STRAIGHTENING RULE FOR AN m′-TRUNCATED POLYNOMIAL RING

KAY JIN LIM

Abstract. We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of symmetric group and Schur algebra generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiple of the odd prime p the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pair of partitions. As a result, the combinatorial number, for each partition, the number of compositions which can be rearranged to the partition and whose all partial sums are not divisible by p arises naturally.

1. Introduction

The ring of symmetric functions plays an important role in both combinatorics and representation theory of the symmetric groups (see [13, §I.7] and [12]). In the characteristic zero case, the characteristic map is an isometric isomorphism between the ring of symmetric functions and the Green ring for symmetric group where Schur functions correspond to Specht modules, the complete and elementary symmetric functions correspond to the trivial and signature representations for the symmetric groups such that the multiplication of two symmetric functions correspond to the induction of the outer tensor product of the two respective modules. In the positive characteristic case, the ring of symmetric functions is isomorphic to the Green ring of the symmetric group generated by Young permutation modules.

Let k be a field of odd characteristic p. In [4], Donkin showed that the isomorphism classes of the signed Young permutation modules

M(α|β) = Ind_{𝔖_{α}×𝔖_{β}}^{𝔖_{n}}(k(α) ⊠ sgn(β)),

where (α|β) is a pair of compositions such that the sizes of α and β sum up to n and k(α), sgn(β) are the trivial and sign k𝔖_{α}- and k𝔖_{β}-modules respectively, are parametrised by the set 𝒫_{p}^{2}(n) consisting of pairs of partitions of the form (λ|pμ) such that |λ|+p|μ| = n. Furthermore, M(λ|pμ) has a distinguished indecomposable summand the signed Young module Y (λ|pμ) of multiplicity one such that any other summand of M(λ|pμ) is of the form Y (δ|pξ) for some (δ|pξ) ▷ (λ|pμ), here, ▷ is certain dominance order on 𝒫_{p}^{2}(n). In other words, the Green ring of the symmetric group generated by signed Young permutation modules, denoted by 𝒴 (𝔖), has ℤbases C = {[M(λ|pμ)] : (λ|pμ) ∈ 𝒫_{p}^{2}} and {[Y (λ|pμ)] : (λ|pμ) ∈ 𝒫_{p}^{2}} where 𝒫_{p}^{2} = ⋃_{n∈ℕ_{0}} 𝒫_{p}^{2}(n).

2010 Mathematics Subject Classification. 05E05, 20C30, 20G43.

Let E be the natural module for the Schur algebra S(∞, 1). The mixed powers are defined as

K^{α|β}E = S^{α}E ⊗ ⋀^{β} E.

For each (λ|pμ) ∈ 𝒫_{p}^{2}, the mixed power K^{λ|pμ}E has a distinguished indecomposable summand the listing module List^{λ|pμ}E of multiplicity one such that any other summand of K^{λ|pμ}E is of the form List^{δ|pξ}E for some (δ|pξ) ▷ (λ|pμ). In other words, the Green ring of the Schur algebra generated by mixed powers, denoted by ℒ (S), has ℤ-bases D = {[K^{λ|pμ}E] : (λ|pμ) ∈ 𝒫_{p}^{2}} and {[List^{λ|pμ}E] : (λ|pμ) ∈ 𝒫_{p}^{2}}. In fact, ℒ (S) is isomorphic to 𝒴 (𝔖) induced by the Schur functor where K^{α|β}E and List^{λ|pμ}E are mapped to M(α|β) and Y (λ|pμ) respectively.

In the classical case, the class of non-isomorphic direct summands of Young permutation k𝔖_{n}-modules are known as Young modules and labelled by the set of partitions of n. Determining the multiplicity of a Young module as a direct summand of a Young permutation module is an open problem (see [3, 5, 7, 9, 11]). The numbers are known as the p-Kostka numbers. The signed p-Kostka numbers are generalisation of the p-Kostka numbers which are the multiplicities of signed Young modules as direct summands of signed Young permutation modules (see [6]). While writing a signed Young permutation module as ℤ-linear combination of the signed Young modules in the Green ring is an open problem, in this article, we present an explicit formula writing a signed Young permutation module (respectively, a mixed power) in terms of the basis C (respectively, D). The proof relies on the categorification theorem of by certain quotient Γ^{(p)} of a polynomial ring by both 𝒴 (𝔖) and ℒ (S) proved by Donkin in [4]. Along the way, the combinatorial number, for each partition λ, the number of compositions δ such that δ can be rearranged to λ and all partial sums of δ are not divisible by p arises naturally. The numbers appear in the coefficients of rewriting the sign representations (or the exterior powers) in terms of the basis C (or D).

The article is organised as follows. In the next section, we recall the symmetric functions, certain quotient Γ^{(m)} of a polynomial ring, where m is a positive integer, the Green rings 𝒴 (𝔖) and ℒ (S). In Section 3, we study some properties of the ring Γ^{(m)}. In Section 4, we prove the main result Theorem 4.2 and deduce the formulae of writing a signed Young permutation module and a mixed power in terms of the bases C and D respectively. As a consequence of the formulae, for any pair of composition (α|β), we deduce the ‘canonical’ summand of the signed Young permutation module M(α|β) (respectively, the mixed power K^{α|β}E).

2. Preliminaries

In this section, we fix the notation we shall require throughout in the article and introduce the background material. The standard references are [4, 8, 10, 13].

Let ℤ be the set of integers, let ℕ be the set of positive integers and let ℕ_{0} be the set of non-negative integers. For a finite set A, let 𝔖_{A} denote the symmetric group acting on the set A. For each n ∈ ℕ_{0}, let

[n] = {i ∈ ℕ : 1 ≤ i ≤ n}

and 𝔖_{n} = 𝔖_{[n]}. By convention, [0] = ∅ and 𝔖_{0} is the trivial group.

Let n ∈ ℕ_{0}. A composition λ of n is a sequence of positive integers (λ_{1}, …, λ_{r}) such that ∑_{i=1}^{r} λ_{i} = n. In this case, we write ℓ(λ) = r and |λ| = n. By convention, the unique composition of 0 is denoted as ∅ and ℓ(∅) = 0. The set of all compositions of n is denoted by 𝒞 (n) and we write 𝒞 = ⋃_{n∈ℕ_{0}} 𝒞 (n). The composition λ is called a partition if λ_{1} ≥ ⋯ ≥ λ_{r}. We write 𝒫(n) for the set of partitions of n and 𝒫 = ⋃_{n∈ℕ_{0}} 𝒫(n) for the set of all partitions.

The concatenation of two compositions α and β is the composition

α#β = (α_{1}, …, α_{ℓ(α)}, β_{1}, …, β_{ℓ(β)}).

By convention, α#∅ = α and, similarly, ∅#β = β. Let m ∈ ℕ and μ = (μ_{1}, …, μ_{s}) be a composition. We write mμ for the composition (mμ_{1}, …, mμ_{s}) of m|μ|.

The set of all pairs (α|β) of compositions of n, i.e., α, β are compositions and |α|+|β| = n, is denoted by 𝒞^{2}(n). We write 𝒞^{2} = ⋃_{n∈ℕ_{0}} 𝒞^{2}(n). Similarly, we write 𝒫^{2}(n) for the set of pairs of partitions of n and 𝒫^{2} = ⋃_{n∈ℕ_{0}} 𝒫^{2}(n). For m ∈ ℕ, the subset of 𝒫^{2}(n) consisting of pairs of the form (λ|mμ) is denoted by 𝒫_{m}^{2}(n). Furthermore, we write 𝒫_{m}^{2} = ⋃_{n∈ℕ_{0}} 𝒫_{m}^{2}(n).

Throughout, k is a field of odd characteristic p.

2.1. Symmetric function. Let ℤ⟦X⟧ be the set of formal power series in the set consisting of the countably infinite commuting variables X = {x_{i} : i ∈ ℕ} with coefficients in ℤ such that deg(x_{i}) = 1 for all i ∈ ℕ. For each n ∈ ℕ, the symmetric group 𝔖_{n} acts on ℤ⟦X⟧ by permuting the variables x_{1}, …, x_{n} naturally, i.e., for each σ ∈ 𝔖_{n} and f(X) ∈ ℤ⟦X⟧, the element σ · f(X) is obtained from f(X) by replacing each x_{i} with x_{σ(i)}. The element f(X) is called a symmetric function if it is invariant under the actions 𝔖_{n} for all n ∈ ℕ.

The set of symmetric functions with degrees bounded above in the set of variables X is denoted by Λ(X). Notice that Λ(X) is a commutative graded ring where Λ(X) = ⨁_{n∈ℕ_{0}} Λ^{n}(X) and Λ^{n}(X) consists of homogeneous symmetric functions of degree n. In the theory of the symmetric function, Λ(X) has different ℤ-bases of special interest. In particular, we are interested in the elementary symmetric functions and complete symmetric functions which we shall now describe.

For any n ∈ ℕ, the nth elementary symmetric function and the nth complete symmetric function are

e_{n}(X) = ∑_{i_{1}<···<i_{n}} x_{i_{1}} ⋯ x_{i_{n}},
h_{n}(X) = ∑_{i_{1}≤···≤i_{n}} x_{i_{1}} ⋯ x_{i_{n}},

respectively. By convention, let e_{0}(X) = 1 = h_{0}(X) and h_{n}(X) = 0 = e_{n}(X) if n < 0. For each composition α = (α_{1}, …, α_{r}), let

e_{α}(X) = e_{α_{1}}(X) ⋯ e_{α_{r}}(X),
h_{α}(X) = h_{α_{1}}(X) ⋯ h_{α_{r}}(X).

By convention, e_{∅}(X) = 1 = h_{∅}(X). Clearly, if α can be rearranged to a partition λ then e_{α}(X) = e_{λ}(X) and h_{α}(X) = h_{λ}(X). The following results are well-known.

Theorem 2.1.

(i) The ring of symmetric functions Λ(X) is the polynomial ring in {h_{n}(X) : n ∈ ℕ} and {e_{n}(X) : n ∈ ℕ} respectively over ℤ. Furthermore, the subsets {e_{λ}(X) : λ ∈ 𝒫} and {h_{λ}(X) : λ ∈ 𝒫} are ℤ-bases for Λ(X).

(ii) For each d ∈ ℕ, we have ∑_{i=0}^{d}(−1)^{i}h_{i}(X)e_{d−i}(X) = 0.

2.2. An m′-truncated polynomial ring. Let X = {x_{i} : i ∈ ℕ} and Y = {y_{j} : j ∈ ℕ} be two sets of independent countably infinite commuting variables. Clearly, Λ(X) and Λ(Y) are graded subrings of ℤ⟦X ∪ Y ⟧. Let ⟨Λ(X), Λ(Y)⟩ be the subring of ℤ⟦X ∪ Y ⟧ generated by Λ(X) and Λ(Y). By Theorem 2.1(i), it is clear that ⟨Λ(X), Λ(Y)⟩ is a polynomial ring in {h_{i}(X), e_{j}(Y) : i, j ∈ ℕ} over ℤ and has a ℤbasis {h_{λ}(X)e_{μ}(Y) : (λ|μ) ∈ 𝒫^{2}} whose nth component has ℤ-basis {h_{λ}(X)e_{μ}(Y) : (λ|μ) ∈ 𝒫^{2}(n)}.

Fix a positive integer m. We define the quotient

Γ^{(m)} = ⟨Λ(X), Λ(Y)⟩/I

where I is the ideal of ⟨Λ(X), Λ(Y)⟩ generated by ∑_{i=0}^{d}(−1)^{i}h_{i}(X)e_{d−i}(Y) for every positive integer d such that m ∤ d. For each i, j ∈ ℕ_{0} and α, β ∈ 𝒞, we denote

h_{i} = h_{i}(X) + I, e_{j} = e_{j}(Y) + I, h_{α} = h_{α}(X) + I, e_{β} = e_{β}(Y) + I.

We call Γ^{(m)} the ring of m′-truncated symmetric functions in the variables X ∪ Y.

Since I is a homogeneous ideal of ⟨Λ(X), Λ(Y)⟩, the ring Γ^{(m)} is graded with the nth component as

(Γ^{(m)})^{n} = (⟨Λ(X), Λ(Y)⟩^{n} + I)/I

consisting of the ℤ-linear combinations of the m′-truncated symmetric functions h_{α}e_{β} such that |α| + |β| = n, together with the zero element.

Theorem 2.2 ([4, §4.1(10,11)]). The ring Γ^{(m)} is the polynomial ring in {h_{i}, e_{jm} : i, j ∈ ℕ} and it has a ℤ-basis

B = {h_{λ}e_{mμ} : (λ|mμ) ∈ 𝒫_{m}^{2}}.

Furthermore, there is a graded involution ψ : Γ^{(m)} → Γ^{(m)} given by ψ(h_{i}) = e_{i}.

2.3. Signed Young permutation module. The general theory of the representation theory of the symmetric group can be found, for example, in [10].

Let m, n ∈ ℕ_{0}. We denote 𝔖_{n}^{+m} for the symmetric group acting on the set {i + m : i ∈ [n]} consisting of permutations of the forms τ^{+m} where τ ∈ 𝔖_{n} such that, for all i ∈ [n],

τ^{+m}(i + m) = τ(i) + m.

Furthermore, we identify 𝔖_{m} × 𝔖_{n} with the subgroup 𝔖_{m}𝔖_{n}^{+m} of 𝔖_{m+n}. For a given composition α = (α_{1}, …, α_{r}) of n, we denote the Young subgroup of 𝔖_{n} with respect to α by

𝔖_{α} = 𝔖_{α_{1}} × ⋯ × 𝔖_{α_{r}}.

For each n ∈ ℕ_{0}, the set 𝒫_{p}^{2}(n) is partially ordered by the dominance order ⊵ where (λ|pμ) ⊵ (δ|pξ) if

(a) ∑_{i=1}^{ℓ} λ_{i} ≥ ∑_{i=1}^{ℓ} δ_{i}, and

(b) |λ| + p ∑_{i=1}^{ℓ} μ_{i} ≥ |δ| + p ∑_{i=1}^{ℓ} ξ_{i},

for all ℓ ∈ ℕ, where λ_{i} is treated as 0 if i > ℓ(λ) and similarly for μ_{i}, δ_{i} and ξ_{i}. In the case when (λ|pμ) ⊵ (δ|pξ) and (λ|pμ) ≠ (δ|pξ), we write (λ|pμ) ▷ (δ|pξ).

Recall that k is a field of odd characteristic p. In this article, we denote ⊗ and ⊠ for tensor product and outer tensor product of modules over k.

We write k(n) and sgn(n) for the trivial and signature representations for k𝔖_{n}. For a composition α ∈ 𝒞 (n), k(α) and sgn(α) are the restrictions of k(n) and sgn(n) respectively to the Young subgroup 𝔖_{α}. Let (α|β) ∈ 𝒞^{2}(n). We define the signed Young permutation k𝔖_{n}-module M(α|β) with respect to (α|β) as

M(α|β) = Ind_{𝔖_{α}×𝔖_{β}}^{𝔖_{n}}(k(α) ⊠ sgn(β)).

Following the work of Donkin [4, §2.3], we know that all non-isomorphic indecomposable summands of the signed Young permutation k𝔖_{n}-modules are parametrised by the set 𝒫_{p}^{2}(n). They are called signed Young k𝔖_{n}-modules and denoted by Y (λ|pμ) such that, for each (λ|pμ) ∈ 𝒫_{p}^{2}(n), Y (λ|pμ) is a summand of M(λ|pμ) with multiplicity one and the remaining summands of M(λ|pμ) are of the form Y (δ|pθ) where (δ|pθ) ∈ 𝒫_{p}^{2}(n) and (δ|pθ) ▷ (λ|pμ).

Let G be a group and ℛ(kG) be the Green ring (or representation ring) of kG consisting of formal linear combinations of the isomorphism classes [V] of finite dimensional indecomposable kG-modules V over ℤ such that [V] = [W] if and only if V ≅ W, [V]+[W] = [V ⊕W] and the (inner) product given by [V]·[W] = [V ⊗W]. The contragradient dual induces an automorphism on ℛ(kG) since (V ⊕ W)^{∗} ≅ V^{∗} ⊕ W^{∗} and (V ⊗ W)^{∗} ≅ V^{∗} ⊗ W^{∗}.

For each n ∈ ℕ_{0}, we have the Green ring ℛ(k𝔖_{n}) where, by convention, ℛ(k𝔖_{0}) ≅ ℤ has ℤ-basis consisting only the trivial k𝔖_{0}-module. Let V and W be k𝔖_{m}- and k𝔖_{n}-modules respectively. We obtain a k𝔖_{m+n}-module given by Ind_{𝔖_{m}×𝔖_{n}}^{𝔖_{m+n}}(V ⊠W).

It is easily checked that Ind_{𝔖_{m}×𝔖_{n}}^{𝔖_{m+n}}(V ⊠W) ≅ Ind_{𝔖_{n}×𝔖_{m}}^{𝔖_{m+n}}(W ⊠V). We define an outer product by

[V] × [W] = [Ind_{𝔖_{m}×𝔖_{n}}^{𝔖_{m+n}}(V ⊠ W)],

and hence [V] × [W] = [W] × [V].

Since [M(α|β)] = [M(λ|μ)] when λ, μ are rearrangements of α, β respectively and [M(α|β)] × [M(γ|δ)] = [M(α#γ|β#δ)], the subset 𝒴 (𝔖) of ℛ_{k}(𝔖) spanned by the isomorphism classes of indecomposable summands of signed Young permutation modules is a graded subring where

𝒴 (𝔖) = ⨁_{n∈ℕ_{0}} 𝒴 (𝔖_{n})

and 𝒴 (𝔖_{n}) is spanned by [M(α|β)] one for each (α|β) ∈ 𝒫^{2}(n).

Theorem 2.3. Both {[Y (λ|pμ)] : (λ|pμ) ∈ 𝒫_{p}^{2}} and

C = {[M(λ|pμ)] : (λ|pμ) ∈ 𝒫_{p}^{2}}

are ℤ-bases for 𝒴 (𝔖) and there is a graded involution ϖ : 𝒴 (𝔖) → 𝒴 (𝔖) defined by ϖ([V]) = [V ⊗ sgn(n)] for each [V] ∈ 𝒴 (𝔖_{n}).

2.4. Listing module. We refer the reader to [8] for the classical theory of Schur algebras.

Fix r ∈ ℕ and Ω be a set. The symmetric group 𝔖_{r} acts on Ω^{r} by place permutation and hence it induces a diagonal action on the set Ω^{r} × Ω^{r}. The set of orbits of the action of 𝔖_{r} on Ω^{r} × Ω^{r} is denoted by O(Ω^{r}). Let f, g, h ∈ O(Ω^{r}), (p, q) ∈ f and

λ_{g,h}^{f} = |{j ∈ Ω^{r} : (p, j) ∈ g and (j, q) ∈ h}|.

Let k(Ω^{r}) be the formal vector space over k with a basis {ξ_{f} : f ∈ O(Ω^{r})} and the multiplication be defined by

ξ_{g}ξ_{h} = ∑_{f∈O(Ω^{r})}(λ_{g,h}^{f} · 1_{k})ξ_{f}.

Notice that S(n, r) = k([n]^{r}) is the classical Schur algebra. We denote the algebra k(ℕ^{r}) by S(∞, r). By convention, we set S(∞, 0) = kξ_{0} ≅ k as algebras. Furthermore, we denote

S(∞) = ⨁_{r∈ℕ_{0}} S(∞, r).

The algebra S(∞) is a bialgebra as follows. Let (i, j) ∈ ℕ^{r}×ℕ^{r} and (p, q) ∈ ℕ^{s}×ℕ^{s} and f, g be the orbits containing them, respectively. We define the orbit containing (i#p, j#q) ∈ ℕ^{r+s} × ℕ^{r+s} as f #g. Define the comultiplication Δ : S(∞) → S(∞) ⊗ S(∞) and counit ε : S(∞) → k by

Δ(ξ_{f}) = ∑_{\begin{matrix}g∈O(ℕ^{r}), h∈O(ℕ^{s}) \\ f=g#h \end{matrix}} ξ_{g} ⊗ ξ_{h},  ε(ξ_{f}) = \{ \begin{matrix}1 & f = 0, \\ 0 & otherwise. \end{matrix}

If V, W are S(∞, r)- and S(∞, s)-modules respectively, then V ⊗W is an S(∞, r+s)- module.

Consider the natural S(∞, 1)-module E where E is k-vector space with a basis {e_{i} : i ∈ ℕ} and with the action defined by

ξ_{f}e_{j} = \{ \begin{matrix}e_{i} & if f = \{(i, j)\}, \\ 0 & otherwise, \end{matrix}

for all j ∈ ℕ and f ∈ O(ℕ). For each r ∈ ℕ, let S^{r}E and ⋀^{r} E be the symmetric and exterior powers of E respectively. By convention, S^{0}E ≅ k ≅ ⋀^{0} E. For α ∈ 𝒞 (n), we set

S^{α}E = S^{α_{1}}E ⊗ ⋯ ⊗ S^{α_{ℓ(α)}}E,  ⋀^{α} E = ⋀^{α_{1}} E ⊗ ⋯ ⊗ ⋀^{α_{ℓ(α)}} E,

so that S^{α}E and ⋀^{α} E are S(∞, n)-modules. For (α|β) ∈ 𝒞^{2}(n), we define the mixed power the S(∞, n)-module K^{α|β}E := S^{α}E ⊗ ⋀^{β} E. Notice that K^{δ|ξ}E ≅ K^{α|β}E if δ, ξ are rearrangements of α, β respectively. Following Donkin [4, §4], the class of non-isomorphic indecomposable summands of mixed powers for S(∞, n) are parametrised by the set 𝒫_{p}^{2}(n). They are called listing modules denoted by List^{λ|pμ}E such that, for each (λ|pμ) ∈ 𝒫_{p}^{2}(n), List^{λ|pμ}E is a summand of K^{λ|pμ}E with multiplicity one and any other summand of K^{λ|pμ}E are of the form List^{δ|pθ}E where (δ|pθ) ∈ 𝒫_{p}^{2}(n) and (δ|pθ) ▷ (λ|pμ). Furthermore, under the Schur functor f, we have f(K^{α|β}E) ≅ M(α|β) and f(List^{λ|pμ}E) ≅ Y (λ|pμ).

Let ℒ (S) be the graded subring of the representation ring of S(∞) generated by [S^{n}E] and [⋀^{n} E] for every n ∈ ℕ_{0} with the gradation

ℒ (S) = ⨁_{n∈ℕ_{0}} ℒ (S(∞, n))

where ℒ (S(∞, n)) is spanned by the mixed powers [K^{α|β}E] such that (α|β) ∈ 𝒞^{2}(n).

Theorem 2.4. Both {[List^{λ|pμ}E] : (λ|pμ) ∈ 𝒫_{p}^{2}} and

D = {[K^{λ|pμ}E] : (λ|pμ) ∈ 𝒫_{p}^{2}}

are ℤ-bases for ℒ (S). Furthermore, the ring ℒ (S) has a graded involution ω : ℒ (S) → ℒ (S) defined by ω(S^{n}E) = ⋀^{n} E.

The following categorification theorem of the rings 𝒴 (𝔖) and ℒ (S) is essential throughout this article.

Theorem 2.5 ([4, §4.1(10,13)]). We have a commutative diagram

\xymatrix{ℒ (S) \ar[rr]^{f} &  & 𝒴 (𝔖) \\  & Γ^{(p)} \ar@{-->}[ur]_{ϕ} \ar@{-->}[ul]^{θ} \\ ℒ (S) \ar[rr]^{f} \ar[uu]^{ω} &  & 𝒴 (𝔖) \ar[uu]^{ϖ} \\  & Γ^{(p)} \ar@{-->}[ur]_{ϕ} \ar@{-->}[ul]^{θ} \ar[uu]^{ψ}}

where the maps f, θ, ϕ are isomorphisms of graded rings where, for each (α|β) ∈ 𝒞^{2}, f([K^{α|β}E]) = [M(α|β)], θ(h_{α}e_{β}) = [K^{α|β}E] and ϕ(h_{α}e_{β}) = [M(α|β)].

We end this section with a remark.

Remark 2.6. Let ℳℳℳ be the Mullineux map on p-restricted partitions (see, for example, [1, §6]) and let λ(0) be the p-restricted partition such that λ = λ(0)+pξ for some partition ξ (here, λ(0)+pξ has the obvious meaning componentwise summation of partitions). Using Theorem 2.5 and [2, Theorem 3.21], for each (λ|pμ) ∈ 𝒫_{p}^{2}, since ϖ([Y (λ|pμ)]) = [Y (ℳℳℳ (λ(0)) + pμ|λ − λ(0))], we have

ω([List^{λ|pμ}E]) = [List^{ℳℳℳ (λ(0))+pμ|λ−λ(0)}E].

Similarly, in Γ^{(p)}, we have

ψ(h_{λ}e_{pμ}) = h_{ℳℳℳ (λ(0))+pμ}e_{λ−λ(0)}.
3. The ring Γ^{(m)}

Throughout this section, we fix m ∈ ℕ. In this section, we derive some properties about the graded ring Γ^{(m)}. Recall that Γ^{(m)} is the quotient ring ⟨Λ(X), Λ(Y)⟩/I where I is the ideal generated by ∑_{i=0}^{d}(−1)^{i}h_{i}(X)e_{d−i}(Y) for every positive integer d such that m ∤ d. Furthermore, Γ^{(m)} is the polynomial ring on {h_{i}, e_{jm} : i, j ∈ ℕ} where h_{n} = h_{n}(X) + I and e_{n} = e_{n}(Y) + I, and it has a ℤ-basis

B = {h_{λ}e_{mμ} : (λ|mμ) ∈ 𝒫_{m}^{2}}.

We begin with a lemma.

Lemma 3.1.

(i) For each d ∈ ℕ, e_{d} is a ℤ-linear combination of h_{λ}e_{sm} for some partitions λ and s ∈ ℕ_{0} such that |λ| + sm = d.

(ii) Let

a_{ij} = \{ \begin{matrix}h_{1−i+j} & if m ∤ j, \\ (−1)^{j+1}e_{j} & if j = sm and i = 1, \\ 1 & if j = sm and i = sm + 1, \\ 0 & otherwise. \end{matrix}

Then, for each d ≥ 1, we have e_{d} = det(a_{ij})_{1≤i,j≤d}. (iii) If we set x_{i} = y_{i} for each i ∈ ℕ then I = 0 and Γ^{(m)} = Λ(X) where h_{λ} = h_{λ}(X)

and e_{μ} = e_{μ}(X).

Proof. The proof of part (i) is straightforward by using induction on d and the relation ∑_{i=1}^{d}(−1)^{i}h_{i}e_{d−i} = 0 when m ∤ d. Alternatively, it follows from part (ii).

For part (ii), let A_{d} = (a_{ij})_{1≤i,j≤d}. Notice that A_{1} = (e_{1}) and hence e_{1} = det(A_{1}). Fix a d ∈ ℕ. By induction, suppose that e_{j} = det(A_{j}) for all 1 ≤ j ≤ d − 1. Let w_{d} be the ((d − 1) × 1)-column matrix with the first entry (−1)^{d+1}e_{d} and 0 elsewhere, let z_{i} be the (i × 1)-column matrix whose entries are h_{d}, h_{d−1}, …, h_{d−i+1} read from the top and let v_{j} be the (1 × j)-row matrix whose last entry is 1 and 0 elsewhere. If d = ℓm then, expand along the last column,

det(A_{d}) = det (\begin{matrix}A_{d−1} & w_{d} \\ v_{d−1} & 0 \end{matrix}) = (−1)^{ℓm+1} · (−1)^{ℓm+1}e_{ℓm} = e_{d}.

Suppose that m ∤ d. In particular, m ≠ 1 and thus h_{1} = e_{1}. Let B_{s} = (\begin{matrix}A_{d−s} & z_{d−s} \\ v_{d−s} & h_{s} \end{matrix}).

We claim that, for all 1 ≤ s ≤ d−1, det(B_{s}) = ∑_{i=s}^{d}(−1)^{i−s}h_{i}e_{d−i}. When s = d−1, we have

det(B_{d−1}) = det (\begin{matrix}A_{1} & z_{1} \\ v_{1} & h_{d−1} \end{matrix}) = e_{1}h_{d−1} − h_{d} = e_{1}h_{d−1} − e_{0}h_{d}.

Expand det(B_{s}) along the last row, by induction, we have

det(B_{s}) = h_{s} det(A_{d−s}) − det(B_{s+1}) = h_{s}e_{d−s} − ∑_{i=s+1}^{d}(−1)^{i−(s+1)}h_{i}e_{d−i}

= ∑_{i=s}^{d}(−1)^{i−s}h_{i}e_{d−i}.

Notice that A_{d} = B_{1}. Therefore, we conclude that

det(A_{d}) = det(B_{1}) = ∑_{i=1}^{d}(−1)^{i−1}h_{i}e_{d−i} = e_{d}.

For part (iii), if x_{i} = y_{i} for each i ∈ ℕ, since ∑_{i=0}^{d}(−1)^{i}h_{i}(X)e_{d−i}(X) = 0 for all d by Theorem 2.1(ii), we have I = 0. So h_{i} = h_{i}(X), e_{i} = e_{i}(X) and Γ^{(m)} = Λ(X). □

In view of Lemma 3.1(i), we introduce the following notation. For each n ∈ ℕ, let

e_{n} = ∑_{n=|μ|+sm} d_{n,μ}h_{μ}e_{sm}

where d_{n,μ} ∈ ℤ with respect to the basis B so that s ∈ ℕ_{0} in the above summation. Notice that if n = ℓm for some ℓ then d_{ℓm,μ} = 0 for all μ ≠ ∅ and d_{ℓm,∅} = 1.

Corollary 3.2. Let μ ∈ 𝒫(n). For all r ∈ ℕ_{0}, we have

d_{n+rm,μ} = d_{n,μ}.

Proof. For each (α|mβ) ∈ 𝒫_{m}^{2}, we denote the coefficient of h_{α}e_{mβ} in z ∈ Γ^{(m)} as (z, h_{α}e_{mβ}). Then d_{n,μ} = (e_{n}, h_{μ}) and d_{n+rm,μ} = (e_{n+rm}, h_{μ}e_{rm}). Let A_{d} = (a_{ij})_{1≤i,j≤d} be defined as in Lemma 3.1(ii) and, by which, we have both e_{n} = det(A_{n}) and e_{n+rm} = det(A_{n+rm}). By virtue of the entries of A_{d}, an element of the form h_{α}e_{jm} only appears in e_{jm}(A_{d})^{(1,jm)} where (A_{d})^{(1,jm)} is the (1, jm)-minor of the matrix A_{d}.

Let C_{d} be the matrix obtained from A_{d} by replacing all entries of the form (−1)^{ℓ+1}e_{ℓ} by 0. Expand along the rmth column of A_{n}, we have

(e_{n+rm}, h_{μ}e_{rm}) = (det(A_{n+rm}), h_{μ}e_{rm})

= (e_{rm}(A_{n+rm})^{(1,rm)}, h_{μ}e_{rm})

= (e_{rm} det (\begin{matrix}J & B \\ 0 & C_{n} \end{matrix}) , h_{μ}e_{rm})

= (det(C_{n}), h_{μ})

where J is an upper unitriangular matrix and B is some suitable matrix. By similar calculation, we have (e_{n}, h_{μ}) = (det(C_{n}), h_{μ}). So we conclude that d_{n+rm,μ} = d_{n,μ}.

□

In view of Corollary 3.2, we have the following notation.

Notation 3.3. Let n ∈ ℕ_{0}. We denote

𝒫(n; m) = {μ ∈ 𝒫 : n = |μ| + sm for some s ∈ ℕ_{0}}.

For each partition μ, we write d_{μ} = d_{n,μ} for the common number for all n = |μ|+rm where r ∈ ℕ_{0} so that, for all n ∈ ℕ_{0}, we have

e_{n} = ∑_{n=|μ|+sm} d_{μ}h_{μ}e_{sm} = ∑_{μ∈𝒫(n;m)} d_{μ}h_{μ}e_{n−|μ|}.

4. The straightening rule

Recall the ℤ-bases B, C and D of the rings Γ^{(m)}, 𝒴 (𝔖) and ℒ (S) denoted in Theorems 2.2, 2.3 and 2.4 respectively. Our main result in this section provides an explicit formula to write, for each (α|β) ∈ 𝒞^{2}(n), the element h_{α}e_{β} in terms of B. As a consequence, we have the explicit formulae to write [M(α|β)] and [K^{α|β}E] as ℤ-linear combinations in terms of the bases C and D respectively.

We begin with a list of notation.

Notation 4.1. Fix a positive integer m, let λ ∈ 𝒫(n) and let b = ⌊\frac{n}{m}⌋.

(i) Let 𝒞 (λ) be the set of compositions of the form δ = (δ_{1}, …, δ_{ℓ(λ)}) such that its rearrangement is λ and let c_{λ} = |𝒞 (λ)|, i.e.,

c_{λ} = \frac{n!}{∏_{i∈ℕ} n_{i}!}

where n_{i} is the number of parts of λ of size i.

(ii) Let c_{λ}^{(m)} be the number of compositions δ = (δ_{1}, …, δ_{ℓ(λ)}) ∈ 𝒞 (λ) such that

δ_{1} + δ_{2} + ⋯ + δ_{j} ≢ 0 mod m

for all 1 ≤ j ≤ ℓ(λ). (iii) Let ε_{λ} = (−1)^{|λ|−ℓ(λ)}. (iv) For a finite sequence A = (δ^{(1)}, …, δ^{(r)}) of partitions, we write c_{A} = ∏_{i=1}^{r} c_{δ^{(i)}}

and ℓ(A) = r. Similarly, we write c_{A}^{(m)} = ∏_{i=1}^{r} c_{δ^{(i)}}^{(m)} and ε_{A} = ∏_{i=1}^{r} ε_{δ^{(i)}}.

(v) If δ, ξ are compositions such that δ#ξ can be rearranged to λ, we write λ = δ∪ξ (or λ = ξ ∪ δ). The notation λ = ξ^{(1)} ∪ ⋯ ∪ ξ^{(r)} has the obvious meaning.

(vi) Let W (λ) be the set consists of sequences (δ^{(1)}, …, δ^{(r)}, ξ) of partitions for some r ∈ ℕ_{0} such that ξ ∪ δ^{(1)} ∪ ⋯ ∪ δ^{(r)} = λ, δ^{(i)} ≠ ∅ and m | |δ^{(i)}| for all 1 ≤ i ≤ r.

(vii) For each 1 ≤ i ≤ b, let W_{i}(λ) = {A ∈ W (λ) : ℓ(A) = i + 1}. (viii) For each 1 ≤ i ≤ b, let P_{i} be the subset of 𝒞 (λ) consisting of all compositions

(μ_{1}, …, μ_{ℓ(λ)}) such that μ_{1} + μ_{2} + ⋯ + μ_{r} = mi for some 1 ≤ r ≤ ℓ(λ). (ix) Let ξ be a partition and β be a composition. Suppose that s = ℓ(β). We

denote by V (β; (ξ, mμ)) the set consisting of the s-tuples

(ξ^{(1)}, …, ξ^{(s)}) ∈ 𝒫(β_{1}; m) × ⋯ × 𝒫(β_{s}; m)

such that ξ = ξ^{(1)} ∪ ⋯ ∪ ξ^{(s)} and mμ = (β_{1} − |ξ^{(1)}|) ∪ ⋯ ∪ (β_{s} − |ξ^{(s)}|). Furthermore, we write

c_{β;(ξ,mμ)}^{(m)} = ∑_{A∈V (β;(ξ,mμ))} ε_{A}c_{A}^{(m)}.

By convention, c_{∅} = 1, c_{∅}^{(m)} = 1, ε_{∅} = 1, W (∅) = {(∅)} and c_{β;(ξ,mμ)}^{(m)} = 0 if V (β; (ξ, mμ)) = ∅.

For example, let λ = (3, 2, 1, 1) and m = 3. Then the compositions (μ_{1}, μ_{2}, μ_{3}, μ_{4}) ∈ 𝒞 (λ) with the property that 3 ∤ ∑_{i=1}^{j} μ_{i} for all 1 ≤ j ≤ 4 are

(1, 1, 3, 2),  (1, 3, 1, 2),  (1, 1, 2, 3).

Thus c_{(3,2,1,1)}^{(3)} = 3. Furthermore,

W (λ) = {(λ), ((3), (2, 1, 1)), ((2, 1), (3, 1)), ((3), (2, 1), (1)), ((2, 1), (3), (1))}.

For another example, if λ ≠ ∅ and m | |λ|, then c_{λ}^{(m)} = 0. Also,

W ((m)) = {((m)), ((m), ∅)}.

We may now state our main theorem.

Theorem 4.2. Fix a positive integer m. For each (α|β) ∈ 𝒞^{2}(n),

h_{α}e_{β} = ∑ c_{β;(ξ,mμ)}^{(m)}h_{λ}e_{mμ}

where the sum runs over all (λ|mμ) ∈ 𝒫_{m}^{2}(n) such that λ = α∪ξ for some partition ξ. In particular, we have d_{λ} = ε_{λ}c_{λ}^{(m)} for each partition λ.

We illustrate the theorem with an example. Let m = 3. Then

e_{2} = ∑_{2=|λ|+3s} ε_{λ}c_{λ}^{(3)}h_{λ}e_{3s} = −h_{(2)} + h_{(1,1)}

since c_{(2)}^{(3)} = 1, ε_{(2)} = −1, c_{(1,1)}^{(3)} = 1 and ε_{(1,1)} = 1. Similarly, we have

e_{4} = −h_{(4)} + h_{(3,1)} + h_{(2,2)} − h_{(2,1,1)} + h_{(1)}e_{(3)}.

So, for example, we get

h_{(5)}e_{(4,3,2)} = h_{(5)} (−h_{(4)} + h_{(3,1)} + h_{(2,2)} − h_{(2,1,1)} + h_{(1)}e_{(3)}) e_{(3)} (−h_{(2)} + h_{(1,1)})

= h_{5,4,2}e_{3} − h_{5,4,1,1}e_{3} − h_{5,3,2,1}e_{3} + h_{5,3,1,1,1}e_{3} − h_{5,2,2,2}e_{3} + 2h_{5,2,2,1,1}e_{3}

− h_{5,2,1,1,1,1}e_{3} − h_{5,2,1}e_{3,3} + h_{5,1,1,1}e_{3,3},

here, after the last equality, we have deliberately removed the parentheses. To prove Theorem 4.2, we need the following two lemmas.

Lemma 4.3. For each partition λ,

d_{λ} = −ε_{λ} ( ∑_{A∈W (λ)}(−1)^{ℓ(A)}c_{A}) .

Proof. Let b = ⌊\frac{n}{m}⌋. Recall that the elementary symmetric functions can be written in terms of the complete symmetric functions as

e_{n}(X) = ∑_{λ∈𝒫(n)} ε_{λ}c_{λ}h_{λ}(X)

with respect to the set of variables X = {x_{i} : i ∈ ℕ} (see [13, Chapter I.2 Example 20]). Similarly, e_{n}(Y) = ∑_{λ∈𝒫(n)} ε_{λ}c_{λ}h_{λ}(Y). Let x_{i} = y_{i} for each i ∈ ℕ. Using Lemma 3.1(iii) and Notation 3.3, we have

∑_{μ∈𝒫(n;m)} d_{μ}h_{μ}(X) ( ∑_{δ∈𝒫(n−|μ|)} ε_{δ}c_{δ}h_{δ}(X)) = ∑_{λ∈𝒫(n)} ε_{λ}c_{λ}h_{λ}(X).

Fix a partition λ ∈ 𝒫(n). If λ = μ ∪ δ with μ, δ ∈ 𝒫 then μ uniquely determines δ and vice versa. By Theorem 2.1(i), comparing the coefficients of h_{λ}(X), we have

(1)  ε_{λ}c_{λ} = ∑_{i=0}^{b} ∑_{\begin{matrix}δ∈𝒫(mi), \\ δ∪μ=λ \end{matrix}} ε_{δ}d_{μ}c_{δ}.

We prove the equality in the statement by using induction on the size of λ. If λ = ∅ then d_{∅} = 1 and −ε_{∅}((−1)^{ℓ((∅))}c_{∅} = 1. Using Equation 1 and induction, we have

ε_{λ}c_{λ} = d_{λ} + ∑_{i=1}^{b} ∑_{\begin{matrix}δ∈𝒫(mi), \\ δ∪μ=λ \end{matrix}} ε_{δ}d_{μ}c_{δ} = d_{λ} − ∑_{i=1}^{b} ∑_{\begin{matrix}δ∈𝒫(mi), \\ δ∪μ=λ \end{matrix}} ε_{δ}ε_{μ}c_{δ} ( ∑_{A∈W (μ)}(−1)^{ℓ(B)}c_{B})

= d_{λ} + ε_{λ} ∑_{i=1}^{b} ∑_{\begin{matrix}δ∈𝒫(mi), \\ δ∪μ=λ \end{matrix}} ( ∑_{A∈W (μ)}(−1)^{ℓ(B)+1}c_{δ}c_{B})

Notice that W (λ)\{(λ)} consists of sequences of partitions, one for each i ∈ [b], (δ, δ^{(1)}, …, δ^{(j)}, ξ) such that δ ∪ μ = λ, |δ| = im and (δ^{(1)}, …, δ^{(j)}, ξ) ∈ W (μ). So we get

d_{λ} = ε_{λ} (c_{λ} − ∑_{A∈W (λ)\{(λ)}}(−1)^{ℓ(A)}c_{A}) = −ε_{λ} ( ∑_{A∈W (λ)}(−1)^{ℓ(A)}c_{A}) .

The proof is now complete. □

Lemma 4.4. Let λ ∈ 𝒫(n) and b = ⌊\frac{n}{m}⌋. For each j ∈ [b], have

∑_{1≤i_{1}<···<i_{j}≤b} |P_{i_{1}} ∩ ⋯ ∩ P_{i_{j}}| = ∑_{A∈W_{j}(λ)} c_{A}.

Proof. In the proof, we denote the disjoint union of sets S_{1}, …, S_{a} by ⨆_{i=1}^{a} S_{i}. Let

ψ : ⨆_{1≤i_{1}<···<i_{j}≤b} P_{i_{1}} ∩ ⋯ ∩ P_{i_{j}} → ⨆_{(δ^{(1)},...,δ^{(j)},ξ)∈W_{j}(λ)} 𝒞 (δ^{(1)}) × ⋯ × 𝒞 (δ^{(j)}) × 𝒞 (ξ)

be defined as follows. For each η ∈ P_{i_{1}} ∩ ⋯ ∩ P_{i_{j}}, we have positive integers 1 ≤ s_{1} < ⋯ < s_{j} ≤ ℓ(λ) such that ∑_{t=1}^{s_{r}} η_{t} = mi_{r} for each 1 ≤ r ≤ j. Therefore, for each 1 ≤ t ≤ j + 1,

η^{(t)} = (η_{s_{t−1}+1}, η_{s_{t−1}+2}, …, η_{s_{t}}) ∈ 𝒞 (δ^{(t)})

for some partition δ^{(t)} of size m(i_{t} − i_{t−1}) where i_{0} = 0 = s_{0} and s_{j+1} = ℓ(λ). Then

ψ(η) = ∏_{t=1}^{j+1} η^{(t)} ∈ 𝒞 (δ^{(1)}) × ⋯ × 𝒞 (δ^{(j)}) × 𝒞 (ξ)

since (δ^{(1)}, …, δ^{(j)}, ξ) ∈ W_{j}(λ). The map ψ is invertible with the inverse ϕ given as follows. For each (δ^{(1)}, …, δ^{(j)}, ξ) ∈ W_{j}(λ) and

h = (η^{(1)}, …, η^{(j)}, η^{(j+1)}) ∈ 𝒞 (δ^{(1)}) × ⋯ × 𝒞 (δ^{(j)}) × 𝒞 (ξ),

we define ϕ(h) = η^{(1)}# ⋯ #η^{(j)}#η^{(j+1)} ∈ P_{i_{1}} ∩ ⋯ ∩ P_{i_{j}} where mi_{r} = ∑_{t=1}^{r} |η^{(t)}| for 1 ≤ r ≤ j. The bijection ψ shows that

∑_{1≤i_{1}<···<i_{j}≤b} |P_{i_{1}}∩⋯∩P_{i_{j}}| = ∑_{(δ^{(1)},...,δ^{(j)},ξ)∈W_{j}(λ)} |𝒞 (δ^{(1)})×⋯×𝒞 (δ^{(j)})×𝒞 (ξ)| = ∑_{A∈W_{j}(λ)} c_{A}.

□

We may now prove our main theorem.

Proof of Theorem 4.2. Let b = ⌊\frac{n}{m}⌋. By the inclusion-exclusion principle and Lemmas 4.3 and 4.4, we have

c_{λ}^{(m)} = |⋂_{i=1}^{b}(𝒞 (λ)\P_{i})| = |𝒞 (λ)| − ∑_{j=1}^{b}(−1)^{j+1} ( ∑_{1≤i_{1}<···<i_{j}≤b} |P_{i_{1}} ∩ ⋯ ∩ P_{i_{j}}|)

= |𝒞 (λ)| − ∑_{j=1}^{b}(−1)^{j+1} ( ∑_{A∈W_{j}(λ)} c_{A})

= − ( ∑_{A∈W (λ)}(−1)^{ℓ(A)}c_{A})

= ε_{λ}d_{λ},

i.e., d_{λ} = ε_{λ}c_{λ}^{(m)}. This proves the particular case. Let s = ℓ(β). For the general case, for each 1 ≤ i ≤ s, we have e_{β_{i}} = ∑_{ξ^{(i)}∈𝒫(β_{i};m)} ε_{ξ^{(i)}}c_{ξ^{(i)}}^{(m)}h_{ξ^{(i)}}e_{β_{i}−|ξ^{(i)}|}. So

e_{β} = ∏_{i=1}^{s} e_{β_{i}} = ∑_{(ξ|mμ)∈𝒫_{m}^{2}(|β|)} c_{β;(ξ,mμ)}^{(m)}h_{ξ}e_{mμ}.

Therefore,

h_{α}e_{β} = ∑_{(ξ|mμ)∈𝒫_{m}^{2}(|β|)} c_{β;(ξ,mμ)}^{(m)}h_{α∪ξ}e_{mμ} = ∑_{\begin{matrix}(λ|mμ)∈𝒫_{m}^{2}(n), \\ λ=ξ∪α, ξ∈𝒫 \end{matrix}} c_{β;(ξ,mμ)}^{(m)}h_{λ}e_{mμ}.

□

Using Theorems 2.5 and 4.2, we obtain the following immediate corollaries.

Corollary 4.5. Let (α|β) ∈ 𝒞^{2}(n). Then

(i) [M(α|β)] = ∑ c_{β;(ξ,pμ)}^{(p)}[M(λ|pμ)], and

(ii) [K^{α|β}E] = ∑ c_{β;(ξ,pμ)}^{(p)}[K^{λ|pμ}E]

where both of the sums above run over all (λ|pμ) ∈ 𝒫_{p}^{2}(n) such that λ = α ∪ ξ for some partition ξ.

We demonstrate Corollary 4.5 with an example. When p = 3, as we have computed earlier, using the maps θ and ϕ as in Theorem 2.5, we have

[M(∅|(2))] = −[M((2)|∅)] + [M((1, 1)|∅)],
[M(∅|(4))] = −[M((4)|∅)] + [M((3, 1)|∅)] − [M((2, 1, 1)|∅)] + [M((1)|(3))],
[⋀^{2} E] = −[S^{2}E] + [S^{(1,1)}E],
[⋀^{4} E] = −[S^{4}E] + [S^{(3,1)}E] − [S^{(2,1,1)}E] + [E ⊗ ⋀^{3} E].

Let (M(α|β) : Y (δ|pξ)) denote the multiplicity of Y (δ|pξ) as a direct summand of M(α|β). Similarly, we have the multiplicity (K^{α|β}E : List^{δ|pξ}E). The following corollary is clear.

Corollary 4.6. Let (α|β) ∈ 𝒞^{2}(n) and (δ|pξ) ∈ 𝒫_{p}^{2}(n). Then

(i) (M(α|β) : Y (δ|pξ)) = ∑ c_{β;(ξ,pμ)}^{(p)}(M(λ|pμ) : Y (δ|pξ)), and

(ii) (K^{α|β}E : List^{δ|pξ}E) = ∑ c_{β;(ξ,pμ)}^{(p)}(K^{λ|pμ}E : List^{δ|pξ}E)

where both of the sums above run over all (λ|pμ) ∈ 𝒫_{p}^{2}(n) such that λ = α ∪ ξ for some partition ξ.

Recall that Y (λ|pμ) (respectively, List^{λ|pμ}E) is a summand of M(λ|pμ) (respectively, K^{λ|pμ}E) with multiplicity one and any other summand Y (α|pβ) (respectively, List^{α|pβ}E) necessarily satisfies (α|pβ)▷(λ|pμ). Applying our result, we have a similar description when (λ|pμ) is replaced by a general pair of compositions (α|β).

Corollary 4.7. Let (α|β) ∈ 𝒞^{2}(n) and s = ℓ(β). For each 1 ≤ i ≤ s, let β_{i} = pη_{i}+r_{i} such that 0 ≤ r_{i} ≤ p − 1 and let r = ∑_{i=1}^{s} r_{i}. Suppose that α#(1^{r}) and (η_{1}, …, η_{s}) are rearranged to the partitions λ and μ respectively.

(i) The signed Young permutation module M(α|β) has the summand Y (λ|pμ) such that (M(α|β) : Y (λ|pμ)) = 1 and, if Y (δ|pθ) is a summand of M(α|β), then (δ|pθ) ⊵ (λ|pμ).

(ii) The mixed power K^{α|β}E has the summand List^{λ|pμ}E such that (K^{α|β}E : List^{λ|pμ}E) = 1 and, if List^{δ|pθ}E is a summand of K^{α|β}E, then (δ|pθ) ⊵ (λ|pμ).

Proof. Since M(α|β) ≅ M(ζ|ν) if ζ, ν are rearrangements of α, β respectively, we may assume that α, β are partitions so that λ = α#(1^{r}) and μ = (η_{1}, …, η_{s}) in the statement. For each 0 ≤ d ≤ p − 1, we have ε_{(1^{d})}c_{(1^{d})}^{(p)} = 1. It is easy to check that

V (β; ((1^{|β|−p|μ|}), pμ)) = {((1^{r_{1}}), …, (1^{r_{s}}))}

and hence c_{β;((1^{|β|−p|μ|}),pμ)}^{(p)} = 1. If c_{β;(ξ,pθ)}^{(p)} ≠ 0 as in Corollary 4.5(i), then δ = α ∪ ξ, |ξ| ≥ r and θ_{i} ≤ μ_{i} for all i. Suppose that Y (λ|pμ) is a summand of M(δ|pθ). We have (λ|pμ) ⊵ (δ|pθ). In particular,

|α| + r = |λ| ≥ |δ| = |α| + |ξ|.

Hence |ξ| = r, |λ| = |δ| and α#(1^{r}) = λ ⊵ δ = α ∪ ξ where, here, ⊵ is the usual dominance order on 𝒫(n). Therefore, ξ = (1^{r}). Furthermore, p|θ| = p|μ| and hence we conclude that θ = μ. This shows that any summand Y (δ|pθ) of M(α|β) satisfies (δ|pθ) ⊵ (λ|pμ) and (M(α|β) : Y (λ|pμ)) = 1. Part (ii) can be proved in the similar fashion or follows from part (i) using Theorem 2.5. □

In [6, Proposition 7.1], Giannelli, O’Donovan, Wildon and the author found the label of signed Young module when M(α|β) is indecomposable in the odd characteristics case, i.e., the pair of partitions (δ|pθ) such that M(α|β) ≅ Y (δ|pθ). Direct application of Corollary 4.7 gives an alternative proof.

Corollary 4.8 ([6, Proposition 7.1]). Let a, b ∈ ℕ_{0}, a ≠ 0 and b = sp + r where 0 ≤ r ≤ p − 1. Then M(∅|(b)) ≅ Y ((1^{r})|p(s)) and, if p | a + b, M((a)|(b)) ≅ Y ((a, 1^{r})|p(s)).

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[13] I. G. Macdonald, ‘Symmetric functions and Hall polynomials’, 2nd edn., Oxford Science Publications.

(K. J. Lim) Division of Mathematical Sciences, Nanyang Technological University, SPMS-PAP-03-01, 21 Nanyang Link, Singapore 637371.

E-mail address, K. J. Lim: limkj@ntu.edu.sg