defined to be the sequence obtained from β by removing j.

Let λ ⊢_{l} n and let 𝔅_{(s,h)}(λ) be an associated shifted symbol. Let μ ⊢_{l} n and let 𝔅_{(s,h)}(μ) be an associated shifted symbol (so we assume that h ≥ max(hc^{λ}, hc^{μ}) + 1.) Then we write

λ ⟶_{c}^{j} μ,

if for all d ∈ {0, 1, …, l − 1}, we have

𝔅_{(s,h)}(μ)^{d} = \{ \begin{matrix}𝔅_{(s,h)}(λ)^{d} & if d ≠ c, \\ (𝔅_{(s,h)}(λ)^{d} \ \{j + h − 1\})) ∪ \{j + h\} & if d = c. \end{matrix}

For k ∈ ℕ, we write

λ ⟶_{(c_{1},...,c_{k})}^{j:k} μ,

if there exists a sequence of multipartitions

λ := λ[1], …, λ[k], λ[k + 1] := μ,

ON THE REGULARIZATION PROCESS FOR ARIKI-KOIKE ALGEBRAS 7

such that for all i = 1, …, k, we have

λ[i] ⟶_{c_{i}}^{j} λ[i + 1].

Hence, we have

𝔅_{(s,h)}(μ)^{d} = \begin{cases}𝔅_{(s,h)}(λ)^{d} & if d ≠ c_{i} for all i = 1, …, k \\ (𝔅_{(s,h)}(λ)^{d} \ \{j + h − 1\})) ∪ \{j + h\} & if d = c_{i} for some i ∈ \{1 … k\}. \end{cases}

We also write λ ⟶^{j:k} μ if there exists a sequence (c_{1}, …, c_{k}) ∈ {0, 1, …, l − 1}^{k} such that λ ⟶_{(c_{1},...,c_{k})}^{j:k} μ. Note that in this case, the sequence (c_{1}, …, c_{k}) ∈ {0, 1, …, l − 1}^{k} is uniquely determined. Finally, given λ, μ and j ∈ ℤ such that λ ⟶_{(c_{1},...,c_{k})}^{j:k} μ, we define the following number:

N_{j}(λ, μ) = ∑_{1≤i≤k} (♯{integers equal to j + h − 1 in 𝔅_{(s,h)}(μ)^{c} with c ≤ c_{i}}

−♯{integers equal to j + h in 𝔅_{(s,h)}(λ)^{c} with c ≤ c_{i}}) .

It is clear that all these notions do not depend on the choice of h. Example 4.3. Consider the 4-partition λ = (3.1, ∅, ∅, 6.2) and s = (0, 0, 1, 1). For h = 3, we get the symbol

𝔅_{((0,0,1,1),3)}(λ) = ( \begin{matrix}0 & 1 & 4 & 9 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 2 & 5 \end{matrix} ) .

We have

(3.1, ∅, ∅, 6.2) ⟶_{(0,1)}^{0:2} (3.2, 1, ∅, 6.2),

where the symbol of μ is

𝔅_{((0,0,1,1),3)}(μ) = ( \begin{matrix}0 & 1 & 4 & 9 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 3 \\ 0 & 3 & 5 \end{matrix} ) .

and we have N_{0}(λ, μ) = 0. 4.4. For our purpose, we only need to describe the action of the Chevalley generators f_{j} for j ∈ ℤ and their divided power f_{j}^{(r)} := \frac{f_{j}^{r}}{[r]_{v}!}, for r ∈ ℕ. This is given as follows. Let λ ⊢_{l} n, let r ∈ ℕ and let j ∈ ℤ, then we have

f_{j}^{(r)}.λ = ∑_{λ⟶^{j:r}μ}v^{N_{j}(λ,μ)}μ.

It is known that the U_{v}(𝔰𝔩_{∞})-submodule V (s) of F^{s} generated by the empty multipartition is an irreducible highest weight module for U_{v}(𝔰𝔩_{∞}).

4.5. Let U_{v}(𝔰𝔩_{∞})^{−} be the subalgebra of U_{v}(𝔰𝔩_{∞}) generated by the f_{i}’s. We have a ring automorphism x ↦ \overline{x} of U_{v}(𝔰𝔩_{∞})^{−} such that

\overline{f}_{j} = f_{j} (j ∈ ℕ),  and  \overline{v} = v^{−1},

which induces a ℂ-linear map v ↦ \overline{v} on V (s) defined by

\overline{v.∅} = \overline{v}.∅.

Using this, one can define the canonical basis elements of V (s). They are elements of V (s) which are parametrized by the set of cylindric multipartitions

{b_{λ} | λ ∈ Φ_{s}},

and characterized by the following property:

for all λ ∈ Φ_{s}, we have \overline{b_{λ}} = b_{λ}, and b_{λ} = λ + ∑_{μ⊢_{l}n} d_{μ,λ}(v)μ,

for elements d_{μ,λ}(v) ∈ vℚ[v] with μ ⊢_{l} n such that λ ≠ μ.

4.6. We now briefly explain an algorithm for computing the canonical bases of irreducible highest weight U_{v}(𝔰𝔩_{∞})-modules. An analogue of this algorithm has already been described in [7] (and the proofs can be found therein) in a more general setting but it can be simplified here. One can also use the algorithm described in [13] using different conventions. Let λ = (λ^{0}, …, λ^{l−1}) ⊢_{l} n be a nonempty cylindric multipartition and let 𝔅_{(s,h)} := 𝔅_{(s,h)}(λ) be an associated shifted symbol. Let 𝔅_{(s,h)} = (𝔅^{0}, …, 𝔅^{l−1}). From this symbol, we define a new multipartition λ^{−} ⊢_{l} n′ < n together with two integers j(λ) and r(λ).

(1) Let c(λ) be the minimal integer such that λ^{c(λ)} is nonempty. This means that there exists i(λ) ∈ ℕ such that

𝔅_{i(λ)}^{c(λ)} > 𝔅_{i(λ)+1}^{c(λ)} + 1.

Assume that i(λ) is minimal with this property and set

j(λ) := 𝔅_{i(λ)}^{c(λ)}.

(2) Let r(λ) ∈ ℕ be maximal such that

𝔅_{i(λ)}^{c(λ)} = 𝔅_{i(λ)+s_{c(λ)+1}−s_{c(λ)}}^{c(λ)+1} = … = 𝔅_{i(λ)+s_{c(λ)+r(λ)−1}−s_{c(λ)}}^{c(λ)+r(λ)−1} = j(λ),

then we have r(λ) ≥ 1 and because λ is cylindric, by Proposition 3.4, we deduce that

j(λ) > 𝔅_{i(λ)+s_{c(λ)+k−1}−s_{c(λ)}+1}^{c(λ)+k−1} + 1

for all k = 1, …, r(λ).

(3) Take the symbol obtained by replacing in 𝔅_{(s,h)} all the elements

𝔅_{i(λ)}^{c(λ)} = 𝔅_{i(λ)+s_{c(λ)+1}−s_{c(λ)}}^{c(λ)+1} = … = 𝔅_{i(λ)+s_{c(λ)+r(λ)−1}−s_{c(λ)}}^{c(λ)+r(λ)−1} = j(λ)

by j(λ) − 1. This is a well-defined shifted symbol 𝔅′_{(s,h)}. Thus, there exists a unique l-partition λ^{−} ⊢_{l} n − r(λ) such that 𝔅_{(s,h)}(λ^{−}) = 𝔅′_{(s,h)}.

(4) By construction, 𝔅′_{(s,h)} is semistandard so λ^{−} is cylindric.

Using this, we can produce a sequence of cylindric multipartitions λ[k] with k = 1, …, m + 1 such that λ[1] = λ, λ[m + 1] = ∅ and λ[k + 1] = λ[k]^{−} for k = 1, …, m. Set j_{k} := j(λ[k]) − h and a_{k} := r(λ[k]) for k = 1, …, m. Then note that we have

λ[m + 1] ⟶^{j_{m}:a_{m}} λ[m] ⟶^{j_{m−1}:a_{m−1}} … ⟶^{j_{1}:a_{1}} λ[1].

We have thus defined two sequences of integers (j_{m}, …, j_{1}) and (a_{m}, …, a_{1}). Then we can define

a_{λ} := f_{j_{1}}^{(a_{1})}f_{j_{2}}^{(a_{2})} … f_{j_{m}}^{(a_{m})}.∅.

By induction, our construction implies that we have

a_{λ} = λ + ∑_{λ▷μ}b_{μ,λ}(v)μ

for elements b_{μ,λ}(v) ∈ ℕ[v, v^{−1}] (the proof is exactly the same as in [6, Prop. 4.6].) Note that if b_{μ,λ}(v) ≠ 0 then the multiset of elements appearing in 𝔅_{(s,h)}(μ) is the same as the one of 𝔅_{(s,h)}(λ) (for h large enough). When λ runs over the set of all cylindric multipartitions, these elements provide a basis for V (s) and an algorithm for the computation of the canonical basis (see below or [5, Ch. 6]). This, in turn, implies that if d_{μ,λ}(v) ≠ 0 then the multiset of elements appearing in 𝔅_{(s,h)}(μ) is the same as the one of 𝔅_{(s,h)}(λ) (for h large enough) and we have

b_{λ} = λ + ∑_{λ▷μ}d_{μ,λ}(v)μ,

and all the multipartitions appearing in this decomposition have the same rank. Alternatively, these properties are a consequence of the fact that both the a_{λ}’s and the b_{λ}’s are weight vectors, and that weight spaces of the Fock spaces are parametrized by symbols with a fixed distribution of entries.

Example 4.7. Let s = (0, 0, 1) and λ = (2.2, 2.2, 2.2.1), for h = 3, we get the following symbol:

𝔅_{((0,0,1),3)}(λ) = ( \begin{matrix}0 & 2 & 4 & 5 \\ 0 & 3 & 4 \\ 0 & 3 & 4 \end{matrix} )

which is semistandard. We begin with c(λ) = 0 and j(λ) = 3. Then we have r(λ) = 2. The new symbol is

( \begin{matrix}0 & 2 & 4 & 5 \\ 0 & 2 & 4 \\ 0 & 2 & 4 \end{matrix} )

which is the symbol 𝔅_{((0,0,1),3)}(λ[2]) with λ[2] = (2.1, 2.1, 2.2.1). Then we have c(λ[2]) = 0, j(λ[2]) = 4, r(λ[2]) = 3 and we get the symbol

( \begin{matrix}0 & 2 & 3 & 5 \\ 0 & 2 & 3 \\ 0 & 2 & 3 \end{matrix} )

which is the symbol 𝔅_{((0,0,1),3)}(λ[3]) with λ[3] = (1.1, 1.1, 2.1.1). Continuing in this way, keeping the notation of the above remark, we obtain

λ[4] = (1.1, 1.1, 1.1.1),  λ[5] = (1, 1, 1.1),  λ[6] = (∅, ∅, 1),  λ[7] = ∅

and we have

∅ ⟶_{(2)}^{1:1} (∅, ∅, 1) ⟶_{(0,1,2)}^{0:3} (1, 1, 1.1) ⟶_{(0,1,2)}^{−1:3} (1.1, 1.1, 1.1.1) ⟶_{(2)}^{2:1} (1.1, 1.1, 2.1.1)

(1.1, 1.1, 2.1.1) ⟶_{(0,1,2)}^{1:3} (2.1, 2.1, 2.2.1) ⟶_{(0,1)}^{0:2} (2.2, 2.2, 2.2.1).

4.8. Following [14, § 6.25], the above construction provides an algorithm for the computation of the canonical basis of V (s): The (modified) LLT algorithm. This is done recursively as follows. Let n ∈ ℕ_{>0}. For all k < n, assume that we have constructed all the canonical basis elements

{b_{λ} | λ ∈ Φ_{s}(k)}.

Now, let λ ∈ Φ_{s}(n). We want to compute b_{λ}.

(1) We set c_{λ} = f_{j(λ)−h}^{(r(λ))}b_{λ}^{−} and we have \overline{c_{λ}} = c_{λ}.

(2) The element b_{λ}^{−} is known by induction. Again, by the above construction, we

have

c_{λ} = λ + ∑_{λ▷ν} d̂_{ν,λ}(v)ν,

for Laurent polynomials d̂_{ν,λ}(v) with d̂_{λ,λ}(v) = 1 and we obtain

b_{λ} = c_{λ} − ∑_{λ▷ν}α_{ν,λ}(v)b_{ν},

for some Laurent polynomials α_{ν,λ}(v) such that α_{ν,λ}(v) = α_{ν,λ}(v^{−1}).

(3) We find the greatest multipartition ν ≠ λ with respect to ▷ such that d̂_{ν,λ}(v) ≠ 0. If no such multipartition exists, then we have c_{λ} = b_{λ}. Otherwise, α_{ν,λ}(v) is the unique bar-invariant Laurent polynomial such that the coefficients of α_{ν,λ}(v) and d̂_{ν,λ}(v) associated to the v^{i} with i ≤ 0 are the same. We then replace c_{λ} with the bar-invariant element c_{λ} − α_{ν,λ}(v)b_{ν} and we repeat the last step until all of the coefficients d̂_{ν,λ}(v) belong to vℤ[v] for λ ≠ ν.

5. The main results

The first main result of this paper is the following theorem concerning the computation of the coefficients of the canonical basis elements for irreducible highest weight U_{v}(𝔰𝔩_{∞})-modules.

Theorem 5.1. Assume that μ is a fixed cylindric l-partition of rank n. Then for any l-partition λ of rank n, we have d_{λ,μ}(v) = 0 unless μ ⊵ λ^{R} while d_{λ,λ^{R}}(v) = v^{R(λ)}.

The strategy for the proof of this theorem is modeled on the one presented in [4]. However, the proofs of the preparatory results we need here are of course different. The first lemma is the analogue of [4, Lemma 2.3].

Lemma 5.2. Let λ = (λ^{0}, …, λ^{l−1}) ⊢_{l} n and μ = (μ^{0}, …, μ^{l−1}) ⊢_{l} n. Assume that μ is cylindric and that μ^{−} ⟶^{j:k} μ. Let ν be a partition of n − k such that ν ⟶^{j:k} λ and μ^{−} ⊵ ν^{R}. Then μ ⊵ λ^{R} with equality only if μ^{−} = ν^{R}.

Proof. Assume that we have

μ^{0} = … = μ^{c(μ)−1} = ∅  and  μ^{c(μ)} ≠ ∅.

Then, by construction, we have

𝔅_{(s,h)}(μ)^{d} = (𝔅_{(s,h)}(μ^{−})^{d} \ {j + h − 1})) ∪ {j + h}

for d = c(μ), …, c(μ)+k −1. Similarly, by the definition of the regularization, we have

𝔅_{(s,h)}(λ^{R})^{d} = (𝔅_{(s,h)}(ν^{R})^{d} \ {j + h − 1})) ∪ {j + h}

for elements d ∈ {r_{1}, …, r_{k}} ⊂ {0, …, l − 1}. Since we have μ^{−} ⊵ ν^{R}, we deduce that

(ν^{R})^{0} = … = (ν^{R})^{c(μ)−1} = ∅.

We have to check that r_{i} ≥ c(μ) for all i ∈ {1, …, k}. So let us assume that we have r_{1} < c(μ) (without loss of generality). Thus, we have

𝔅_{(s,h)}(ν^{R})_{1}^{r_{1}} = 0 − 1 + s_{r_{1}} + h = j + h − 1

which implies that j = s_{r_{1}}. Thus, we deduce that j + h = s_{r_{1}} + h ∈ 𝔅_{(s,h)}(μ)^{c(μ)}. Since j − 1 ∉ 𝔅_{(s,h)}(μ)^{c(μ)} by the construction of μ, we deduce that i(μ) > s_{c(μ)} − s_{r_{1}} + 1. However, this implies that 𝔅_{(s,h)}(μ) is not semistandard. Indeed, in these case we have 𝔅_{(s,h)}(μ)_{i(μ)}^{c(μ)} > 𝔅_{(s,h)}(μ)_{i(μ)+s_{r_{1}}−s_{c(μ)}}^{r_{1}}. This is a contradiction because μ is cylindric.

Assume now that we have the equality μ = λ^{R}, this implies that we have

{r_{1}, …, r_{k}} = {c(μ), …, c(μ) + k − 1}.

Thus, we also have μ^{−} = ν^{R} which concludes the proof.

□

The second lemma is the analogue of [4, Prop. 2.5].

Lemma 5.3. Let λ be a cylindric multipartition and let μ be a multipartition such that μ^{R} = λ. Let μ^{−} be a multipartition of n − k such that μ^{−} ⟶^{j:k} μ for some j ∈ ℤ and k ∈ ℕ and such that (μ^{−})^{R} = λ^{−} then the coefficient of μ in f_{j}^{(k)}μ^{−} is v^{R(μ)−R(μ^{−})}.

Proof. Consider a shifted symbol 𝔅_{(s,h)}(μ^{−}). Let (i_{1}, c_{1}), …, (i_{m}, c_{m}) be all the elements of ℕ × {0, 1, …, l − 1} such that

𝔅_{(s,h)}(μ^{−})_{i_{1}}^{c_{1}} = … = 𝔅_{(s,h)}(μ^{−})_{i_{m}}^{c_{m}} = j + h − 1.

By hypothesis, we have m ≥ k and one can assume without loss of generality that

𝔅_{(s,h)}(μ)_{i}^{c} = \{ \begin{matrix}j + h & if (i_{t}, c_{t}) = (i, c) for one t ∈ \{1, …, k\}, \\ 𝔅_{(s,h)}(μ^{−})_{i}^{c} & otherwise. \end{matrix}

First, we easily see that for all (i, c) such that 𝔅_{(s,h)}(μ^{−})_{i}^{c} ∉ {j + h − 1, j + h}, we have

R(μ^{−})_{(i,c)} = R(μ)_{(i,c)}.

Let now (i, c) be such that

𝔅_{(s,h)}(μ^{−})_{i}^{c} = j + h − 1.

• Assume that c ≠ c_{t} for all t ∈ {1, …, k}. Then we have 𝔅_{(s,h)}(μ)_{i}^{c} = j + h − 1. We claim that we have

R(μ)_{(i,c)} − R(μ^{−})_{(i,c)} = ♯{integers t ∈ {1, …, k} such that c < c_{t}}.

We need to show that if c < c_{t} then we have

𝔅(μ^{−})_{i+s_{c_{t}}−s_{c}}^{c_{t}} > 𝔅(μ^{−})_{i}^{c}.

Assume to the contrary that 𝔅(μ^{−})_{i+s_{c_{t}}−s_{c}}^{c_{t}} ≤ 𝔅(μ^{−})_{i}^{c}, then by the process of regularization, in the construction of 𝔅_{(s,h)}(λ^{−}), the number 𝔅_{(s,h)}(μ^{−})_{i_{t}}^{c_{t}} = j+h−1 is sent to a row c′ of the symbol which is greater than the row containing 𝔅_{(s,h)}(μ^{−})_{i}^{c} = j + h − 1. This is impossible by the construction of λ^{−}.

• Assume that c = c_{t} for t ∈ {1, …, k}. Then we have 𝔅_{(s,h)}(μ)_{i}^{c} = j + h. We

claim that we have

R(μ)_{(i,c)} = R(μ^{−})_{(i,c)}.

This comes from the fact that if there exists c′ > c such that c′ ≠ c_{s} for s ∈ {1, …, k} and

𝔅(μ^{−})_{i+s_{c′}−s_{c′}}^{c′} > 𝔅(μ^{−})_{i}^{c},

then j + h − 1 ∉ 𝔅_{(s,h)}(μ^{−})^{c′} which follows from the construction of λ^{−}. Now assume that (i, c) is such that

𝔅_{(s,h)}(μ^{−})_{i}^{c} = j + h.

We follow the same kind of reasoning as above by showing that if c_{t} > c then we have

𝔅(μ^{−})_{i+s_{c_{t}}−s_{c_{t}}}^{c_{t}} > 𝔅(μ^{−})_{i}^{c}.

We deduce that

R(μ^{−})_{(i,c)} − R(μ)_{(i,c)} = ♯{integers t ∈ {1, …, k} such that c < c_{t}}

Now we use the formula in §4.2 to deduce the result. Namely, we have

N_{j}(μ^{−}, μ) = ∑_{1≤i≤k} (♯{integers equal to j + h − 1 in 𝔅_{(s,h)}(μ)^{c} with c ≤ c_{i}}
−♯{integers equal to j + h in 𝔅_{(s,h)}(μ^{−})^{c} with c ≤ c_{i}})

= ∑_{c∈{0,...,l−1}} ♯{integers t ∈ {1, …, k} s.t. c < c_{t} and j + h − 1 ∈ 𝔅_{(s,h)}(μ)^{c} }

− ∑_{c∈{0,...,l−1}} ♯{integers t ∈ {1, …, k} s.t. c < c_{t} and j + h ∈ 𝔅_{(s,h)}(μ^{−})^{c} }

= R(μ) − R(μ^{−}).

□

Remark 5.4. In [12], B. Leclerc and H. Miyachi have given an explicit closed formula for the elements of the canonical bases for irreducible highest weight modules of level l = 2. This formula also uses the notion of symbols. When l > 2, these elements are more difficult to compute because they are non-monomial in terms of the Chevalley operators (contrary to the case l = 2). When l = 2, it is easy to check that our formulas are consistent with the ones of Leclerc-Miyachi in this particular case.

5.5. Proof of Theorem 5.1. The proof is exactly the same as in [4, Thm. 2.2]. We give it for the convenience of the reader. We argue by induction on n ∈ ℕ and on the dominance order. When n = 0 or when μ is minimal with respect to ⊵, there is nothing to do. So let us assume that n > 0 and that we have a cylindric multipartition μ of rank n. By induction, the result holds when μ is replaced with μ^{−}. Assume that we have μ^{−} ⟶^{j:k} μ for j ∈ ℤ and k ∈ ℕ. Then we have

f_{j}^{(k)}b_{μ}^{−} = μ + ∑_{μ▷λ} d̂_{λ,μ}(v)λ.

We will prove the property of the theorem for the numbers d̂_{λ,μ}(v) and then deduce the result for the numbers d_{λ,μ}(v). Assume that d̂_{λ,μ}(v) ≠ 0. Then there exists ν ⊢_{l} n − k such that d_{ν,μ^{−}}(v) ≠ 0 and such that ν ⟶^{j:k} λ. By induction, this implies that μ^{−} ⊵ν^{R}. We can then apply Lemma 5.2 which implies that μ ⊵ λ^{R}.

Now let us assume that λ^{R} = μ. If ν ⊢_{l} n is such that d_{ν,μ^{−}}(v) ≠ 0 and ν ⟶^{j:k} λ then by induction, we have ν^{R} ⊵μ^{−}. Then we obtain ν^{R} = μ^{−}. Thus, the coefficient d̂_{λ,μ}(v) equals d̂_{ν,ν^{R}}(v) times the coefficient of λ in f_{j}^{(m)}ν. By Lemma 5.3 and by induction, this coefficient is v^{R(λ)}. Hence, we have shown that d̂_{λ,μ}(v) is zero unless μ ⊵ λ^{R} and d̂_{λ,λ^{R}}(v) = v^{R(λ)}. The LLT algorithm in §4.8 implies that

d_{λ,μ}(v) = d̂_{λ,μ}(v) + ∑_{ξ◁μ}α_{ξ,μ}(v)d_{λ,ξ}(v).

Now assume that μ is not greater than λ^{R} with respect to the dominance order. This implies that for all ξ ⊢_{l} n such that ξ ◁ μ, ξ is not greater than λ. By induction, we have d_{λ,ξ}(v) = 0. Thus, we obtain d_{λ,μ}(v) = d̂_{λ,μ}(v) and the result follows. This concludes the proof.

6. Regularization for Ariki–Koike algebras

We now present consequences for the representation theory of Ariki–Koike algebras.

6.1. Let s = (s_{0}, …, s_{l−1}) ∈ S^{l} and let η ∈ ℂ. We consider the associative ℂ-algebra H(s) generated by T_{0}, …, T_{n−1} subject to the relations (T_{0} − η^{s_{0}}) … (T_{0} − η^{s_{l−1}}) = 0, (T_{i} − η)(T_{i} + 1) = 0, for 1 ≤ i ≤ n and the type B braid relations

(T_{0}T_{1})^{2} = (T_{1}T_{0})^{2}, T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1} for 1 ≤ i < n,
T_{i}T_{j} = T_{j}T_{i} for j ≥ i + 2.

H(s) is called the Ariki–Koike algebra. Set e = +∞ if η is not a root of 1. Otherwise, e is the order of η as a root of 1. We assume that e ≠ 1. H(s) is non-semisimple in general and its representation theory is usually studied through its decomposition matrix which we now define. Following Dipper, James and Mathas [2], one can construct a certain set of finite dimensional H(s)-modules called Specht modules. We denote them by

{S^{λ} | λ ⊢_{l} n} .

The simple H(s)-modules are indexed by a certain set of multipartitions called FLOTW multipartitions Φ_{s,e}(n) (see [5, § 5.7])

{D^{μ} | μ ∈ Φ_{s,e}(n)} .

If e = ∞, we have Φ_{s,e}(n) = Φ_{s}(n), the set of cylindric multipartitions.

6.2. If λ ⊢_{l} n then one can consider the composition multiplicities [S^{λ} : D^{μ}], with μ ∈ Φ_{s,e}(n). The resulting matrix,

D := ([S^{λ} : D^{μ}])_{λ⊢_{l}n,μ∈Φ_{s,e}(n)},

is called the decomposition matrix of H(s). The problem of computing the decomposition matrix has been solved by Ariki [1] by proving a generalization of a conjecture by Lascoux, Leclerc and Thibon. The theorem asserts that the decomposition numbers [S^{λ} : D^{μ}] with μ ∈ Φ_{s,e}(n) corresponds to the coefficients of the canonical bases for an irreducible highest weight U_{v}(𝔰̂𝔩_{e})-module (realized as a submodule of the Fock space) evaluated at v = 1.

In the case e = +∞, this decomposition numbers are thus the polynomials d_{λ,μ}(v) evaluated at v = 1. We obtain the following result which is deduced from the above discussion and Theorem 5.1.

Theorem 6.3. Assume that e = +∞ and that μ is a fixed cylindric l-partition of rank n. Then, for any l-partition λ of rank n, we have [S^{λ} : D^{μ}] = 0 unless μ ⊵ λ^{R} while [S^{λ} : D^{λ^{R}}] = 1.

By results of several authors, the polynomials d_{λ,μ}(v) also have an interpretation in terms of the representation theory of Ariki–Koike algebras. They correspond to graded decomposition numbers (see [10]). Thus, Theorem 5.1 can be interpreted as a graded analogue of the above regularization theorem.

6.4. It is natural to ask what happens in the case where e ∈ ℕ. Here, the main problem is to find a natural order on the set of multipartitions which is the analogue of the dominance order on partitions. A natural choice for it is the one used in [7]. Now, the decomposition matrices for Ariki–Koike algebras can be computed using the algorithm described in this paper and implemented in [8]. Let e = 2, l = 2 , s = (0, 1) and n = 6 then for the 2-partitions λ = (3, 3), μ^{1} = (4, 2) and μ^{2} = (2, 4), we have [S^{λ} : D^{μ^{1}}] = [S^{λ} : D^{μ^{2}}] = 1 and there are no 2-partition ν such that [S^{λ} : D^{ν}] ≠ 0 which are less than μ^{1} and μ^{2}. Thus, an analogue of Theorem 6.3 is not available for these choices.

References

[1] S. Ariki, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series, vol. 26, American Mathematical Society, Providence, RI, 2002. Translated from the 2000 Japanese edition and revised by the author.

[2] R. Dipper, G. James, A. Mathas, Cyclotomic q-Schur Algebras, Mathematische Zeitschrift 229 (1998), 385–416.

[3] M. Fayers, Regularisation and the Mullineux map, Electron. J. Combin. 15(1) (2008), Research Paper 142, 15 pp.

[4] M. Fayers, q-analogues of regularisation theorems for linear and projective representations of the symmetric group, J. Algebra 316 (2007), 346–367.

[5] M. Geck and N. Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, vol. 15, Springer-Verlag, London, 2011.

[6] N. Jacon, On the parametrization of the simple modules for Ariki–Koike algebras at roots of unity, J. Math. Kyoto Univ. 44 (2004), 729–767.

[7] N. Jacon, An algorithm for the computation of the decomposition matrices for Ariki–Koike algebras, J. Algebra 292 (2005), 100–109.

[8] N. Jacon, GAP program for the computation of the decomposition matrices for Ariki–Koike

algebras, https://lmb.univ-fcomte.fr/IMG/zip/jacon_arikikoike.g.zip

[9] G. James, On the decomposition matrices of the symmetric groups II, J. Algebra 43 (1976), 45–54.

[10] A. Kleshchev, Representation theory of symmetric groups and related Hecke algebras, Bull. Amer. Math. Soc. 47 (2010), 419–481

[11] A. Lascoux, B. Leclerc and J-Y Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.

[12] B. Leclerc and H. Miyachi, Constructible characters and canonical bases, J. Algebra 277 (2004), 298–317.

[13] B. Leclerc and P Toffin, A simple algorithm for computing the global crystal basis of an irreducible U_{q}(sl_{n})-module, Internat. J. Algebra Comput. 10 (2000), 191–208.

[14] A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University lecture series, 15, American Mathematical Society (Providence, R.I.), 1999.

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