A SPECHT FILTRATION OF AN INDUCED SPECHT MODULE

ANDREW MATHAS

To John Cannon and Derek Holt on the occasions of their significant birthdays, in recognition of their distinguished contributions to mathematics.

Abstract. Let ℋ_{n} be a (degenerate or non-degenerate) Hecke algebra of type G(ℓ, 1, n), defined over a commutative ring R with one, and let S(μ) be a Specht module for ℋ_{n}. This paper shows that the induced Specht module S(μ)⊗_{ℋ_{n}} ℋ_{n+1} has an explicit Specht filtration.

1. Introduction

The Ariki-Koike algebras, and their rational degenerations, are interesting algebras which appear naturally in the representation theory of affine Hecke algebras, quantum groups, symmetric groups and general linear groups; see [14, 18] for details. They include as special cases the group algebras of the Coxeter groups of type A (the symmetric groups) and the Coxeter groups of type B (the hyperoctahedral groups).

Let ℋ_{n} be an Ariki-Koike algebra, or a degenerate cyclotomic Hecke algebra, of type G(ℓ, 1, n), for integers ℓ, n ≥ 1. For each multipartition μ of n there is a Specht module S(μ), which is a right ℋ_{n}-module. (All of the undefined terms and notation, here and below, can be found in section 2.) When ℋ_{n} is semisimple the Specht modules give a complete set of pairwise non-isomorphic irreducible ℋ_{n}-modules as μ runs through the multipartitions of n. In general, the Specht modules are not irreducible however every irreducible ℋ_{n}-module arises, in a unique way, as the simple head of some Specht module.

The Hecke algebra ℋ_{n} embeds into ℋ_{n+1} so there are natural induction and restriction functors, Ind and Res, between the categories of finite dimensional ℋ_{n}-modules and ℋ_{n±1}-modules. By [2, Proposition 1.9], in the Ariki-Koike case the restriction of the Specht module S(μ) to ℋ_{n−1} has a Specht filtration of the form

0 = R_{0} ⊂ R_{1} ⊂ ⋯ ⊂ R_{r} = Res S(μ),

such that R_{j}/R_{j−1} ≅ S(μ − ρ_{j}), where ρ_{1} > ρ_{2} > ⋯ > ρ_{r} are the removable nodes of μ. Consequently, if ℋ_{n+1} is semisimple then by Frobenius reciprocity

Ind S(μ) ≅ S(μ ∪ α_{1}) ⊕ ⋯ ⊕ S(μ ∪ α_{a}),

where α_{1}, …, α_{a} are the addable nodes of μ. This note generalizes this result to the case when ℋ_{n} is not necessarily semisimple. More precisely, we prove the following:

Main Theorem. Suppose that ℋ_{n} is an Ariki-Koike algebra or a degenerate cyclotomic Hecke algebra of type G(ℓ, 1, n) and let μ be a multipartition of n. Then, as an ℋ_{n+1}- module, the induced module Ind S(μ) has a filtration

0 = I_{0} ⊂ I_{1} ⊂ ⋯ ⊂ I_{a} = Ind S(μ),

2000 Mathematics Subject Classification. 20C08, 20C30, 05E10.

Key words and phrases. Cyclotomic Hecke algebras, Specht modules, representation theory.

This research was supported, in part, by the Australian Research Council.

such that I_{j}/I_{j−1} ≅ S(μ ∪ α_{j}), where α_{1} > α_{2} > ⋯ > α_{a} are the addable nodes of μ.

This result is part of the folklore for the representation theory of these algebras, however, we have been unable to find a proof of it in the literature when ℓ > 1. If ℓ = 1 then our Main Theorem is an old result of James [12, §17] in the degenerate case (that is, for the symmetric group), and it can be deduced from [9, Theorem 7.4] in the non-degenerate case (the Hecke algebra of the symmetric group). We prove our Main Theorem by giving an explicit construction of Ind S(λ); see Corollary (3.7). Our argument is similar in spirit to that originally used by James [12] for the symmetric groups in that we identify the induced module as a quotient of the corresponding permutation module. Our approach, which uses cellular basis techniques, gives an explicit Specht filtration of the induced module; in contrast, James’ approach is recursive.

Suppose now that ℋ_{n} is defined over a field of characteristic p ≥ 0, or a suitable discrete valuation ring. Then by projecting onto the blocks of ℋ_{n} the induction functor Ind can be decomposed as a direct sum of subfunctors

Ind = ⨁_{i∈I} i-Ind,

where I = ℤ/pℤ, in the degenerate case, and I = { q^{a}Q_{s} | a ∈ ℤ and 1 ≤ s ≤ r } in the non-degenerate case. (If the parameters Q_{1}, …, Q_{r} are all non-zero then, up to Morita equivalence, it is enough to consider the cases where Q_{1}, …, Q_{r} are all powers of q by the main result of [11]. In this case we can take I = ℤ/eℤ where e is the smallest positive integer such that 1 + q + ⋯ + q^{e−1} = 0.) The functor i-Ind is a natural generalization of Robinson’s i-induction functor; see [2, 1.11] and [14, §8] for the precise definitions.

(1.1). Corollary. Suppose that μ is a multipartition of n and i ∈ I. Then i-Ind S(μ) has a filtration

0 = I_{0} ⊂ I_{1} ⊂ ⋯ ⊂ I_{b} = i-Ind S(μ),

such that I_{j}/I_{j−1} ≅ S(μ ∪ α_{j}), where α_{1} > α_{2} > ⋯ > α_{b} are the addable i-nodes of μ.

Proof. By [15] and [4], the Specht modules S(μ ∪ α) and S(μ ∪ β) are in the same block if and only if α and β have the same residue. By the Main Theorem and the definition of the functor i-Ind, the Specht module S(μ ∪ α) is a subquotient of i-Ind S(μ) if and only if α is an i-node (cf. [2, Cor. 1.12]). This implies the result. □

Recently Brundan and Kleshchev [5] have shown that ℋ_{n} is naturally ℤ-graded and Brundan, Kleshchev and Wang [8] have shown that S(μ) admits a natural grading. There should be a graded analogue of our induction theorem; see [8, Remark 4.12] for a precise conjecture. Unfortunately, the arguments of this paper do not automatically lift to the graded setting because it is not clear how to use our results to find a homogeneous basis of the induced module.

2. Ariki-Koike algebras

In order to make this note self-contained, this section quickly recalls the definitions and results that we need from the literature and, at the same time, sets our notation. We concentrate on the non-degenerate case as the degenerate case follows in exactly the same way, with only minor changes of notation, using the results of [3, §6]. See the remarks at the end of this section for more details.

Throughout this note we fix positive integers ℓ and n and let 𝔖_{n} be the symmetric group of degree n. For 1 ≤ i < n let s_{i} = (i, i + 1) ∈ 𝔖_{n}. Then s_{1}, …, s_{n−1} are the standard Coxeter generators of 𝔖_{n}.

Let R be a commutative ring with 1 and let q, Q_{1}, …, Q_{ℓ} be elements of R with q invertible. The Ariki–Koike algebra ℋ_{n} = ℋ_{R,ℓ,n}(q, Q_{1}, …, Q_{ℓ}) is the associative unital R–algebra with generators T_{0}, T_{1}, …, T_{n−1} and relations

(T_{0} − Q_{1}) … (T_{0} − Q_{ℓ}) = 0,
(T_{i} − q)(T_{i} + 1) = 0,  for 1 ≤ i ≤ n − 1,
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},
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.

Using the relations it follows that there is a unique anti-isomorphism ∗: ℋ_{n} ⟶ ℋ_{n} such that T_{i}^{∗} = T_{i}, for 0 ≤ i < n.

Ariki and Koike [1, Theorem 3.10] showed that ℋ_{n} is free as an R-module with basis { L_{1}^{a_{1}} … L_{n}^{a_{n}}T_{w} | 0 ≤ a_{1}, …, a_{n} < ℓ and w ∈ 𝔖_{n} } where L_{1} = T_{0} and L_{i+1} = q^{−1}T_{i}L_{i}T_{i} for i = 1, …, n−1, and T_{w} = T_{i_{1}} … T_{i_{k}} if w = s_{i_{1}} … s_{i_{k}} ∈ 𝔖_{n} is a reduced expression (that is, k is minimal).

The Ariki-Koike basis theorem implies that there is a natural embedding of ℋ_{n} in ℋ_{n+1} and that ℋ_{n+1} is free as an ℋ_{n}-module of rank ℓ(n + 1). If M is an ℋ_{n}-module let

Ind M = M ⊗_{ℋ_{n}} ℋ_{n+1}

be the corresponding induced ℋ_{n+1}-module. Note that induction is an exact functor since ℋ_{n+1} is free as an ℋ_{n}-module.

We will need to the following easily proved property of the basis elements [10, 2.1].

(2.1). Suppose that 1 ≤ k ≤ n, a ∈ R and w ∈ 𝔖_{k} × 𝔖_{n−k}. Then

(L_{1} − a) … (L_{k} − a)T_{w} = T_{w}(L_{1} − a) … (L_{k} − a).

The algebra ℋ_{n} has another basis which is crucial to this note. In order to describe it recall that a partition of n is a weakly decreasing sequence λ = (λ_{1} ≥ λ_{2} ≥ …) of nonnegative integers such that |λ| = ∑_{i} λ_{i} = n. A multipartition, or ℓ-partition, of n is an ordered ℓ-tuple λ = (λ^{(1)}, …, λ^{(ℓ)}) of partitions such that |λ| = |λ^{(1)}| + ⋯ + |λ^{(ℓ)}| = n. Let Λ_{ℓ,n}^{+} be the set of multipartitions of n. If λ, μ ∈ Λ_{ℓ,n}^{+} then λ dominates μ, and we write λ ⊵ μ, if

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

for 1 ≤ s ≤ ℓ and for all k ≥ 1. Dominance is a partial order on Λ_{ℓ,n}^{+}.

If λ is a multipartition let 𝔖_{λ} = 𝔖_{λ^{(1)}} × ⋯ × 𝔖_{λ^{(ℓ)}} be the corresponding parabolic subgroup of 𝔖_{n} and set a_{s}^{λ} = ∑_{t=1}^{s−1} |λ^{(t)}|, for 1 ≤ s ≤ ℓ, and put a_{ℓ+1}^{λ} = n − 1. Define m_{λ} = u_{λ}^{+}x_{λ} where

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

Then u_{λ}^{+}x_{λ} = m_{λ} = x_{λ}u_{λ}^{+} by (2.1).

Let λ be a multipartition (of n). The diagram of λ is the set of nodes

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

More generally a node is any element of ℕ × ℕ × {1, …, ℓ}, which we consider as a partially ordered set where (r, c, s) ≥ (r′, c′, s′) if either s > s′, or s = s′ and r < r′. For the sake of Corollary (1.1) only, define the residue of the node (r, c, s) to be q^{c−r}Q_{s}.

An addable node of λ is any node α ∉ [λ] such that [λ] ∪ {α} is the diagram of some multipartition. Let λ ∪ α be the multipartition such that [λ ∪ α] = [λ] ∪ {α}. Similarly, a removable node of λ is a node ρ ∈ [λ] such that [λ] − {ρ} is the diagram of a multipartition; let λ − ρ be this multipartition. Note that the set of addable and removable nodes for λ are both totally ordered by >.

If X is a set then an X-valued λ-tableau is a function T : [λ]⟶X. If T is a λ-tableau then we write Shape(T) = λ. For convenience we identify T = (T^{(1)}, …, T^{(ℓ)}) with a labeling of the diagram [λ] by elements of X in the obvious way. Thus, we can talk of the rows, columns and components of T.

A standard λ-tableau is a map 𝔱: [λ] ⟶ {1, 2, …, n} such that for s = 1, …, ℓ the entries in each row of 𝔱^{(s)} increase from left to right and the entries in each column of 𝔱^{(s)} increase from top to bottom. Let T^{Std}(λ) be the set of standard λ-tableaux.

Let 𝔱^{λ} be the standard λ-tableau such that the entries in 𝔱^{λ} increase from left to right along the rows of 𝔱^{λ^{(1)}}, …, 𝔱^{λ^{(ℓ)}} in order. If 𝔱 is a standard λ-tableau let d(𝔱) ∈ 𝔖_{n} be the unique permutation such that 𝔱 = 𝔱^{λ}d(𝔱). Define m_{𝔰𝔱} = T_{d(𝔰)}^{∗}m_{λ}T_{d(𝔱)}, for 𝔰, 𝔱 ∈ T^{Std}(λ). By [10, Theorem 3.26], the set

{ m_{𝔰𝔱} | 𝔰, 𝔱 ∈ T^{Std}(λ) and λ ∈ Λ_{ℓ,n}^{+} }

is a cellular basis of ℋ_{n}. Consequently, if ℋ_{n}(λ) is the R-module spanned by

{ m_{𝔰𝔱} | 𝔰, 𝔱 ∈ T^{Std}(μ) for some μ ∈ Λ_{ℓ,n}^{+} with μ ▷ λ } ,

then ℋ_{n}(λ) is a two-sided ideal of ℋ_{n}.

The Specht module S(λ) is the submodule of ℋ_{n}/ℋ_{n}(λ) generated by m_{λ} +ℋ_{n}(λ). It follows from the general theory of cellular algebras that S(λ) is free as an R-module with basis { m_{𝔱} | 𝔱 ∈ T^{Std}(λ) }, where m_{𝔱} = m_{𝔱^{μ}𝔱} + ℋ_{n}(λ) for 𝔱 ∈ T^{Std}(λ).

Let M be an ℋ_{n}-module. Then M has a Specht filtration if there exists a filtration

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{k} = M

and multipartitions λ_{1}, …, λ_{k} such that M_{i}/M_{i−1} ≅ S(λ_{i}), for i = 1, …, k.

For each multipartition μ ∈ Λ_{ℓ,n}^{+} let M(μ) = m_{μ}ℋ_{n}. The final result that we will need gives an explicit Specht filtration of M(μ). The proof of our Main Theorem is inspired by this filtration.

Given two tuples (i, s) and (j, t) write (i, s) ⪯ (j, t) if either s < t, or s = t and i ≤ j.

(2.2). Definition ( [10, Definition 4.4]). Suppose that λ, μ ∈ Λ_{ℓ,n}^{+} and let T : [λ] ⟶ ℕ × {1, 2, …, ℓ} be a λ-tableau. Then:

a) T is a tableau of type μ if μ_{i}^{(s)} = # { x ∈ [λ] | T(x) = (i, s) }, for all i ≥ 1 and 1 ≤ s ≤ ℓ.

b) T is semistandard if the entries in each component T^{(s)}, for 1 ≤ s ≤ ℓ, of T are: i) weakly increasing from left to right along each row (with respect to ⪯); ii) strictly increasing from top to bottom down columns; and, iii) (j, t) appears in T^{(s)} only if t ≥ s. Let T_{μ}^{SStd}(λ) be the set of semistandard λ–tableau of type μ and let T_{μ}^{SStd}(Λ_{ℓ,n}^{+}) = ⋃_{λ∈Λ_{ℓ,n}^{+}} T_{μ}^{SStd}(λ) be the set of all semistandard tableaux of type μ.

Let 𝔱 be a standard λ–tableau. Define μ(𝔱) to be the tableau obtained from 𝔱 by replacing each entry j in 𝔱 with (i, s) if j appears in row i of 𝔱^{μ^{(s)}}. The tableau μ(𝔱) is a λ–tableau of type μ; it is not necessarily semistandard. Finally, if S ∈ T_{μ}^{SStd}(λ) and 𝔱 ∈ T^{Std}(λ) set

m_{S𝔱} = ∑_{\begin{matrix}𝔰∈T ^{Std}(λ) \\ μ(𝔰)=S \end{matrix}} m_{𝔰𝔱}.

(2.3) ( [10, Theorem 4.14 and Corollary 4.15]). Suppose that λ, μ ∈ Λ_{ℓ,n}^{+}. Then: a) M(μ) is free as an R–module with basis

{ m_{S𝔱} | S ∈ T_{μ}^{SStd}(λ), 𝔱 ∈ T^{Std}(λ) for someλ ∈ Λ_{ℓ,n}^{+} } .

b) Suppose that T_{μ}^{SStd}(Λ_{ℓ,n}^{+}) = {S_{1}, …, S_{m}} ordered so that i ≤ j whenever λ_{i} ⊵ λ_{j}, where λ_{i} = Shape(S_{i}). Let M_{i} be the R-submodule of M(μ) spanned by the elements { m_{S_{j}𝔱} | j ≤ i and ∈ T^{Std}(λ_{j}) }, Then

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{m} = M(μ)

is an ℋ_{n}-module filtration of M(μ) and M_{i}/M_{i−1} ≅ S(λ_{i}), for 1 ≤ i ≤ m.

(2.4). Remark. Very few changes need to be made to the results above in the degenerate case. The analogue of the cellular basis {m_{𝔰𝔱}} in the degenerate case is constructed in [3, §6]. Using this basis of the degenerate Hecke algebra, the construction of the Specht filtration of the ideals M(μ) follows easily using the arguments of [10, §4]; cf. [7, Cor. 6.13]. The arguments in the next section, modulo minor differences in the meaning of the symbols, applies to both the degenerate and non-degenerate cases.

3. Inducing Specht modules

We are now ready to start proving the Main Theorem. Fix a multipartition μ ∈ Λ_{ℓ,n}^{+}. As in (2.3) we let T_{μ}^{SStd}(Λ_{ℓ,n}^{+}) = {S_{1}, …, S_{m}} be the set of semistandard tableau of type μ ordered so that i ≤ j whenever λ_{i} ⊵ λ_{j}, where λ_{i} = Shape(S_{i}) for 1 ≤ i ≤ m. So, in particular, S_{m} = T^{μ} = μ(𝔱^{μ}) is the unique semistandard μ-tableaux of type μ.

Throughout this section we freely identify ℋ_{n} with its image under the natural embedding ℋ_{n} ↪ ℋ_{n+1}. In particular, we will think of the basis element m_{𝔰𝔱} as an element of ℋ_{n+1}, for standard λ-tableaux 𝔰, 𝔱 ∈ T^{Std}(λ) with λ ∈ Λ_{ℓ,n}^{+}. This embedding also identifies Ind M(μ) with a submodule of ℋ_{n+1}.

The following simple Lemma contains the idea which drives our proof.

(3.1). Lemma. Suppose that μ is a multipartition of n and let ω be the lowest addable node of μ (that is, α ≥ ω whenever α is an addable node of μ). Then :

a) Ind M(μ) = M(μ ∪ ω).

b) The induced module Ind M(μ) has a filtration

0 = N_{0} ⊂ N_{1} ⋯ ⊂ N_{m} = Ind M(μ)

such that N_{i}/N_{i−1} ≅ Ind S(λ_{i}), where λ_{i} = Shape(S_{i}) for 1 ≤ i ≤ m.

Proof. By definition, m_{μ} = m_{μ∪ω} using the embedding ℋ_{n} ↪ ℋ_{n+1}. Therefore,

Ind M(μ) = m_{μ}ℋ_{n} ⊗_{ℋ_{n}} ℋ_{n+1} = m_{μ}ℋ_{n+1} = m_{μ∪ω}ℋ_{n+1} = M(μ ∪ ω),

proving (a). As induction is exact, part (b) follows from part (a) and (2.3)(b). □

If μ = ((n), (0), …, (0)) then S(μ) = M(μ). The Main Theorem in this special case is just part (b) of the Lemma. To prove the theorem when μ ≠ ((n), (0), …, (0)) we explicitly describe the filtration of Ind M(μ) given by the Lemma in terms of the basis of M(μ ∪ ω) from (2.3).

Let ω be the lowest addable node of μ. Then ω = (z, 1, ℓ), where z ≥ 1 is minimal such that (z, 1, ℓ) ∉ [μ]. Suppose that S ∈ T_{μ}^{SStd}(λ), for some λ ∈ Λ_{ℓ,n}^{+}, and that β is an addable node of λ. Let S ∪ β be the semistandard (λ ∪ β)-tableau given by

(S ∪ β)(η) = \begin{cases}S(η), & if η ∈ [λ], \\ (z, ℓ), & if η = β. \end{cases}

Thus S ∪ β is the semistandard (λ ∪ β)-tableau of type μ ∪ ω obtained by adding the node β to S with label (z, ℓ). Let T_{μ∪ω}^{SStd}(S) be the set of semistandard tableau of type μ ∪ ω obtained in this way from S as β runs over the addable nodes of λ. It is easy to see that every semistandard tableau of type μ ∪ ω arises uniquely in this way, so

(3.2)  T_{μ∪ω}^{SStd}(Λ_{ℓ,n+1}^{+}) = ∐_{S∈T_{μ}^{SStd}(Λ_{ℓ,n}^{+})} T_{μ∪ω}^{SStd}(S).

Armed with this notation, observe that if S ∈ T_{μ}^{SStd}(λ) then m_{S𝔱^{λ}} = m_{(S∪β)𝔱^{λ∪β}}, as an element of ℋ_{n+1}, where β is the lowest addable node of λ.

Suppose that 1 ≤ a ≤ b < n. Let 𝔖_{a,b} be the symmetric group on {a, a+1, …, b} and set s_{b,a} = (b, b + 1) … (a, a + 1) ∈ 𝔖_{n} and T_{b,a} = T_{s_{b,a}} = T_{b} … T_{a}. For convenience, we set T_{b,a} = 1 if b < a. The following useful identity is surely known.

(3.3). Lemma. Suppose that 1 ≤ a < b ≤ n. Then

( ∑_{w∈𝔖_{a,b}} T_{w})T_{b,a} = T_{b,a}( ∑_{v∈𝔖_{a+1,b+1}}T_{v}).

Proof. It is easy to check that 𝔖_{a,b}s_{b,a} = s_{b,a}𝔖_{a+1,b+1} and that s_{b,a} is a distinguished (𝔖_{a,b}, 𝔖_{a+1,b+1})-double coset representative (in the sense of [16, Prop. 4.4], for example). Therefore, if w ∈ 𝔖_{a,b} and v = s_{b,a}ws_{b,a} ∈ 𝔖_{a+1,b+1} then T_{w}T_{b,a} = T_{ws_{b,a}} = T_{s_{b,a}v} = T_{b,a}T_{v} by [16, Prop. 3.3]. This implies the Lemma. □

(3.4). Lemma. Suppose that λ ∈ Λ_{ℓ,n}^{+} and ν = λ ∪ β, where β = (r, c, e) is an addable

node of λ. Then T_{n−1,a+1}m_{ν} ∈ m_{λ}ℋ_{n+1}, where a = a_{1}^{λ} + ⋯ + a_{e}^{λ} + λ_{1}^{(e)} + ⋯ + λ_{r}^{(e)}.

Proof. Let D_{d,a} = 1 + T_{a} + T_{a,a−1} + ⋯ + T_{a,d}, where d = a − λ_{r}^{(e)} + 1. Then D_{d,a} is the sum of distinguished right coset representatives for 𝔖_{d,a} in 𝔖_{d,a+1}. Therefore, x_{λ}T_{n−1,a+1}D_{d,a} = T_{n−1,a+1}x_{ν} by Lemma (3.3). On the other hand, it follows directly from the definitions that u_{ν}^{+} = u_{λ}^{+}(L_{a_{ℓ}^{λ}+1} −Q_{ℓ}) … (L_{a_{e+1}^{λ}+1} −Q_{e+1}). Therefore, writing m_{λ} = x_{λ}u_{λ}^{+} and using (2.1) we see that

m_{λ}( ∏_{s=ℓ,...,e+1} T_{a_{s+1}^{λ},a_{s}^{λ}+1}(L_{a_{s}^{λ}+1} − Q_{s}))T_{a_{e+1}^{λ},a+1}D_{d,a} = T_{n−1,a+1}m_{ν},

where the product on the left-hand side is read in order, from left to right, with decreasing values of s. (Recall that, for convenience, a_{ℓ+1}^{λ} = n − 1 and T_{n−1,n} = 1.) □

Let ≤ be the Bruhat order on 𝔖_{n}; see, for example, [16, p.30]. If S is a semistandard λ-tableau of type μ let Ṡ be the unique standard λ-tableau such that μ(Ṡ) = S and d(Ṡ) ≤ d(𝔰) whenever 𝔰 ∈ T^{Std}(λ) and μ(𝔰) = S. Such a tableau Ṡ exists by [13, Lemma 3.9].

(3.5). Lemma. Suppose that S ∈ T_{μ}^{SStd}(λ) and that U ∈ T_{μ∪ω}^{SStd}(S). Let ν = Shape(U). Then m_{U𝔱^{ν}} ∈ m_{S𝔱^{λ}}ℋ_{n+1}.

Proof. Definition, m_{S𝔱^{λ}} = ∑_{𝔰} m_{𝔰𝔱^{λ}} where d(𝔰) runs over a set of right 𝔖_{μ}-coset representatives in the double coset 𝔖_{λ}d(Ṡ)𝔖_{μ}. Therefore, m_{S𝔱^{λ}} = h_{S}T_{d(Ṡ)}^{∗}m_{λ} for some h_{S} ∈ ℋ_{q}(𝔖_{μ}). (Explicitly, h_{S} = ∑_{d} T_{d} where d runs over the set of distinguished left coset representatives of 𝔖_{μ} ∩ d(Ṡ)𝔖_{λ}d(Ṡ)^{−1} in 𝔖_{μ}.)

As in Lemma (3.4), write ν = λ ∪ β, where β = (r, c, e) and set a = a_{1}^{λ} + ⋯ + a_{e}^{λ} + λ_{1}^{(e)} + ⋯ + λ_{r}^{(e)}. Then U = S ∪ β. Therefore, d(U̇) = s_{n−1,a+1}^{−1}d(Ṡ), so that m_{U𝔱^{ν}} = hT_{d(Ṡ)}^{∗}T_{n−1,a+1}m_{ν}.

Finally, T_{n−1,a+1}m_{ν} = m_{λ}h_{ν,a}, for some h_{ν,a} ∈ ℋ_{n+1}, by Lemma (3.4). Therefore,

m_{U𝔱^{ν}} = h_{S}T_{d(Ṡ)}^{∗}T_{n−1,a+1}m_{ν} = h_{S}T_{d(Ṡ)}^{∗}m_{λ}h_{ν,a} = m_{S𝔱^{λ}}h_{ν,a} ∈ m_{S𝔱^{λ}}ℋ_{n+1},

as required. □

We can now make the filtration of Lemma (3.1)(b) explicit. As a result we will show that we can obtain a basis for the induced module by adding a node labeled (z, ℓ) to the basis elements of M(μ) in all possible ways.

(3.6). Theorem. Suppose that μ ∈ Λ_{ℓ,n}^{+} and order T_{μ}^{SStd}(Λ_{ℓ,n}^{+}) = {S_{1}, …, S_{m}} as above, with λ_{i} = Shape(S_{i}). Let N_{i} be the R-submodule of M(μ ∪ ω) spanned by the elements

{ m_{U𝔳} | U ∈ T_{μ∪ω}^{SStd}(S_{j}), 𝔳 ∈ T^{Std}(Shape(U)) for 1 ≤ j ≤ i } ,

for i = 0, 1, …, m. Then N_{i} is an ℋ_{n+1}-submodule of Ind M(λ) and

Ind S(λ_{i}) ≅ N_{i}/N_{i−1},

for 1 ≤ i ≤ m.

Proof. By Lemma (3.1)(a), Ind M(μ) = M(μ ∪ ω) and by (3.2) the set of elements

{ m_{U𝔳} | U ∈ T_{μ∪ω}^{SStd}(S_{j}), 𝔳 ∈ T^{Std}(Shape(U)) for 1 ≤ j ≤ m }

is precisely the basis of M(μ ∪ ω) given by (2.3), so M(μ ∪ ω) = N_{m}. Moreover, since ℋ_{n+1}(ν) is a two-sided ideal of ℋ_{n+1} for all ν ∈ Λ_{ℓ,n+1}^{+}, the action of ℋ_{n+1} on the basis {m_{U𝔳}} respects dominance, so N_{i} is a submodule of M(μ ∪ ω), for 0 ≤ i ≤ m.

Recall the filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{m} = M(λ) of M(λ) given in (2.3). By Lemma (3.1)(b), to prove the Theorem it is enough to show by induction on i that Ind M_{i} = N_{i}, for 0 ≤ i < m. This is trivially true when i = 0 so we may assume that i > 0.

To show that Ind M_{i} ⊆ N_{i} note that m_{S_{i}𝔱^{^{λ}i}} ∈ N_{i} and m_{S_{i}𝔱^{^{λ}i}} + M_{i−1} generates M_{i}/M_{i−1} as an ℋ_{n}-module. Therefore, Ind M_{i} ⊆ N_{i} by induction on i.

To prove the reverse inclusion, suppose that U ∈ T_{μ∪ω}^{SStd}(S_{i}) and let ν = Shape(U). Then m_{U𝔱^{ν}} ∈ m_{S_{i}𝔱^{^{λ}i}} ℋ_{n+1} ⊆ Ind M_{i} by Lemma (3.5). Therefore, m_{U𝔳} ∈ Ind M_{i}, for any 𝔳 ∈ T^{Std}(ν). It follows by induction that N_{i} ⊆ Ind M_{i} as required. □

For each addable node β of μ let N^{β} be the submodule of M(μ ∪ ω) spanned by

{ m_{U𝔳} | U ∈ T_{μ∪ω}^{SStd}(λ), 𝔳 ∈ T^{Std}(λ) where λ ∈ Λ_{ℓ,n+1}^{+} and λ ▷ μ ∪ β } + N_{m−1},

where N_{m−1} is the submodule of M(μ ∪ ω) defined in Theorem (3.6). Note, in particular, that N^{α} = N_{m−1}.

We can now prove a more explicit version of the Main Theorem of this paper.

(3.7). Corollary. Suppose that μ is a multipartition of n and let α_{1} = α > ⋯ > α_{a} = ω be the addable nodes of μ. Then Ind S(μ) ≅ M(μ∪ω)/N^{α} is a free R-module with basis

{ m_{U𝔳} + N^{α} | U ∈ T_{μ∪ω}^{SStd}(μ ∪ α_{j}), 𝔳 ∈ T^{Std}(μ ∪ α_{j}), for 1 ≤ j ≤ a } .

In particular, Ind S(μ) has a filtration 0 = I_{0} ⊂ I_{1} ⊂ ⋯ ⊂ I_{a} = Ind S(μ) such that I_{j}/I_{j−1} ≅ S(μ ∪ α_{j}), for j = 1, …, a.

Proof. That Ind S(μ) ≅ M(μ ∪ ω)/N^{α} is a special case of Theorem (3.6). The second claim follows from (2.3) by setting I_{j} = N^{α_{j+1}}/N^{α}, for 0 ≤ j < a. To prove that S(μ ∪ α_{j}) ≅ I_{j}/I_{j−1}, for 1 ≤ j ≤ a, observe that the bijective map

S(μ ∪ α_{j}) ⟶ I_{j}/I_{j−1}; m_{𝔰} ↦ m_{(T^{μ}∪α_{j})𝔰} + I_{j−1}, for 𝔰 ∈ T^{Std}(μ ∪ α_{j}),

commutes with the action of ℋ_{n+1}. (Here, T^{μ} = μ(𝔱^{μ}) is the unique semistandard μtableau of type μ.) □

(3.8). Remark. Maintain the notation of Theorem (3.6) and define integers a_{i} and multipartitions λ_{i,j} by writing {λ_{i,1} ▷ ⋯ ▷ λ_{i,a_{i}}} = { Shape(U) | U ∈ T_{μ∪ω}^{SStd}(S_{i}) }, for i = 1, …, m. Theorem (3.6) then implies, just as in the proof of Corollary (3.7), that M(μ∪ω) has a Specht filtration

0 ⊂ I_{1,1} ⊂ ⋯ ⊂ I_{1,a_{1}} ⊂ I_{2,1} ⊂ ⋯ ⊂ I_{m,a_{m}} = M(μ ∪ ω),

with I_{i,a}/I_{i,a}^{<} ≅ S(λ_{i,a}), where I_{i,a} is the submodule of M(μ ∪ ω) with basis

{ m_{U𝔳} | U ∈ T_{μ∪ω}^{SStd}(λ_{j,b}), 𝔳 ∈ T^{Std}(λ_{j,b}) where j < i, or j = i and b ≤ a }

and where I_{i,a}^{<} = I_{i,a−1} if a > 1, I_{i,1}^{<} = I_{i−1,a_{i−1}} if i > 1 and I_{1,1}^{<} = 0.

Fred Goodman has pointed out that this filtration of M(μ ∪ ω) is, in general, different to that given by (2.3) because the order in which the Specht modules appear does not have to be compatible with the dominance ordering–note, however, that the Specht modules in each ‘layer’ N_{i}/N_{i−1} are totally ordered by dominance. For example, suppose that ℓ = 1 and let μ = (3^{2}, 1) so that μ ∪ α = (4, 3, 1) and μ ∪ ω = (3^{2}, 1^{2}). Then

U = 1 1 1 2 2 2 3 4

is a semistandard ν-tableau of type μ ∪ ω, where ν = (4, 2^{2}). (As ℓ = 1 we can label semistandard tableaux with the integers 1, …, n.) However, μ∪α▷ν even though ν ≠ μ∪β for any addable node β of μ.

As induction and restriction are both exact functors the main result of this note, together with [2, Prop. 1.9] (and the corresponding argument for the degenerate case), shows that the full subcategory of ℋ_{n}-mod which consists of modules which have a Specht filtration is closed under induction and restriction.

(3.9). Corollary. Suppose that M has a Specht filtration. Then the modules Res M and Ind M both have Specht filtrations.

In [17, Theorem 3.6] and [6, Theorem 4.6] it is shown that for each multipartition μ ∈ Λ_{ℓ,n}^{+} there exists an indecomposable ℋ_{n}-module Y (μ), a Young module, such that

M(μ) ≅ Y (μ) ⊕ ⨁_{λ▷μ} Y (λ)^{⊕c_{λμ}}

for some non-negative integers c_{λμ}. Each Young module Y (μ) has a Specht filtration. Therefore, by Corollary (3.9), Res Y (μ) and Ind Y (μ) both have Specht filtrations.

Acknowledgments

I thank Fred Goodman and Alexander Kleshchev for (independently) asking the question that led to this paper. I also thank them for their comments and suggestions which improved our exposition.

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School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia.

E-mail address: a.mathas@usyd.edu.au