On signed Young permutation modules and signed p-Kostka numbers

Schur algebras

Our results may be applied to obtain corollaries on modules for the Schur algebra. Fix n,d∈ℕ with d≥n and let GLd⁢(F) be the general linear group of d×d matrices over F. Let ρ:GLd⁢(F)→GLm⁢(F) be a representation of GLd⁢(F) of dimension m. We say that ρ is a polynomial representation of degree n if the matrix coefficients ρ⁢(X)i⁢j for each i,j∈{1,…,m} are polynomials of degree n in the coefficients of the matrix X. Given a polynomial representation ρ:GLd⁢(F)→GL⁢(V) of degree n, the image of V under the Schur functorf is the subspace of V on which the diagonal matrices diag⁢(a1,…,ad)∈GLd⁢(F) act as a1⁢…⁢an. It is easily seen that f⁢(V) is preserved by the permutation matrices in GLd⁢(F) that fix the final d-n vectors in the standard basis of Fd. Thus f⁢(V) is a module for F⁢𝔖n.

The category of polynomial representations of GLd⁢(F) of degree n is equivalent to the category of modules for the Schur algebra SF⁢(d,n). We refer the reader to [13] for the definition of SF⁢(d,n) and further background. In this setting, the Schur functor may be defined by V↦e⁢V, where e∈SF⁢(d,n) is an idempotent such that e⁢SF⁢(d,n)⁢e≅F⁢𝔖n. It follows that f is an exact functor from the category of polynomial representations of GLd⁢(F) of degree n to the category of F⁢𝔖n-modules.

Let E denote the natural GLd⁢(F)-module. Given α∈𝒫⁢(n), let Symα⁢(E) and ⋀β(E) denote the corresponding divided symmetric and exterior powers of E, defined as quotient modules of E⊗n. The mixed powersSymα⁢E⊗⋀βE for (α|β)∈𝒫2(n) generate the category of GLd⁢(F)-modules of degree n. In [7], Donkin defines a listing module to be an indecomposable direct summand of a mixed power. (As the nautical parlance suggests, listing modules generalize tilting modules). By [7, Proposition 3.1 c], for each (λ|μ)∈𝒫2(n) there exists a unique listing module List(λ|pμ) such that f(List(λ|pμ))≅Y(λ|pμ). By [7, Proposition 3.1a], we have

Moreover, by [7, Proposition 3.1b], the Schur functor induces an isomorphism

Thus each of our three main theorems has an immediate translation to a result on multiplicities of listing modules in certain mixed powers. For example Theorem 1.4 classifies the indecomposable GLd⁢(F)-mixed powers and shows that each has an endomorphism algebra, as a GLd⁢(F)-module, of dimension at most 2. It is also worth noting that many of Gill’s results from [12] are reproved in greater generality in the Schur algebra setting in a recent paper of Donkin [8].

Steinberg tensor product formula

As Gill remarks in [12], some of his results can be obtained using weight spaces and the Steinberg Tensor Product Theorem for irreducible representations of the group GLd⁢(F). We explain the connection here, since this remark is also relevant to this work. Let α be a composition of n where d≥n and let ξα∈SF⁢(d,n) be the idempotent defined in [13, Section 3.2] such that ξα⁢V is the α-weight space, denoted Vα, of the SF⁢(d,n)-module V; the idempotent e defining the Schur functor is ξ(1n). For λ a partition of n, let L⁢(λ) denote the irreducible representation of GLd⁢(F) with highest weight λ, thought of as a module for SF⁢(d,n). Let Proj⁢(λ) be the projective cover of L⁢(λ). By James’ original definition of Young modules (this is shown to be equivalent to ours in Section 5.1), we have Yλ=f⁢(Proj⁢(λ)); moreover,

(Here Symα⁢(E)⊆E⊗n is the contravariant dual, as defined in [13, 2.7a], of the quotient module Symα⁢(E) of E⊗n.) Thus

As an example of this relationship between p-Kostka numbers and dimensions of weight spaces, we use (1.2) to deduce Theorem 1 in [12]. By the Steinberg Tensor Product Theorem, L⁢(p⁢λ)=L⁢(λ)ℱ, where ℱ is the Frobenius map, acting on representing matrices by sending each entry to its pth power. Clearly there is a canonical vector space isomorphism (L⁢(λ)ℱ)p⁢α≅L⁢(λ)α. Therefore

as required.

Schur superalgebras

Our Theorem 1.2 generalizes the result just proved, so it is natural to ask if it can be proved in a similar way, replacing the Schur algebra with the Schur superalgebra defined in [7]. Let a,b∈ℕ. Given (λ|pμ)∈𝒫2(n) where λ has at most a parts and μ has at most b parts, let L(λ|pμ) denote the irreducible module of highest weight (λ|pμ) for the Schur superalgebra S(a|b,n), defined in [7, p. 661]. By [7, Section 2.3], we have

generalizing (1.2).

Let GL(a|b) denote the super general linear group defined in [4, Section 2]. As E⊗n is a generator for the category of polynomial representations of GL(a|b) of degree n, it follows from [7, p. 660, (1)] that the category of such modules is equivalent to the module category of S(a|b,n). Taking the even degree part of GL(a|b) recovers GLa⁢(F)×GLb⁢(F). (More precisely, the even degree part is isomorphic to the product of the affine group schemes corresponding to these two general linear groups.) The Frobenius map is identically zero on the odd degree part of GL(a|b), so induces a map ℱ:GL(a|b)→GLa(F)×GLb(F). Let ℱ⋆ be the corresponding inflation functor, sending modules for GLa⁢(F)×GLb⁢(F) to modules for GL(a|b). By [4, Remark 4.6 (iii)] we have

where ⊠ denotes an outer tensor product. Taking weight spaces we get

By (1.3) we have [M(pα|pβ):Y(pλ|pμ)]=dimFL(λ)αdimFL(μ)β. Replacing μ with p⁢μ and applying the Steinberg Tensor Product Formula, this implies the asymptotic stability of signed p-Kostka numbers mentioned after Theorem 1.2.

Stated for GL(a|b)-modules, the remaining part of Theorem 1.2 becomes

This does not follow from the results mentioned so far, or from the version of the Steinberg Tensor Product Theorem for GL(a|b)-modules proved in [22], because the module on the right-hand side is not an inflation. Moreover, translated into this setting, a special case of Example 6.4 shows that dimFL((1)|∅)(∅|(1))=1 whereas dimFL((p)|∅)(∅|(p))=dimFL((p))∅dimFL(∅)(p)=0, so it is certainly not the case that equality always holds. (Further examples of this type are given by the general case of Example 6.4.) Whether or not a proof using supergroups is possible, the authors believe that since Theorem 1.2 can be stated within the context of symmetric groups, it deserves a proof in this setting.

Klyachko’s multiplicity formula

Klyachko’s multiplicity formula [21, Corollary 9.2] expresses the p-Kostka number [Mα:Yλ] in terms of p-Kostka numbers for p-restricted partitions. Our Corollary 5.2 gives a generalization to signed Young modules. Specializing this result we obtain a symmetric group proof of Klyachko’s formula in the form

where λ=∑i=0rpi⁢λ⁢(i) is the p-adic expansion of λ, ρ is the partition defined in Theorem 1.1 (iii) and the set Λ((α|∅),ρ) is as defined in Notation 3.8.

Outline

In Section 2 we recall the main ideas concerning the Brauer construction for p-permutation modules and set up our notation for symmetric group modules and modules for wreath products. In Section 3 we find the Broué quotients of signed Young permutation modules. In Section 4 we use these results, together with James’ Submodule Theorem, to define Young modules and signed Young modules in the symmetric group setting. We then prove Donkin’s Theorem 1.1. We give some immediate corollaries of this theorem in Section 5. In Sections 6 and 7, we prove Theorems 1.2, 1.3 and 1.4.

2.1 Indecomposable summands

Let G be a finite group. Let M and N be FG-modules. We write N∣M if N is isomorphic to a direct summand of M. We have already used the notation [M:N] for the number of summands in a direct sum decomposition of M that are isomorphic to the indecomposable module N. This multiplicity is well defined by the Krull–Schmidt Theorem (see [1, Section 4, Lemma 3]). The proof of the following lemma is easy.

2.2 Broué correspondence

Let G be a finite group. An FG-module V is said to be a p-permutation module if for every Sylow p-subgroup P of G there exists a linear basis of V that is permuted by P. A useful characterization of p-permutation modules is given by the following theorem (see [3, (0.4)]).

It easily follows that the class of p-permutation modules is closed under restriction and induction and under taking direct sums, direct summands and tensor products.

We now recall the definition and the basic properties of Brauer quotients. Given an FG-module V and P a p-subgroup of G, the set of fixed points of P on V is denoted by

It is easy to see that VP is an F⁢NG⁢(P)-module on which P acts trivially. For Q a proper subgroup of P, the relative trace map TrQP:VQ→VP is the linear map defined by

where the sum is over a complete set of left coset representatives for Q in P. The definition of this map does not depend on the choice of the set of representatives. We observe that

is an F⁢NG⁢(P)-module on which P acts trivially. We define the Brauer quotient of V with respect to P to be the F⁢[NG⁢(P)/P]-module

If V is an indecomposable FG-module and P is a p-subgroup of G such that V⁢(P)≠0, then P is contained in a vertex of V. Broué proved in [3] that the converse holds for p-permutation modules.

Here Pg denotes the conjugate g⁢P⁢g-1 of P. If V is an FG-module with p-permutation basis ℬ with respect to a Sylow p-subgroup P~ of G and P≤P~, then, taking for each orbit of P on ℬ the sum of the basis elements in that orbit, we obtain a basis for VP. Each sum over an orbit of size p or more is a relative trace from a proper subgroup of P. Hence V⁢(P) is isomorphic to the F-span of

Thus Theorem 2.3 has the following corollary.

The next result states what is now known as the Broué correspondence.

The following lemma allows the Broué correspondence to be applied to monomial modules such as signed Young permutation modules.

The Brauer quotient of an outer tensor product of p-permutation modules is easily described.

2.3 Partitions and compositions

Let n∈ℕ0. A composition of n is a sequence of non-negative integers α=(α1,…,αr) such that αr≠0 and α1+⋯+αr=n. In this case, we write ℓ⁢(α)=r and |α|=n. The unique composition of 0 is denoted by ∅; we have ℓ⁢(∅)=0. The Young subgroup𝔖α is the subgroup 𝔖α1×⋯×𝔖αr of 𝔖n, where the ith factor 𝔖αi acts on the set {α1+⋯+αi-1+1,…,α1+⋯+αi-1+αi}. Let α=(α1,…,αr) and β=(β1,…,βs) be compositions and let q∈ℕ. We denote by q⁢α and α∙β the compositions of q⁢|α| and |α|+|β| defined by

respectively. We set 0⁢α=∅. We denote by α+β the composition of |α|+|β| defined by

where we have assumed, without loss of generality, that s≤r. We define α-β similarly, in the case when βi≤αi for each i≤s.

A composition α is a partition if it is non-increasing. A partition α is called p-restricted if αi-αi+1<p for all i≥1. We denote the set of compositions, partitions and p-restricted partitions of n by 𝒞⁢(n), 𝒫⁢(n) and ℛ⁢𝒫⁢(n), respectively. A partition α is p-regular if its conjugate α′, defined by αj′=|{i:αi≥j}|, is p-restricted. It is well known that if λ is a partition, then there exist unique p-restricted partitions λ⁢(i) for i∈ℕ0 such that

We call this expression the p-adic expansion of λ.

Let 𝒫2⁢(n), 𝒞2⁢(n) and ℛ⁢𝒫2⁢(n) be the sets consisting of all pairs (λ|ν) of partitions, compositions and p-restricted partitions, respectively, such that |λ|+|ν|=n. Here λ or ν may be the empty composition ∅. For (λ|ν),(α|β)∈𝒫2(n), we say that (λ|ν)dominates(α|β), and write (λ|ν)⊵(α|β), if, for all k≥1, we have

1. ∑i=1kλi≥∑i=1kαi, and

2. |λ|+∑i=1kνi≥|α|+∑i=1kβi.

(As a standing convention we declare that λi=0 whenever λ is a partition and i>ℓ⁢(λ).) This defines a partial order on the set 𝒫2⁢(n) called the dominance order. This order becomes the usual dominance order on partitions when restricted to the subsets {(λ|∅)∈𝒫2(n)} or {(∅|ν)∈𝒫2(n)} of 𝒫2⁢(n).

2.4 Modules for symmetric groups

Let n∈ℕ0, let 𝔖n be the symmetric group on the set {1,…,n} and let 𝔄n be its alternating subgroup. Given a subgroup H of 𝔖n, we denote the trivial representation of H by F⁢(H), and the restriction of the sign representation of 𝔖n to H by sgn⁢(H). In the case when H=𝔖γ for some composition γ of n we write F⁢(γ) and sgn⁢(γ) for F⁢(H) and sgn⁢(H), respectively. If γ=(n), we reduce the number of parentheses by writing F⁢(n) and sgn⁢(n), respectively.

For λ a p-regular partition of n, let Dλ be the F⁢𝔖n-module defined by

where Sλ is the Specht module labelled by λ (see [17, Chapter 4]). By [17, Theorem 11.5] each Dλ is simple, and each simple F⁢𝔖n-module is isomorphic to a unique Dλ. The simple F⁢𝔖n-modules can also be labelled by p-restricted partitions. For λ∈ℛ⁢𝒫⁢(n) we set Dλ=soc⁢(Sλ). The connection between the two labellings is given by Dλ≅Dλ′⊗sgn⁢(n). For λ∈ℛ⁢𝒫⁢(n), let Pλ denote the projective cover of the simple F⁢𝔖n-module Dλ.

Finally, for γ∈𝒫⁢(n), let χγ denote the ordinary irreducible character of Sγ, defined over the rational field.

2.5 Modules for wreath products

Let m∈ℕ and let G be a finite group. Recall that the multiplication in the group G≀𝔖m is given by

for (g1,…,gm;σ),(g1′,…,gm′;σ′)∈G≀𝔖m. (Our notation for wreath products is taken from [19, Section 4.1].) Let M be an FG-module. The m-fold tensor product of M becomes an F⁢[G≀𝔖m]-module with the action given by

for (g1,…,gm;σ)∈G≀𝔖m, v1,…,vm∈M. We denote this module by M^⊗m. Note that we have twisted the action of the top group 𝔖m by the sign representation. Thus, in the notation of [19, 4.3.14], we have

The 1-dimensional module sgn⁢(k)^⊗n will be important to us. In our applications k will be a p-power, and so odd. Since a transposition in the top group 𝔖n acts on {1,…,k⁢n} as a product of k disjoint transpositions, and so has odd sign, there is a simpler definition of this module, as Res𝔖k≀𝔖n𝔖k⁢n⁡sgn⁢(k⁢n). More generally, given α∈𝒞⁢(n) and an odd number k, we define

For use in the proof of Proposition 4.5 we briefly recall the character theory of the group C2≀𝔖n. Let χλ denote the irreducible character of 𝔖n labelled by λ∈𝔖n. For (λ|μ)∈𝒫2(n), with |λ|=m1 and |μ|=m2, we define χ(λ|μ) to be the ordinary character of the following module for C2≀𝔖n:

A standard Clifford theory argument (see for instance [19, Theorem 4.34]) shows that the characters χ(λ|μ) for (λ|μ)∈𝒫2(n) are precisely the irreducible characters of C2≀𝔖n.

2.6 Sylow p-subgroups of 𝔖n

Let Pp be the cyclic group 〈(1,2,…,p)〉≤𝔖p of order p. Let P1={1} and, for d≥1, set

By [19, 4.1.22, 4.1.24], Ppd is a Sylow p-subgroup of 𝔖pd.

Let n∈ℕ. Let n=∑i=0rni⁢pi, where 0≤ni<p for i∈{0,…,r}, and let nr≠0 be the p-adic expansion of n. By [19, 4.1.22, 4.1.24], the Sylow p-subgroups of 𝔖n are each conjugate to the direct product ∏i=0r(Ppi)ni. Hence if we define Pn to be a Sylow p-subgroup of the Young subgroup ∏i=0r(𝔖pi)ni, then Pn is a Sylow p-subgroup of 𝔖n. The normalizer N𝔖n⁢(Pn) of Pn in 𝔖n is denoted by Nn.

Whenever ρ=(ρ1,…,ρr)∈𝒞⁢(n), we denote by Pρ a Sylow p-subgroup of 𝔖ρ, defined so that Pρ=∏i=1rPρi. In the special case when

where mi∈ℕ0 for each i, we have Pρ=∏i=0s(Ppi)mi; in particular, the group Pρ has precisely mi orbits of size pi on the set {1,2,…,n} for each i. We write Nρ=N𝔖n⁢(Pρ).