Homomorphisms from Specht Modules to Signed Young Permutation Modules

We construct a class $\Theta_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^{\lambda}$ to a signed permutation module $M_{\mathbb{Z}}(\alpha|\beta)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $\phi \in \operatorname{Hom}_{{\mathbb{Z}}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^\lambda, M_{\mathbb{Z}}(\alpha|\beta)\big)$ lies in the $\mathbb{Q}$-span of $\Theta_{\text{sstd}}$, a subset of $\Theta_{\mathscr{R}}$ corresponding to semistandard $\lambda$-tableaux of type $(\alpha|\beta)$. We also study the conditions for which $\Theta^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ induced by $\Theta_{\mathrm{sstd}}$ - is linearly independent, and show that it is a basis for $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$ when $\mathbb{F}\mathfrak{S}_{n}$ is semisimple.


Introduction
Modular representation theory of finite groups, unlike its ordinary counterpart, is not at all well understood. Even for ubiquitous groups like the symmetric groups, many fundamental questions remain open. It is therefore important to understand the naturally occurring representations, as these may provide more information about the modular representation theory in general.
Among the naturally occurring representations of the symmetric groups, the Young permutation modules are perhaps the most well-known. James [7] studied these modules in detail and exploited their knowledge to obtain important information such as those about the Specht modules.
When the characteristic of the underlying field is not 2, the signed permutation modules are a natural generalisation of the Young permutation modules. Their indecomposable summands, known as signed Young modules, are first studied by Donkin in [3]. Subsequently, signed Young modules are shown to be related to irreducible Specht modules. More specifically, Hemmer [6] showed that irreducible Specht modules are signed Young modules and, recently, Danz and the first author [2] described the label explicitly. Signed permutation modules for Iwahori-Hecke algebras of type A are also studied by Du and Rui in [4].
In this paper, we construct a class Θ R of homomorphisms from the Specht module S λ Z to the signed permutation module M Z (α|β), just like James did for the (unsigned) Young per-This paper is a contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics. The full collection is available at https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html mutation module M µ F . A subset of this class, denoted Θ sstd , is a direct generalisation of James's semistandard homomorphisms. James's semistandard homomorphisms form a basis for Hom FSn S λ F , M µ F , unless char(F) = 2 and λ is 2-singular. It is therefore natural to seek generalisation of this statement for our Θ sstd . We show that Hom ZSn S λ Z , M Z (α|β) is contained in the Q-span of Θ sstd , and provide a sufficient condition for Θ sstd to be linearly independent when reduced modulo p. In particular, we prove that when FS n is semisimple, with char(F) = p, then reducing Θ sstd modulo p does indeed give a basis for Hom FSn S λ F , M F (α|β) . We found many examples in our study which show that the signed permutation modules behave in a much more unpredictable way than the unsigned ones. We include some of these in Examples 3.1, 4.1, 4.2 and 4.3.
The paper is organised as follows. In the next section, we give a quick introduction of the background. In Section 3, we generalise James's construction to obtain homomorphisms from the Specht module S λ Z to the signed permutation module M Z (α|β), and in Section 4, we look into the conditions for which a subset of our constructed homomorphisms, corresponding to the semistandard λ-tableaux of type (α|β), will be a basis for Hom FSn S λ F , M F (α|β) .

Preliminaries
In this section, we provide the necessary background and introduce some notations that we shall use in this paper. Throughout, we fix a field F of arbitrary characteristic.

Symmetric groups
Let X be a finite set. The symmetric group S X on X is the group of bijections from X to X under composition of functions. By convention, S ∅ is the trivial group. When Y is a non-empty subset of X, we view S Y as a subgroup of S X by identifying an element of S Y with its extension that sends x to x for all x ∈ X \ Y . Let X ⊆ Z + , and k ∈ Z + . Define the k-translated subset X +k of Z + by Let S +k X = f +k : f ∈ S X where f +k : X +k → X +k is defined by f +k (x + k) = f (x) + k for all x ∈ X. Clearly, f → f +k is a group isomorphism from S X to S +k X . For any S ⊆ S X , write S +k for {s +k : s ∈ S}.
For n ∈ Z + , we write S n for S {1,2,...,n} , the usual symmetric group on n letters.
To a composition λ of n, we associate the Young subgroup S λ of S n , where Let (α|β) be a bicomposition of n, i.e., α and β are compositions and |α| + |β| = n. One may view (α|β) as a composition of n by concatenating α with β, and so we have the associated Young subgroup

Young diagrams
For a partition λ, its Young diagram [λ] is defined as The conjugate partition of λ, denoted λ , is the partition whose Young diagram

Tableaux
Let λ be a partition of n. Define formally a λ-tableau t as a bijective function t : [λ] → {1, 2, . . . , n}. We usually view this as a labelling of the elements of [λ] by numbers 1, 2, . . . , n, such that each number appears exactly once. Denote the set of λ-tableaux by T (λ).
Through a fixed λ-tableau t we obtain a group isomorphism and C [λ] be the row and column stabilizers of [λ], i.e., Under the above group isomorphism, R [λ] corresponds to the row stabilizer R t of t, i.e., R t = t • R [λ] • t −1 ⊆ S n , and similarly, C [λ] corresponds to the column stabilizer C t of t, i.e., Let t λ denote the initial λ-tableau, defined by t λ (i, j) = λ 1 + · · · + λ i−1 + j for all (i, j) ∈ [λ]. Observe that R t λ = S λ .
Post-composition of λ-tableaux by elements of the symmetric group S n gives a well-defined, faithful and transitive left action of S n on T (λ), i.e., σ · t = σ • t for all σ ∈ S n and t ∈ T (λ). Observe that and similarly, C σ·t = σC t σ −1 . In particular, the row stabilizers of the λ-tableaux are conjugate subgroups of S λ .
A λ-tableau t is standard if it is increasing along each row and down each column, i.e., we have both t(i, j) < t(i, j ) and t(i, j) < t(i , j) for all (i, j), (i, j ), (i , j) ∈ [λ] with j < j and i < i . Write T std (λ) for the set of all standard λ-tableaux.
We also have a transitive left action on T (λ, (α|β)) by S n through t 0 as follows: while the node labelled σ −1 (a) by t 0 is coloured c k (respectively, d k ) by T, then σ · T colours (i, j) with c k (respectively, d k ). Observe that under this action, the stabiliser of T 0 is S α|β .

One-dimensional representations of symmetric groups
For the remainder of this section, let O be either F or Z. The signature representation of the symmetric group S n over O is denoted by sgn : S n → {±1} ⊆ O. We shall abuse notation and also write sgn for the OS n -module associated to it. In this latter context, sgn is O-free of rank 1, with basis { }. We denote the trivial OS n -module that is O-free of rank 1 as O, with basis {1}. Thus, σ · 1 = 1 and σ · = sgn(σ) for all σ ∈ S n .
When H is a subgroup of S n , we write the respective restricted OH-modules as O H and sgn H . Sometimes, we shall abuse notation and write them as O and sgn when there is no confusion.

Young permutation modules and Specht modules
Let µ be a composition of n. The Young permutation module M µ O is the permutation module associated to the left regular action of S n on the left cosets of S µ in S n . In other words, , and σ · (dS µ ) = (σd)S µ for all σ ∈ S n and dS µ ∈ S n /S µ . Whenμ is a composition of n obtained by rearranging some parts of µ, the Young subgroups S µ and Sμ are conjugate in S n , so that M µ Let λ be a partition. Given a λ-tableau t, let d t = t • t λ −1 . Then d t ∈ S n and d t · t λ = d t • t λ = t. We define the polytabloid It is not difficult to see that τ · e t = e τ ·t for τ ∈ S n and t ∈ T (λ), so that S λ O is an OS n -submodule of M λ O . Let O = F and M * denote the contragradient dual of an FS n -module M . The following isomorphism is well known (see [7,Theorem 8.15]): Furthermore, FS n is semisimple as an algebra if and only if either char(F) = 0 or char(F) > n, in which case, the Specht modules S λ F , as λ runs over all the partitions of n, give a complete list of pairwise non-isomorphic irreducible FS n -modules.

Signed permutation modules
Let (α|β) be a bicomposition of n. We define the signed permutation module M O (α|β) to be Observe that ifα andβ are compositions obtained by rearranging some parts of α and β respectively, then . As such, signed permutation modules generalise Young permutation modules.
Theorem 2.2 (signed Young's rule). Let (α|β) be a bicomposition of n. The signed permutation module M F (α|β) has a Specht filtration in which, for a partition λ of n, the multiplicity of S λ F as a factor of the filtration equals |T sstd (λ, (α|β))|. Dually, the signed permutation module M F (α|β) has a dual Specht filtration in which, for a partition λ of n, the multiplicity of S λ,F ∼ = S λ F * as a factor of the filtration equals |T sstd (λ, (α|β))|.
Proof . This is essentially proved by Du and Rui in [4]; we review their proof here. In the proof of [4, Proposition 5.2], they showed that there is a Specht filtration for the signed qpermutation module of the Iwahori-Hecke algebra of type A over the ring Z[v, v −1 ] (where v is an indeterminate and q = v 2 ) with the correct multiplicity for each Specht factor. As such, under specialization to any field F, we get a Specht filtration for the signed q-permutation module with the correct multiplicity for each Specht factor. Since the symmetric group algebra is an Iwahori-Hecke algebra of type A with v = 1 = q, the result follows. Remark 2.3. In a private communication with the authors, Andrew Mathas constructed explicitly a Specht filtration and a dual Specht filtration of the signed permutation module, with the correct multiplicity for each Specht factor, for the Iwahori-Hecke algebra of type A.
Proof . (i) It is well known that the Specht module S λ F has dimension |T std (λ)| (see, for example, [11, Theorem 1.1]). Taking the dimension of M F (α|β) and applying Theorem 2.2 thus yield part (i).
(ii) When F = Q, the Specht modules are the irreducible modules, so that the Specht filtration in Theorem 2.2 is in fact a composition series. Thus here, [M : S] denotes the composition multiplicity of an irreducible module S in M .
(iii, iv) These are proved in a manner similar to but easier than part (ii), using the isomor-

Homomorphisms
Let λ be a partition of n and µ be a composition of n. In [7, Section 13], James constructed , unless char(F) = 2 and λ is 2-singular. In particular, this shows that every homomorphism in Hom In this section, we shall generalise these homomorphisms to obtain homomorphisms between Specht modules and signed permutation modules. The next example shows that not every homomorphism in Hom FSn S λ F , M F (α|β) is the restriction of a homomorphism in Hom FSn M λ F , M F (α|β) , illustrating the difficulty of such generalisation. On the other hand, Hom FSn M λ F , M F (α|β) ∼ = Hom FSn (FS n , sgn) has dimension 1 with a basis {θ : FS n → sgn}, where θ(1 Sn ) = . For any σ ∈ S n , we have We have This example shows that the map Hom As such, to generalise the homomorphisms constructed by James for signed permutation modules, we should not attempt to generalise θ T and take its restriction to S λ F , but have to generaliseθ T directly instead. In other words, we need to understandθ T (e t ).

James's construction
Let λ be a partition of n and µ be a composition of n. Let θ T : M λ F → M µ F be the FS n -module homomorphism defined in [7,Section 13]. In this subsection, we study how θ T acts on the polytabloids in M λ F . Fix a λ-tableau t 0 , so that S n acts on T (λ, µ) via t 0 , as described in Section 2.4. Let T 0 be the canonical λ-tableau of type µ associated to t 0 ; recall that stab Sn (T 0 ) = S µ . Then each left coset of S µ in S n corresponds to a λ-tableau of type µ; let d T S µ be the left coset corresponding to T. Recall also the initial λ-tableau t λ , and let d t We summarise this below.
Theorem 3.2. Let λ be a partition of n and µ be a composition of n, and let T ∈ T (λ, µ). The

Generalization of James's construction
Fix a partition λ of n and a bicomposition (α|β) of n. In this subsection, we generalise James's construction of homomorphisms between Specht modules and Young permutation modules to obtain, for each d in a subset R of S n to be defined below (see Definition 3.4), a ZS n -module Theorem 3.3). As before, we fix a λ-tableau t 0 , so that S n acts on T (λ, (α|β)) through t 0 , and denote the canonical λ-tableau of type (α|β) associated to t 0 by T 0 . The following is the main theorem of this section: Theorem 3.3. Let λ be a partition of n, (α|β) be a bicomposition of n and Γ be a fixed left By 'reducing modulo char(F)' the coefficients a ρ −1 t d,d inθ d (e t ), we obtain an FS n -module homomorphismθ F d : S λ F → M F (α|β). Comparing this withθ T in Theorem 3.2, one can see that the former is indeed a generalisation of the latter. The appearance of the map ε d in Theorem 3.3 also explains why our maps are indexed by elements of S n instead of left cosets of S α|β (or equivalently, λ-tableaux of type (α|β)), since ε d depends on d and not on dS α|β . A little thought should convince the reader that to ensure that ε d is well-defined, d may only run over a carefully chosen subset R of S n .
The remainder of this section is devoted to the proof of Theorem 3.3.
Definition 3.4. Define subsets of S n as follows These sets R and C of course depend on t 0 and (α|β).
Proof . For part (i), firstly, This proves the equivalence of (a) and (b).
Next, observe that a transposition (a b) lies in R t 0 if and only if a and b label nodes in the same row of t 0 , i.e., (t 0 ) −1 (a) = (i, j) and (t 0 ) −1 (b) = (i, j ) for some i. Furthermore, is an intersection of conjugates of Young subgroups, it is generated by the transpositions it contains. Thus, stab Rt 0 (T d ) is generated by Hence (b) and (c) are equivalent.
For part (ii), observe that We have analogous statements and proofs for C too.
Lemma 3.6. Let d ∈ S n .
(i) The following statements are equivalent: (ii) If d ∈ C , then τ dξ ∈ C for all τ ∈ C t 0 and ξ ∈ S α|β . Lemmas 3.5 and 3.6 give the following immediate corollary.
In order to generalise the coefficient a d,d T in Theorem 3.2 to a d,d in Definition 3.9, we need the following lemma which also explains the choice of the set R.
Remark 3.10. We give another description of Ω d,d here. Restrict the left regular action of the symmetric group S n on S n /S α|β to the subgroups R t 0 and C t 0 , which partition S n /S α|β into R t 0 -orbits and into C t 0 -orbits respectively. Then σ ∈ Ω d,d if and only if σ ∈ C t 0 and σ·(dS α|β ) ∈ R t 0 ·(dS α|β ). As such, and is therefore a left coset of stab Ct 0 (dS α|β ) = C t 0 ∩ dS α|β d −1 . In particular, Ω d,d is a union of some left cosets of C t 0 ∩ dS α|β d −1 .
As such, the contributions to the sum in a d,d by σ and σρ cancel each other out. Consequently, a d,d = 0 and the proof of part (ii) is now complete. For part (iii), for each i, let Ω (i) = {σ ∈ Ω d,d : σd ∈ d (i) S α|β }, so that Ω d,d is a disjoint union of the Ω (i) 's (see Remark 3.10). Fix i. There exist τ i ∈ R t 0 and γ Consequently, where the final equality is given by the following bijection: fix σ (i) ∈ Ω (i) , we have a bijection between the sets C t 0 ∩ dS α|β d −1 and Ω (i) given by σ → σ (i) σ .
We are now ready to define, for each d ∈ R, a map ϑ d which will eventually lead toθ d .
Definition 3.13. Let λ be a partition of n. Recall that there is a natural left S n -action on the set T (λ) of λ-tableaux, and let ZT (λ) denote the associated permutation ZS n -module. Let (α|β) be a bicomposition of n and let Γ be a fixed left transversal of S α|β in S n . For each d ∈ R, we define a Z-linear map ϑ d : ZT (λ) → M Z (α|β) as follows The following properties of the map ϑ d follow easily from its definition and Lemma 3.11(i).
(i) The map ϑ d is independent of the choice of the left transversal Γ, i.e., if Γ is any left transversal of S α|β in S n , then Next, we aim to show that each of the maps ϑ d : ZT (λ) → M Z (α|β) induces a ZS n -module homomorphismθ d : S λ Z → M Z (α|β). For this, we will show that ϑ d is a ZS n -module homomorphism and that the kernel of the natural map ψ : ZT (λ) → S λ Z given by ψ(t) = e t is contained in ker(ϑ d ) and hence ϑ d inducesθ d as desired. For Proposition 3.15. Let d ∈ R. Then (i) ϑ d is a ZS n -module homomorphism, and (ii) for any t ∈ T (λ), π ∈ C t and Garnir transversal ∆, we have that ker(ϑ d ) contains both π · t − sgn(π)t and G t ∆ · t.
Proof . Let t ∈ T (λ) and x ∈ S n . We have Next we turn to G t ∆ · t. We have Fix d ∈ Γ, and let b(γ, σ) = sgn(γ)sgn(σ)ε d σ(γρ t ) −1 d . We need to show that Let the Garnir transversal ∆ be a left transversal of S X S Y in S X∪Y , where X and Y are subsets of the jth and j th column of [λ], with |X| + |Y | > |C j (λ)|. Since |X| + |Y | > |C j (λ)|, there exists 1 ≤ i ≤ (λ) such that (i, j) ∈ X and (i, j ) ∈ Y , and we may choose i to be the least such. Let η be the transposition (t 0 (i, j) t 0 (i, j )). Then η ∈ R t 0 .
The next result is well known when the underlying ring is a field (see, for example, [5, Section 7.4, Corollary, p. 101]); we are however unable to find its generalisation to Z in the existing literature.
Proof . That ψ is a ZS n -module epimorphism is clear, so we only need to justify the assertion about its kernel. Let K = Z(G ∪ H). It is straightforward to verify that both G and H are invariant under the action of S n , and that ψ(G ∪ H) = {0} [8, 7.2.1 and Theorem 7.2.3], so that K is a ZS n -submodule of ZT (λ), and K ⊆ ker(ψ). It remains to show that ker(ψ) ⊆ K.
Let x ∈ ker(ψ). Then there exists b t ∈ Z for each t ∈ T std (λ) such that Thus there exists k ∈ K such that Hence x + k = 0 and so x ∈ K, and our proof is complete.
We are now ready to prove Theorem 3.3.
Let F be a field. Since As we mentioned in the paragraph following Theorem 3.3, ourθ F d 's generalise theθ T 's constructed by James.

Basis
Let λ be a partition of n and let (α|β) be a bicomposition of n. As before, fix a λ-tableau t 0 so that S n acts on T (λ, (α|β)) through t 0 . Fix a left transversal Γ of S α|β in S n , and write Recall from Corollary 3.7 that we have Γ sstd ⊆ R ∩ C . It is easy to see that |Γ sstd | = |T sstd (λ, (α|β))|. In the last section, we constructed for each d ∈ R an FS n -homomorphism In James's classical case, i.e., when β = ∅, theθ T 's, as T runs over all the semistandard λ tableaux of type α, form a basis for Hom FSn S λ F , M α F , unless F has characteristic 2 and λ is 2-singular (see [7,Theorem 13.13]).
In this section, we investigate the circumstances under which Θ F sstd is a basis for Hom FSn S λ F , M F (α|β) .

Linear independence
While James'sθ T is always nonzero irrespective of the characteristic of the ground field, ourθ d may be zero in some characteristic in view of Lemma 3.11(iii). As such, it is certainly possible for Θ F sstd to be linearly dependent. In fact, even when all elements of Θ F sstd are nonzero, it is still possible for Θ F sstd to be linearly dependent: Example 4.3. We continue with Example 3.12, where λ = (2, 1 p+2 ) and (α|β) = ∅| p, 2 2 . Suppose that p = char(F) is an odd prime. Then Lemma 3.11(i). In this subsection, we obtain a sufficient condition for Θ F sstd to be linearly independent. We first introduce a pre-order ¤ on T (λ, (α|β)), which induces another on S n . For each T ∈ T (λ, (α|β)), write C j (T) for the multi-set associated to the jth column of T, i.e., Let T, T ∈ T (λ, (α|β)). Suppose that C j (T) = {y 1 , . . . , y r } and C j (T ) = {z 1 , . . . , z r }, where y 1 ≤ y 2 ≤ · · · ≤ y r and z 1 ≤ z 2 ≤ · · · ≤ z r . Write C j (T) £ C j (T ) if and only if there exists k such that y s = z s for all 1 ≤ s < k and y k > z k . We write T ¤ T if and only if C j (T) = C j (T ) for all j, or there exists t such that C j (T) = C j (T ) for all j < t and C t (T) £ C t (T ). It is easy to check that ¤ is a pre-order on T (λ, (α|β)) (i.e., it is a reflexive and transitive binary relation on T (λ, (α|β))). In addition, write T ∼ T if and only if T ¤ T and T ¤ T (equivalently, C j (T) = C j (T ) for all j), and T £ T if and only if T ¤ T but T T (equivalently, there exists t such that C j (T) = C j (T ) for all j < t and C t (T) £ C t (T )).
Recall   (ii) If T d is row semistandard, and d ∈ R t 0 dS α|β , then either T d = T d or T d £ T d .
We can now state a sufficient condition for Θ F sstd to be linearly independent.
by Lemma 4.4(ii). Applying Lemma 3.11(iii) completes the proof since stab Ct 0 ( Combining the last two results, we get the following immediate corollary.

Main results
Similar to how ourθ d inducesθ F d , every φ ∈ Hom ZSn S λ Z , M Z (α|β) also induces φ F ∈ Hom FSn S λ F , M F (α|β) . Write Proposition 4.8. Let Ψ F be a linearly independent subset of Φ F of size |T sstd (λ, (α|β)|. Then (i) Ψ F is a basis for Φ F , and (ii) if S λ F lies in a block of FS n which is simple as an algebra, then Proof . If S λ F lies in a block of FS n which is simple as an algebra, then the composition multiplicity of S λ F in M F (α|β) equals dim F Hom FSn S λ F , M F (α|β) , which is in turn equal to |T sstd (λ, (α|β)| by Theorem 2.2. Since Ψ F is linearly independent with the same cardinality, it is a basis for Hom FSn (S λ F , M F (α|β)). In particular, since Hom FSn S λ F , M F (α|β) = F-span Ψ F ⊆ Φ F ⊆ Hom FSn S λ F , M F (α|β) , we must have equality throughout, proving part (ii).
For part (i), let Ψ F = φ F 1 , . . . , φ F k , where k = |T sstd (λ, (α|β)|, and let φ F 0 ∈ Φ F . Since QS n is semisimple as an algebra, we apply the previous paragraph and obtain that Hom QSn (S λ Q , M Q (α|β)) has dimension k. Thus there is a non-trivial relation on φ Q 0 , φ Q 1 , . . . , φ Q k , which we can write as k j=0 c j φ Q j = 0, where the c j 's are coprime integers. This yields the nontrivial linear relation k j=0 c F j φ F j = 0. Since Ψ F is linearly independent, this implies that φ F 0 lies in the F-span of Ψ F . Remark 4.9. As is well-known, S λ F lies in a simple block if and only if λ is a p-core partition (in other words, λ has no rimhook of size p) where p = char(F).
The following is our first main result: Theorem 4.11. Suppose that FS n is semisimple as an algebra (or equivalently, char(F) = 0 or char(F) > n). Then Θ F sstd is a basis for Hom FSn S λ F , M F (α|β) . Proof . Since FS n is semisimple, every block of FS n is simple. Thus, Θ F sstd is a basis for Hom FSn S λ F , M F (α|β) by Corollary 4.7 and Proposition 4.8. Our next main result provides a sufficient condition for Θ F sstd to be a basis for Φ F when FS n is not semisimple. Note that Example 4.2 shows that this condition is insufficient in ensuring that Θ F sstd spans Hom FSn S λ F , M F (α|β) . Theorem 4.13. Suppose that char(F) = p > 0. If no column of T has p or more nodes of the same colour for every T ∈ T sstd (λ, (α|β)), then Θ F sstd is a basis for the Φ F .
Proof . Let d ∈ Γ sstd . Observe that p |stab Ct 0 (T d )|, if and only if no column of T d has p or more nodes of the same colour. Thus, this follows from Corollary 4.7 and Proposition 4.8 immediately.
We end our paper with the following remark and leave the details to the reader.
Remark 4.15. Recall from Definition 3.4 that the set R and hence the sets Θ R and Θ sstd depend on the fixed λ-tableau t 0 and (α|β) we chose. Suppose that we had chosen another λ-tableau t 0 and arrived at the sets R , Θ R and Θ sstd . It is easy to verify that On the other hand, let {1, 2, . . . , n} = giving us a subset Θ A|B R A|B of Hom ZSn S λ Z , M Z (A|B) . Let g ∈ S n , and let A i = g −1 (A i ), B j = g −1 (B j ) for all i = 1, . . . , r and j = 1, . . . , s. Then where g is the natural ZS n -module isomorphism M Z (A|B) → M Z (A |B ) defined by g (x ⊗ 1 ⊗ ) = xg ⊗ 1 ⊗ for all x ∈ S n .