ON COMPARING THE COHOMOLOGY OF GENERAL LINEAR AND SYMMETRIC GROUPS

ALEXANDER S. KLESHCHEV AND DANIEL K. NAKANO

1 Introduction

1.1 Let k be an algebraically closed field, GL_{n}(k) be the general linear group over k, and Σ_{d} be the symmetric group on d letters. For k = ℂ, Frobenius and Schur discovered that the commuting actions of GL_{n}(k) and Σ_{d} on V^{⊗d} can be used to relate the character theory of these two groups. For modular representations of these groups the relationship between their representation theories is not as direct. However, James [12] showed that the decomposition matrix of the symmetric group is a submatrix of the decomposition matrix of the general linear group, and Erdmann [9] used tilting modules to prove that one can also recover the decomposition numbers of the general linear groups from those of the symmetric groups.

Let M(n, d) be the category of polynomial representations of GL_{n}(k) of a fixed degree d ≤ n. The Schur functor F is a certain covariant exact functor from M(n, d) to modules for the group algebra kΣ_{d} ([10]). Some of the aforementioned results can be proved by using F. So it is natural to hope that the Schur functor can also be used to compare the cohomogy theories of GL_{n}(k) and Σ_{d}. In this paper we address the following question. Let M, N ∈ M(n, d). When is it true that

Ext_{GL_{n}(k)}^{i}(M, N) ≅ Ext_{kΣ_{d}}^{i}(F(M), F(N))?  (1.1.1)

Here Ext_{GL_{n}(k)}^{i} is taken in the category M(n, d). By [5, 2.1f], this is equivalent to considering extensions in the category of all rational GL_{n}(k)-modules. The case where i = 1 and M, N are irreducible representations is of particular importance.

The precise relationship between the two cohomology groups is given by the spectral sequence of [7]. It starts with extensions for GL_{n}(k) and converges to extensions for kΣ_{d}. However, to use the spectral sequence one needs to compute the higher right derived functors R^{•}G(−) ≅ Ext_{kΣ_{d}}^{•}(V^{⊗d}, −), where G is a right adjoint functor to F (see Section 2.2). These derived functors are used extensively throughout the paper.

1.2 To describe the results of the paper we need some notation. Let us denote the simple (polynomial) GL_{n}(k) module with highest weight λ by L(λ). We also write Δ(λ) and ∇(λ) for the standard and costandard GL_{n}(k)-modules with highest weight λ, respectively. It is well-known that F(L(λ)) is non-zero if and only if λ is p-restricted, and

{F(L(λ)) | L(λ) ∈ M(n, d), and λ is p-restricted}

is a complete set of the simple kΣ_{d}-modules up to isomorphism. If λ is p-restricted, we denote F(L(λ)) by D_{λ}.

1991 Mathematics Subject Classification. Primary 20C, 20G; Secondary 20J06, 20G10.

Research of both authors was supported in part by NSF (grants DMS-9600124 and DMS-9800960).

The second author thanks the Department of Mathematics at the University of Oregon for the hospitality.

Both authors thank Henning H. Andersen and Jens C. Jantzen for organizing a productive meeting at Aarhus University, where this work was completed.

A simple kΣ_{d}-module is called completely splittable (CS for short) if its restriction to any Young subgroup is semisimple. These modules were introduced in [15], see also [20, 17, 19]. It is possible to say explicitly which of the D_{λ} are CS, see [15] and Section 4.2.

Now we describe the contents of the paper in greater detail. In Section 2, the basic facts about the Schur functor are reviewed. We also define the spectral sequence constructed in [7] and state the elementary properties of the higher right derived functors R^{•}G(−). In the following section the image of G on twisted Specht and twisted Young modules is computed (for p > 3). The main result of the section (Theorem 3.2) can be interpreted as the isomorphism (1.1.1) for important classes of modules. Section 4 deals with CS modules. Our result in Theorem 4.4(a) says that

G(D_{λ}) = L(λ).  (1.2.1)

if D_{λ} is CS and non-trivial (i.e. D_{λ} ≅ k̸). We note that the image of G on the simple kΣ_{d}- modules is generally very complicated. According to [7, 3.3], G(D_{λ}) can be described as the largest submodule of the injective hull of L(λ) in the category M(n, d) whose only p-restricted composition factor is L(λ). Informally, G(D_{λ}) is L(λ) with as many non-p-restricted modules on top as possible. In the light of this description the property (1.2.1) is quite remarkable. Using this result we show in Theorem 4.4(b) that if λ and μ are p-restricted, D_{λ} is CS, and D_{λ} ≅ k̸ then

Ext_{GL_{n}(k)}^{1}(L(μ), L(λ)) ≅ Ext_{kΣ_{d}}^{1}(D_{μ}, D_{λ}).  (1.2.2)

This again provides us with many cases where (1.1.1) is indeed an isomorphism.

Section 5 contains vanishing results about the cohomology of symmetric groups with coefficients in CS modules. This information is later used to prove the vanishing of some R^{•}G(D_{λ}). The results of Section 5 allow us to generalize the isomorphism (1.2.2) to higher Ext-groups in a suitable range of degrees. This result is a portion of two general theorems on stability of extensions given in Sections 6.1 and 6.3, which give further examples where (1.1.1) holds. For example, Corollary 6.1a claims that

H^{i}(Σ_{d}, F(N) ⊗ sgn) ≅ Ext_{GL_{d}(k)}^{i}(δ, N) for 0 ≤ i ≤ p − 3.

Here, sgn is the one-dimensional sign representation for Σ_{d}, δ is the one-dimensional determinant representation for GL_{d}(k), and N ∈ M(n, d). Note that Cline, Parshall and Scott [4, (12.4)] have a similar result in non-describing characteristic relating the cohomology of the finite general linear group with cohomology for the q-Schur algebra. Their approach relies on using the Deodhar complex and is quite different from ours.

One can improve the stability results on the symmetric group cohomology with coefficients in a simple module if one considers the following setting. Let V be the natural n-dimensional GL_{n}(k)-module and S^{d}(V) be the d-th symmetric power of V. Corollary 6.3(b) shows that for n ≥ d and 0 ≤ i ≤ 2(p − 2),

H^{i}(Σ_{d}, D_{μ}) ≅ Ext_{GL_{n}(k)}^{i}(S^{d}(V), L(μ)).

To interpret this statement in the context of (1.1.1) one needs to note that F(S^{d}(V)) ≅ k.

Theorem 6.1 can also be used to prove a conjecture on the cohomology of dual Specht modules made in [2]. In fact, we get an even stronger result (Corollary 6.2) that for any Specht module S^{λ} over kΣ_{d} and 1 ≤ i ≤ p − 3,

H^{i}(Σ_{d}, (S^{λ})^{∗}) = 0.  (1.2.3)

This result can be interpreted as an isomorphism in (1.1.1) with M = ∧^{d}(V) and N being any costandard module. The vanishing result (1.2.3) along with the work in Section 3 is used in Section 6.4 to prove stability results for extensions of Specht modules. This is followed by similar results in Section 6.5 on extensions of Young modules. Finally, in Section 6.6 we obtain some results on cohomology of the alternating groups.

2 Comparing GL_{n}(k) and kΣ_{d}

2.1 The Schur functor. The basic references for this section are [10], [11]. Let k be an algebraically closed field of characteristic p > 0 and let G = GL_{n}(k). The Schur algebra S(n, d) is the finite-dimensional associative k-algebra End_{kΣ_{d}}(V^{⊗d}) where V is the natural representation of G. The category M(n, d) of polynomial G-modules of a fixed degree d ≥ 0 is equivalent to the category of modules for S(n, d) and we do not distinguish between the two categories from now on. We denote by Mod(kΣ_{d}) (resp. mod(kΣ_{d})) the category of all (resp. all finite dimensional) kΣ_{d}-modules.

Throughout the paper we assume that n ≥ d. Let e = ζ_{(1,2,...,d)(1,2,...,d)} be the idempotent in S(n, d) described in [10, (6.1)]. Then eS(n, d)e ≅ kΣ_{d}. The Schur functor F is the covariant exact functor from M(n, d) to Mod(kΣ_{d}) defined on objects by F(M) = eM.

The simple S(n, d)-modules are in bijective correspondence with set of partitions of d. We will denote this set by Λ = Λ^{+}(n, d) and the corresponding simple S(n, d)-module by L(λ) for λ ∈ Λ. Note that one can also identify Λ as the set of dominant polynomial weights of G of degree d. Moreover, if λ ∈ Λ, let P (λ) be the projective cover of L(λ) and T (λ) be the corresponding tilting module. There exists a duality on M(n, d) fixing simple modules called the transpose dual. This duality will be denoted by τ. The duality τ and the usual duality ‘∗’ in mod(kΣ_{d}) are compatible in the sense that eM^{τ} ≅ (eM)^{∗} for any finite dimensional M ∈ M(n, d).

A partition (λ_{1}, λ_{2}, …) is called p-restricted if λ_{i} − λ_{i+1} ≤ p − 1 for all i. As mentioned in Section 1.2, we label the simple kΣ_{d}-modules by the p-restricted partitions λ ∈ Λ. The set of the p-restricted partitions of d will be denoted by Λ_{res}. A partition λ is called p-regular if its transpose λ′ is p-restricted. We denote the set of all p-regular partitions of d by Λ_{reg}. In [11], the simple kΣ_{d}-modules are labelled by the p-regular partitions and denoted by D^{λ}. We will use both parametrizations so note a result from [10, § 6]:

D^{λ} ≅ D_{λ′} ⊗ sgn for any λ ∈ Λ_{reg}.  (2.1.1)

The Specht, Young, and permutation modules over kΣ_{d} corresponding to a partition λ ∈ Λ are denoted by S^{λ}, Y^{λ}, and M^{λ}, respectively. In particular M^{λ} is the module induced from the trivial module over the Young subgroup Σ_{λ}. One has following correspondences between S(n, d)-modules and kΣ_{d}-modules under F (see [10, § 6] and [6, 3.5, 3.6]):

F(∇(λ)) = S^{λ}, F(Δ(λ)) = (S^{λ})^{∗}, F(P (λ)) = Y^{λ}, F(T (λ)) = Y^{λ′} ⊗ sgn.

2.2 Spectral sequences. Let A = S(n, d) and eAe = kΣ_{d} with n ≥ d. The Schur functor F can be represented as a Hom functor and a tensor functor: F(M) ≅ eM ≅ Hom_{A}(Ae, M) ≅ eA ⊗_{A} M. By using this identification, F admits a right adjoint functor, a right adjoint G defined by

G(N) = Hom_{eAe}(eA, N).

Furthermore, G is a left inverse to F. Since Ae ≅ V^{⊗d}, and G takes injective eAe-modules to injective A-modules, one can construct a spectral sequence [7, 2.2]:

Theorem (A). Let M ∈ M(n, d), N ∈ Mod(kΣ_{d}) with n ≥ d. There exists a first-quadrant Grothendieck spectral sequence, with E_{2}-page given by

E_{2}^{i,j} = Ext_{S(n,d)}^{i}(M, Ext_{kΣ_{d}}^{j}(V^{⊗d}, N)) ⇒ Ext_{kΣ_{d}}^{i+j}(eM, N).  (2.2.1)

For M ∈ M(n, d) and S a simple module in M(n, d), let [M : S] be the multiplicity of S as a composition factor of M. The following results from [7] provide information on composition factors of the higher right derived functors of G.

Theorem (B). Let N ∈ mod(kΣ_{d}), and μ ∈ Λ. Then

(i) [R^{j}G(N) : L(μ)] = dim_{k} Ext_{kΣ_{d}}^{j}(Y^{μ}, N) for j ≥ 0.

(ii) In particular, e(R^{j}G(N)) = 0 for j > 0.

2.3 A standard spectral sequence argument yields the following useful result.

Proposition . Let n ≥ d, M ∈ M(n, d), and N ∈ Mod(kΣ_{d}) Suppose that R^{j}G(N) = 0 for 1 ≤ j ≤ t. Then

Ext_{S(n,d)}^{i}(M, G(N)) ≅ Ext_{kΣ_{d}}^{i}(eM, N)

for 0 ≤ i ≤ t.

Proof. Consider the spectral sequence (2.2.1). By assumption, Ext_{kΣ_{d}}^{j}(V^{⊗d}, N) = R^{j}G(N) = 0 for 1 ≤ j ≤ t. Therefore, E_{2}^{i,j} = 0 if j > 0 and 1 ≤ i + j ≤ t. The spectral sequence has differentials d_{r} with bidegree (r, 1 − r). Therefore,

E_{2}^{i,0} ≅ Ext_{S(n,d)}^{i}(M, G(N)) ≅ Ext_{kΣ_{d}}^{i}(eM, N)

for 0 ≤ i ≤ t. _{□}

3 Twisted Specht and Young modules

In this section we prove that the image under the functor G of a twisted Young (resp. Specht) module is a tilting (resp. Weyl) module as long as p > 3. These results for twisted Young modules can be viewed as dual to the results proved in [3, 5.2.4]. The result in [3] holds for p > 2 and can be used to prove its dual version (see [8, 6.2]). But we employ a different approach than the one given in [3].

3.1 We deal with Young modules first.

Lemma . If p > 3 and λ ∈ Λ then G(Y^{λ′} ⊗ sgn) has a good filtration.

Proof. Let M = Δ(μ) and N = Y^{λ′} ⊗sgn. From the five term exact sequence for the spectral sequence (2.2.1), we have

0 → Ext_{S(n,d)}^{1}(Δ(μ), G(Y′ ⊗ sgn)) ↪ Ext_{kΣ_{d}}^{1}((S^{μ})^{∗}, Y^{λ′} ⊗ sgn).

However,

Ext_{kΣ_{d}}^{1}((S^{μ})^{∗}, Y^{λ′} ⊗ sgn) ≅ Ext_{kΣ_{d}}^{1}(Y^{λ′}, S^{μ} ⊗ sgn) ≅ Ext_{kΣ_{d}}^{1}(Y^{λ′}, (S^{μ′})^{∗}).

As Y^{λ′} is a summand of the permutation module M^{λ′}, the Frobenius reciprocity implies that Ext_{kΣ_{d}}^{1}(Y^{λ′}, (S^{μ′})^{∗}) is a summand of Ext_{kΣ_{λ′}}^{1} (k, (S^{μ′})^{∗}↓_{Σ_{λ′}}). Moreover, the restriction (S^{μ′})^{∗}↓_{Σ_{λ′}} has a Specht filtration by [13], so the last Ext-group is zero in view of [2, 2.4] or [8, 4.2]. Hence, Ext_{S(n,d)}^{1}(Δ(μ), G(Y′ ⊗ sgn)) = 0 for all μ ∈ Λ, thus G(Y′ ⊗ sgn) has a good filtration. _{□}

Theorem . If G(Y^{λ′} ⊗ sgn) has a good filtration then G(Y^{λ′} ⊗ sgn) = T (λ). In particular for p > 3 we have G(Y^{λ′} ⊗ sgn) = T (λ) for all λ ∈ Λ.

Proof. As d ≤ n, every standard module has a p-restricted socle [12]. Hence T (λ) also has a p-restricted socle. So by [7, 3.3] there exists an exact sequence of the form

0 → T (λ) → G(Y′ ⊗ sgn) → X → 0

where all composition factors of X are not p-restricted. Since T (λ) and G(Y′ ⊗ sgn) have good filtrations, it follows that X must have a good filtration. Therefore, X = 0 because each ∇(μ) has a p-restricted head by [12] again. _{□}

3.2 We now prove that the image under G of a twisted Specht module is a Weyl module.

Theorem . Let p > 3. Then Δ(λ) = G(S^{λ′} ⊗ sgn) for all λ ∈ Λ.

Proof. By [12], Δ(λ) has a p-restricted socle so in view of [7, 3.3] there exists a short exact sequence of the form

0 → Δ(λ) → G(S^{λ′} ⊗ sgn) → X → 0.

Therefore, it will suffice to prove that dim_{k} Δ(λ) = dim_{k} G(S^{λ′} ⊗ sgn) or equivalently that Δ(λ) and G(S^{λ′} ⊗ sgn) have the same composition factors with multiplicities. According to Theorems 2.2B(i), 3.1 and [6, (3.8)], we have

[G(S^{λ′} ⊗ sgn) : L(μ)] = dim_{k} Hom_{kΣ_{d}}(Y^{μ}, S^{λ′} ⊗ sgn)
= dim_{k} Hom_{kΣ_{d}}((S^{λ′})^{∗}, Y^{μ} ⊗ sgn)
= dim_{k} Hom_{S(n,d)}(Δ(λ′), G(Y^{μ} ⊗ sgn))
= dim_{k} Hom_{S(n,d)}(Δ(λ′), T (μ′))
= [T (μ′) : ∇(λ′)]
= [Δ(λ) : L(μ)].
□

3.3 For p > 2, Ext^{1}(Y^{λ}, Y^{μ}) = 0 for λ, μ ∈ Λ by [3, 4.6.1] or [8, 6.3]. The following theorem deals with extensions of Specht modules and extensions between Specht and Young modules. We note that part (a) was first observed in [3, 3.8.1].

Theorem . Let p > 3, μ ∈ Λ_{res}, and λ ∈ Λ.

(a) Ext_{kΣ_{d}}^{1}(S^{μ}, Y^{λ}) = 0;

(b) Ext_{kΣ_{d}}^{1}(S^{μ}, S^{λ}) ≅ Ext_{S(n,d)}^{1}(Δ(μ′), Δ(λ′));

(c) If μ does not strictly dominate λ then Ext_{kΣ_{d}}^{1}(S^{μ}, S^{λ}) = 0.

(d) In particular, Ext_{kΣ_{d}}^{1}(S^{μ}, S^{μ}) = 0.

Proof. According to [7, 2.4A(ii)], one can let M = Δ(μ′) to get

Ext_{kΣ_{d}}^{1}(S^{μ} ⊗ sgn, N) ≅ Ext_{kΣ_{d}}^{1}((S^{μ′})^{∗}, N) ≅ Ext_{kΣ_{d}}^{1}(Δ(μ′), G(N)).  (3.3.1)

For part (a) set N = Y^{λ} ⊗sgn and use Theorem 3.1. Part (b) follows by setting N = S^{λ} ⊗sgn and using Theorem 3.2. Note that if μ does not strictly dominate λ then λ′ does not strictly dominate μ′, and Ext_{S(n,d)}^{1}(Δ(μ′), Δ(λ′)) = 0 by [14, II 2.14 (3)] proving part (c), of which (d) is a special case. _{□}

4 Completely splittable modules

4.1 Definition. [15] A simple kΣ_{d}-module D is called completely splittable if the restriction D↓_{Σ_{μ}} is completely reducible for any Young subgroup Σ_{μ} of Σ_{d}.

4.2 We now recall two basic results on CS modules proved in [15]. For a partition λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{s} > 0) define

h(λ) := s and χ(λ) := λ_{1} − λ_{s} + s.

The invariant h(λ) is often referred to as the height of the partition. The first theorem provides necessary and sufficient conditions for a simple module to be CS.

Theorem (A). [15] Let λ ∈ Λ_{reg}. Then D^{λ} is completely splittable if and only if χ(λ) ≤ p.

For a removable node A of a partition λ ⊢ n we denote by λ_{A} the partition of n − 1 whose Young diagram is obtained from that of λ by removing A. The second result describes how CS representations decompose upon restriction to Σ_{d−1}.

Theorem (B). [15] Let λ be a p-regular partition of d with χ(λ) ≤ p. Then D^{λ}↓_{Σ_{d−1}} ≅ ⨁ D^{λ_{A}} where the sum is over all removable nodes A of λ such that χ(λ_{A}) ≤ p.

4.3 The following is an easy consequence of the branching rule given in Theorem 4.2B.

Proposition . Let m > d, μ be any p-regular partition of d, and λ be a p-regular partition of m satisfying h(λ) ≤ s and λ_{1} − λ_{s} + s ≤ p. Then D^{μ} appears as a composition factor of D^{λ}↓_{Σ_{d}} only if h(μ) ≤ s and μ_{1} − μ_{s} + s ≤ p.

Proof. If h(λ) = s then χ(λ) = λ_{1} − λ_{s} + s ≤ p, and by Theorem 4.2A, D^{λ} is CS. If h(λ) < s then λ_{s} = 0, and χ(λ) < λ_{1} − λ_{s} + s ≤ p. So by Theorem 4.2A again, D^{λ} is CS in this case, too. Now apply Theorem 4.2B. _{□}

4.4 A kΣ_{d}-module is called non-trivial if it is not isomorphic to the trivial module k. Our next result shows that non-trivial CS modules enjoy the remarkable property that they always give simple S(n, d)-modules upon application of the functor G.

Theorem . Let D_{λ} be a non-trivial completely splittable kΣ_{d}-module and D_{μ} be an arbitrary simple kΣ_{d}-module. Then

(a) G(D_{λ}) = L(λ).

(b) If n ≥ d then Ext_{kΣ_{d}}^{1}(D_{μ}, D_{λ}) ≅ Ext_{S(n,d)}^{1}(L(μ), L(λ)).

Proof. (a) According to Theorem 2.2B(i), we have [G(D_{λ}) : L(ν)] = dim_{k} Hom_{kΣ_{d}}(Y^{ν}, D_{λ}). If ν is p-restricted then Y^{ν} is a projective kΣ_{d}-module with simple head D_{ν}. So the only p-restricted composition factor of G(D_{λ}) is L(λ). Now assume that ν is not p-restricted. Pick γ ∈ Λ_{reg} so that D_{λ} = D^{γ}. Then Y^{ν} is a direct summand of M^{ν} and

Hom_{kΣ_{d}}(Y^{ν}, D^{γ}) ⊆ Hom_{kΣ_{d}}(M^{ν}, D^{γ}) ≅ Hom_{kΣ_{ν}}(k, D^{γ}↓_{kΣ_{ν}}).

We claim that the last space is zero. Indeed, Σ_{ν} ≅ Σ_{ν_{1}} × Σ_{ν_{2}} × …, with ν_{1} ≥ p because ν is not p-restricted. As D^{γ} is non-trivial, it follows that s := h(γ) ≥ 2. Now by Proposition 4.3, D^{γ}↓_{kΣ_{ν_{1}}} does not contain a trivial component. This completes the proof of part (a).

(b) By [7, 4.2(i)], we have Ext_{kΣ_{d}}^{1}(D_{μ}, D_{λ}) ≅ Ext_{S(n,d)}^{1}(L(μ), G(D_{λ})). But by part (a), G(D_{λ}) = L(λ), which completes the proof. _{□}

Martin and the first author conjectured that for any simple module D, Ext_{kΣ_{d}}^{1}(D, D) = 0 as long as p > 2. The next result shows this holds for CS representations. It is a special case of [16, 2.10 and Remark (iv) on page 2], which show that the conjecture is true for D = D^{λ} provided h(λ) ≤ p − 1. However, we give a proof as our methods here are very different.

Corollary . Let p ≥ 3 and D be any CS module over kΣ_{d}. Then Ext_{kΣ_{d}}^{1}(D, D) = 0.

Proof. If D_{λ} ≅ k̸ then Ext_{kΣ_{d}}^{1}(D_{λ}, D_{λ}) = Ext_{S(n,d)}^{1}(L(λ), L(λ)) = 0 by Theorem 4.4b and [14, II.2.14]. On the other hand, if D_{λ} ≅ k then Ext_{kΣ_{d}}^{1}(k, k) ≅ H^{1}(Σ_{d}, k) = 0 as p > 2. _{□}

5 Vanishing results

Theorem 4.4 motivates us to study images of CS modules under the higher derived functors of the functor G. A vanishing result depending on the height will be proved in Section 5.4. But first we need three lemmas on symmetric group cohomology.

5.1 First we calculate cohomology groups of Σ_{p} with coefficients in simple modules.

Lemma . The cohomology of Σ_{p} with coefficients in simple modules is given as follows.

(i) H^{j}(Σ_{p}, D^{λ}) = 0 unless λ is of the form (p − i, 1^{i}) for 0 ≤ i ≤ p − 2.

(ii) For 1 ≤ i ≤ p − 2,

H^{j}(Σ_{p}, D^{(p−i,1^{i})}) ≅ \begin{cases}k & if j is of the form 2m(p − 1) + i or \\ & 2m(p − 1) + (2p − i − 3) for some integer m ≥ 0; \\ 0 & otherwise. \end{cases}

(iii) For i = 0,

H^{j}(Σ_{p}, k) ≅ \begin{cases}k & if j is of the form 2m(p − 1) or 2m(p − 1) − 1 \\ & for some integer m ≥ 0; \\ 0 & otherwise. \end{cases}

Proof. Let P_{i} be the projective cover of D^{(p−i,1^{i})}, 0 ≤ i ≤ p − 2. The module P_{i} admits a Specht filtration with factors S^{(p−i,1^{i})}, S^{(p−i−1,1^{i+1})} (starting from the top). Since P_{i} is selfdual it also has a filtration with factors (S^{(p−i−1,1^{i+1})})^{∗}, (S^{(p−i,1^{i})})^{∗}. Therefore, a minimal projective resolution can be constructed as follows:

0 ← k ← P_{0} ← P_{1} ← ⋯ ← P_{p−2} ← P_{p−2} ← P_{p−3} ← ⋯ ← P_{0} ← …

Since the resolution is minimal and D^{(p−i,1^{i})} is a simple module, we have H^{j}(Σ_{p}, D^{(p−i,1^{i})}) ≅ Hom_{kΣ_{p}}(Q_{j}, D^{(p−i,1^{i})}), where Q_{j} is the jth term of the resolution. The lemma follows. _{□}

5.2 We now prove a vanishing result on cohomology of Σ_{p^{m}} with coefficients in CS modules, using Lemma 5.1 as an induction base. Note that all simple modules for kΣ_{p} are CS.

Lemma . Let m ≥ 1 and D^{λ} be a CS module of Σ_{p^{m}}.

(a) If D^{λ} ≅ k̸, h(λ) ≤ s and λ_{1} − λ_{s} + s ≤ p, then H^{j}(Σ_{p^{m}}, D^{λ}) = 0 for 0 ≤ j ≤ s − 2.

(b) If D^{λ} ≅ k then H^{j}(Σ_{p^{m}}, k) = 0 for 1 ≤ j ≤ 2(p − 2).

Proof. We apply induction on m. For m = 1 the result follows from Lemma 5.1. Let m > 1. The Sylow p-subgroup Syl_{p}(Σ_{p^{m}}) embeds into the wreath product A := Σ_{p^{m−1}} ≀ Σ_{p} < Σ_{p^{m}}. Hence H^{n}(Σ_{p^{m}}, D^{λ}) embeds into H^{n}(A, D^{λ}). Note that A ≅ B ⋊ Σ_{p} where B is the Young subgroup Σ_{((p^{m−1})^{p})} ≅ Σ_{p^{m−1}} × ⋯ × Σ_{p^{m−1}} < Σ_{p^{m}}.

Now we apply the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{i,j} ≅ H^{i}(Σ_{p}, H^{j}(B, D^{λ})) ⇒ H^{i+j}(A, D^{λ}).  (5.2.1)

By the Kunneth formula, we get

H^{j}(B, D^{λ}) ≅ ⨁ ⨁_{j_{1}+···+j_{p}=n^{(j_{1},...,jp)∈(ℤ^{+})^{p}}} H^{j_{1}}(Σ_{p^{m−1}}, D_{1}) ⊗ ⋯ ⊗ H^{j_{p}}(Σ_{p^{m−1}}, D_{p})

where the first sum is over all composition factors D_{1} ⊗ ⋯ ⊗ D_{p} of the restriction D^{λ}↓_{B}. Note that we have used the fact that the restriction is completely reducible. If D^{λ} ≅ k and 1 ≤ j ≤ 2(p − 2) then by inductive hypothesis the last expression is zero. So assume that D^{λ} is as given in part (a) and 0 ≤ j ≤ s − 2. By Proposition 4.3, none of the modules D_{q} in a composition factor D_{1} ⊗ ⋯ ⊗ D_{p} of D^{λ}↓_{B} is trivial. On the other hand, j_{1} + ⋯ + j_{p} = j ≤ s − 2 ≤ p − 3 implies that at least one of the indices j_{q} in the sum above must be 0. So for this q we have H^{0}(Σ_{p^{n−1}}, D_{q}) = 0. This shows that H^{j}(B, D^{λ}) = 0 under our assumptions.

Finally, we prove part (a). As H^{j}(B, D^{λ}) = 0 for 0 ≤ j ≤ s − 2, it follows that E_{2}^{i,j} = 0 for 0 ≤ i + j ≤ s − 2 in the spectral sequence (5.2.1). The differentials in this spectral sequence have bidegree (r, 1 − r), so H^{n}(A, D^{λ}) = 0 for 0 ≤ n ≤ s − 2. Part (b) can be proved by using a similar line of reasoning and the fact that in this case H^{j}(B, D^{λ}) = 0 for 1 ≤ j ≤ 2(p − 2) and E_{2}^{i,0} = H^{i}(Σ_{p}, H^{0}(G, k)) = H^{i}(Σ_{p}, k) = 0 for 1 ≤ i ≤ 2(p − 2). _{□} 5.3 Now we get a general vanishing result on cohomology of CS modules.

Lemma . Let D^{λ} be a CS module of Σ_{d}.

(a) If D^{λ} ≅ k̸, h(λ) ≤ s and λ_{1} − λ_{s} + s ≤ p then H^{j}(Σ_{d}, D^{λ}) = 0 for 0 ≤ j ≤ s − 2.

(b) If D^{λ} ≅ k then H^{j}(Σ_{d}, k) = 0 for 1 ≤ j ≤ 2(p − 2).

Proof. If d = p^{i_{1}} + p^{i_{2}} + … p^{i_{k}} + r for i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{q} > 0 and 0 ≤ r < p, then there is a Sylow p-subgroup of Σ_{d} contained in the Young subgroup G := Σ_{(p^{i_{1}},...,p^{iq})}. This implies that H^{j}(Σ_{d}, D^{λ}) embeds into H^{j}(G, D^{λ}). By the Kunneth formula, we get

H^{j}(G, D^{λ}) ≅ ⨁ ⨁_{j_{1}+···+j_{q}=j^{(j_{1},...,jq)∈(ℤ^{+})^{q}}} H^{j_{1}}(Σ_{p^{i_{1}}} , D^{μ(1)}) ⊗ ⋯ ⊗ H^{j_{q}}(Σ_{p^{iq}} , D^{μ(q)})

where the first sum is over all composition factors D^{μ(1)} ⊗ ⋯ ⊗ D^{μ(q)} of the restriction D^{λ}↓_{G} (we have used the fact that the restriction is completely reducible). Furthermore, by Proposition 4.3, each μ(l) satisfies satisfies h(μ(l)) ≤ s and μ(l)_{1}−μ(l)_{s}+s ≤ p. So, by Lemma 5.2, H^{j}(G, D^{λ}) = 0 for 0 ≤ j ≤ s − 2 if D^{λ} ≅ k̸, and H^{j}(G, k) = 0 for 1 ≤ j ≤ 2(p − 2). _{□}

5.4 Finally, we obtain an information on vanishing of the higher derived functors of G.

Theorem . Let D^{λ} be a CS module of Σ_{d}.

(a) If D^{λ} ≅ k̸, h(λ) ≤ s and λ_{1} − λ_{s} + s ≤ p then R^{j}G(D^{λ}) = 0 for 1 ≤ j ≤ s − 2.

(b) If D^{λ} ≅ k then R^{j}G(k) = 0 for 1 ≤ j ≤ 2(p − 2).

Proof. (a) Using the definition of G and the decomposition of V^{⊗d} as a kΣ_{d}-module, we have

R^{j}G(D^{λ}) ≅ Ext_{kΣ_{d}}^{j}(V^{⊗n}, D^{λ}) ≅ ⨁ Ext_{kΣ_{d}}^{j}(M^{μ}, D^{λ}) ≅ ⨁ Ext_{kΣ_{μ}}^{j}(k, D^{λ}↓_{Σ_{μ}})

where the sums are over all compositions μ of d. For 1 ≤ j ≤ s − 2 and a composition μ = (μ_{1}, …, μ_{a}) of d we prove that Ext_{kΣ_{μ}}^{j}(k, D^{λ}↓_{Σ_{μ}}). Let D^{λ(1)}⊗⋯⊗D^{λ(a)} be a composition factor of D^{λ}↓_{Σ_{μ}}. According to Proposition 4.3, every partition λ(i) satisfies h(λ(i)) ≤ s and λ(i)_{1}−λ(i)_{s}+s ≤ p. Now Ext_{kΣ_{μ}}^{j}(k, D^{λ(1)}⊗⋯⊗D^{λ(a)}) = 0 by Lemma 5.3a and the Kunneth

formula. Hence, Ext_{kΣ_{μ}}^{j}(k, D^{λ}↓_{Σ_{μ}}) = 0.

(b) As in (a), R^{j}G(k) ≅ ⊕ Ext_{kΣ_{μ}}^{j}(k, k) where the sum is over all compositions μ of d. But,

Ext_{kΣ_{μ}}^{j}(k, k) = 0 for 1 ≤ j ≤ 2(p − 2) by the Kunneth formula and Lemma 5.3b. _{□}

Remark . The sign representation is CS and non-trivial if p > 2. It is equal to some D^{ε} with h(ε) = p − 1 if d ≥ p − 1 ([11, 24.5(iii)]). As kΣ_{d} is semisimple for d < p, the theorem above implies

R^{j}G(sgn) = 0 for 1 ≤ j ≤ p − 3.  (5.4.1)

6 Applications

6.1 Stability of Extensions I. The algebra S(n, d) is quasi hereditary and thus has finite global dimension. On the other hand, kΣ_{d} has infinite global dimension. So there is no hope for the extension groups of these algebras to agree in all degrees. Nevertheless, often one can show that the extension groups for S(n, d) and kΣ_{d} coincide for a certain range of degrees.

Theorem . Let D_{λ} = D^{ν} be a non-trivial CS module and let M ∈ M(n, d).

(a) For 0 ≤ i ≤ h(ν) − 2 we have Ext_{S(n,d)}^{i}(M, L(λ)) ≅ Ext_{kΣ_{d}}^{i}(eM, D_{λ}).

(b) In particular for any μ ∈ Λ and 0 ≤ i ≤ h(ν) − 2 we have:

(i) Ext_{S(n,d)}^{i}(L(μ), L(λ)) ≅ \begin{cases}Ext_{kΣ_{d}}^{i}(D_{μ}, D_{λ}) & if μ ∈ Λ_{res}; \\ 0 & otherwise. \end{cases}
(ii) Ext_{S(n,d)}^{i}(∇(μ), L(λ)) ≅ Ext_{kΣ_{d}}^{i}(S^{μ}, D_{λ}).
(iii) Ext_{S(n,d)}^{i}(Δ(μ), L(λ)) ≅ Ext_{kΣ_{d}}^{i}((S^{μ})^{∗}, D_{λ}).
(iv) Ext_{S(n,d)}^{i}(T (μ), L(λ)) ≅ Ext_{kΣ_{d}}^{i}(Y^{μ′}, D^{λ′}).
(v) Ext_{kΣ_{d}}^{i}(Y^{μ}, D_{λ}) ≅ \begin{cases}k & if λ = μ and i = 0; \\ 0 & otherwise. \end{cases}

Proof. (a) By Theorem 4.4a, G(D_{λ}) = L(λ), and by Theorem 5.4, R^{j}G(D_{λ}) = 0 for 1 ≤ j ≤ h(ν) − 2. The result now follows from Proposition 2.3.

(b) follows from (a). Indeed, for part (ii) (resp. (iii)) set M = ∇(μ) (resp. M = Δ(μ)) and using the results stated in Section 2.1. For (i) set M = L(μ) and use the fact that eL(λ) = 0 for λ ∉ Λ_{res}. To prove part (iv), set M = T (μ) and use (2.1.1). For part (v), set M = P (μ) where P (μ) is the projective cover of L(μ). Observe that Ext_{S(n,d)}^{i}(P (μ), L(λ)) is zero if i > 0 or λ ≠ μ and isomorphic to k if i = 0 and λ = μ. _{□}

Remark. (a) Note that even in the case i = 0 the theorem above is saying something new. Part (v) implies that a non-trivial CS module D_{λ} appears in the head of a Young module Y^{μ} if and only if λ = μ. If μ is not p-restricted, the head of Y^{μ} need not be simple.

(b) Parts (iv) and (v) of the theorem can be used to prove some vanishing results for Ext_{S(n,d)}^{i}(T (μ), L(λ)). We leave the formulation of the corresponding results to the reader.

Corollary . Let M ∈ M(d, d) and δ = L(1^{d}) be the determinant representation for S(d, d).

(a) For 0 ≤ i ≤ p − 3, H^{i}(Σ_{d}, eM ⊗ sgn) ≅ Ext_{S(d,d)}^{i}(δ, M).

(b) In particular for any μ ∈ Λ and 0 ≤ i ≤ p − 3 we have: (i) H^{i}(Σ_{d}, D^{μ′}) ≅ Ext_{S(d,d)}^{i}(δ, L(μ)) for μ ∈ Λ_{res}.

(ii) H^{i}(Σ_{d}, S^{μ′}) ≅ Ext_{S(d,d)}^{i}(δ, V (μ)).

(iii) H^{i}(Σ_{d}, (S^{μ′})^{∗}) ≅ \begin{cases}k & if μ = (1^{d}) and i = 0; \\ 0 & otherwise. \end{cases}

(iv) H^{i}(Σ_{d}, Y^{μ} ⊗ sgn) ≅ \begin{cases}k & if μ = (1^{d}) and i = 0; \\ 0 & otherwise. \end{cases}

Proof. (a) Let D_{λ} = sgn. Then by Theorem 6.1a and (5.4.1) we have for 0 ≤ i ≤ p − 3,

H^{i}(Σ_{d}, eM ⊗ sgn) ≅ Ext_{kΣ_{d}}^{i}((eM)^{∗}, sgn) ≅ Ext_{kΣ_{d}}^{i}(eM^{τ} , sgn)
≅ Ext_{S(n,d)}^{i}(M^{τ} , δ) ≅ Ext_{S(n,d)}^{i}(δ, M).

(b) By (2.1.1), D^{μ′} = D_{μ} ⊗ sgn, and (S^{μ′})^{∗} = S^{μ} ⊗ sgn so parts (i)-(iii) follow. Part (iv) follows from Theorem 6.1b(v). _{□}

6.2 BKM conjecture. We now note that a conjecture made in [2, Conj. 6.2] (stated in the corollary below) is a rather special case of Corollary 6.1b(iii).

Corollary . For a fixed i > 0 there exists a constant C depending only on i such that for p > C we have H^{i}(Σ_{d}, (S^{λ})^{∗}) = 0 for all d and λ ∈ Λ. In fact, we can take C = i + 2.

Proof. Let i > 0. Set C = i+2. If p > i+2, or in other words i < p−2 then H^{i}(Σ_{d}, (S^{λ})^{∗}) = 0 by Corollary 6.1b(iii) for all d and λ. _{□} 6.3 Stability of Extensions II. We can improve the range of the stability of extensions as long as one of the modules involved is Δ(d) ≅ S^{d}(V)^{τ}, the contravariant dual of the dth symmetric power of the natural module V.

Theorem . Let M ∈ M(n, d) with n ≥ d.

(a) For 0 ≤ i ≤ 2(p − 2) we have Ext_{S(n,d)}^{i}(M, Δ(d)) ≅ Ext_{kΣ_{d}}^{i}(eM, k).

(b) In particular for any μ ∈ Λ and 0 ≤ i ≤ 2(p − 2) we have:

(i) Ext_{S(n,d)}^{i}(L(μ), Δ(d)) ≅ \begin{cases}Ext_{kΣ_{d}}^{i}(D_{μ}, k) & if μ ∈ Λ_{res}; \\ 0 & otherwise. \end{cases}

(ii) Ext_{S(n,d)}^{i}(∇(μ), Δ(d)) ≅ Ext_{kΣ_{d}}^{i}(S^{μ}, k).

(iii) Ext_{S(n,d)}^{i}(Δ(μ), Δ(d)) ≅ Ext_{kΣ_{d}}^{i}((S^{μ})^{∗}, k).

(iv) Ext_{S(n,d)}^{i}(T (μ), Δ(d)) ≅ Ext_{kΣ_{d}}^{i}(Y^{μ′}, sgn).

(v) Ext_{kΣ_{d}}^{i}(Y^{μ}, k) ≅ \begin{cases}k & if i = 0 and [Δ(d) : L(μ)] ≠ 0; \\ 0 & otherwise. \end{cases}

Proof. (a) We have G(k) = Δ(d) by [7, 5.5] and R^{j}G(k) = 0 for 1 ≤ j ≤ 2(p − 2) by Theorem 5.4. Now use Proposition 2.3.

(b) Parts (i)-(v) follows from (a) by setting M = ∇(μ) (resp. Δ(μ), T (μ), P (μ)) as in the proof of Theorem 6.1b. For (v) we also use the fact that Δ(d) is multiplicity free. _{□}

Corollary . Let M ∈ M(n, d) with n ≥ d.

(a) For 0 ≤ i ≤ 2(p − 2) we have H^{i}(Σ_{d}, eM) ≅ Ext_{S(n,d)}^{i}(∇(d), M).

(b) In particular for any μ ∈ Λ and 0 ≤ i ≤ 2(p − 2) we have: (i) H^{i}(Σ_{d}, D_{μ}) ≅ Ext_{S(n,d)}^{i}(∇(d), L(μ)) for μ ∈ Λ_{res}.

(ii) H^{i}(Σ_{d}, (S^{μ})^{∗}) ≅ Ext_{S(n,d)}^{i}(∇(d), Δ(μ)).

(iii) H^{i}(Σ_{d}, S^{μ}) ≅ Ext_{S(n,d)}^{i}(∇(d), ∇(μ)).

(iv) H^{i}(Σ_{d}, Y^{μ′} ⊗ sgn) ≅ Ext_{S(n,d)}^{i}(∇(d), T (μ)).

(v) H^{i}(Σ_{d}, Y^{μ}) ≅ \begin{cases}k & if i = 0 and [∇(d) : L(μ)] ≠ 0; \\ 0 & otherwise. \end{cases}

Proof. (a) By Theorem 6.3a, one has for 0 ≤ i ≤ 2(p − 2)

H^{i}(Σ_{d}, eM) ≅ Ext_{kΣ_{d}}^{i}(k, eM) ≅ Ext_{kΣ_{d}}^{i}((eM)^{∗}, k) ≅ Ext_{kΣ_{d}}^{i}(eM^{τ} , k)
≅ Ext_{S(n,d)}^{i}(M^{τ} , Δ(d)) ≅ Ext_{S(n,d)}^{i}(∇(d), M).

(b) For parts (i)-(iv) make the appropriate substitutions for M, (v) follows from Theorem 6.3b(v) above by dualizing. _{□}

6.4 Stability of Extensions III. A conjecture of Burichenko, Kleshchev and Martin verified in Section 6.2 will now be used to obtain stability results involving Specht modules.

Theorem . Let p > 3, 0 ≤ i ≤ p − 3, M ∈ M(n, d), and λ, μ ∈ Λ. Then

(a) Ext_{S(n,d)}^{i}(M, Δ(λ)) ≅ Ext_{kΣ_{d}}^{i}(eM, S^{λ′} ⊗ sgn).

(b) In particular, (i) Ext_{S(n,d)}^{i}(L(μ), Δ(λ)) ≅ Ext_{kΣ_{d}}^{i}(D^{μ′}, S^{λ′}).

(ii) Ext_{S(n,d)}^{i}(Δ(μ), Δ(λ)) ≅ Ext_{kΣ_{d}}^{i}(S^{μ′}, S^{λ′}).

(iii) Ext_{S(n,d)}^{i}(∇(μ), Δ(λ)) ≅ Ext_{kΣ_{d}}^{i}(S^{μ}, S^{λ′} ⊗ sgn).

(iv) Ext_{kΣ_{d}}^{i}(Y^{μ}, S^{λ}) = 0.

(v) Ext_{kΣ_{d}}^{i}(S^{λ}, Y^{μ}) = 0.

Proof. (a) By Theorem 3.2, G(S^{λ′} ⊗ sgn) = Δ(λ). Moreover, by Frobenius reciprocity,

R^{j}G(S^{λ′} ⊗ sgn) = Ext_{kΣ_{d}}^{j}(V^{⊗d}, S^{λ′} ⊗ sgn) = ⊕Ext_{kΣ_{μ}}^{j}(k, (S^{λ})^{∗}↓_{Σ_{μ}})

where the last sum is over all compositions μ of d. By [13], (S^{λ})^{∗} ↓_{Σ_{μ}} admits a filtration with sections of the form (S^{λ(1)})^{∗} ⊗ (S^{λ(2)})^{∗} ⊗ ⋯ ⊗ (S^{λ(a)})^{∗}. By the Kunneth formula and Corollary 6.1b(iii), it follows that Ext_{kΣ_{μ}}^{j}(k, (S^{λ(1)})^{∗} ⊗ (S^{λ(2)})^{∗} ⊗ ⋯ ⊗ (S^{λ(a)})^{∗}) = 0 for 1 ≤ j ≤ p − 3. Consequently, by using induction on the length of the filtration on (S^{λ})^{∗}, we have R^{j}G(S^{λ′} ⊗ sgn) = 0 for 1 ≤ j ≤ p − 3. Part (a) now holds by applying Theorem 2.3.

(b) For (i)-(v) substitute M = L(μ), Δ(μ), ∇(μ), T (μ) and P (μ), respectively. _{□}

6.5 Stability of Extensions IV. We now obtain stability results on extensions involving Young modules.

Theorem . Let p > 3, 0 ≤ i ≤ p − 3, M ∈ M(n, d), and λ, μ ∈ Λ.

(a) Ext_{S(n,d)}^{i}(M, T (λ)) ≅ Ext_{kΣ_{d}}^{i}(eM, Y^{λ′} ⊗ sgn).

(b) In particular, (i) Ext_{S(n,d)}^{i}(L(μ), T (λ)) ≅ Ext_{kΣ_{d}}^{i}(D^{μ′}, Y^{λ′}).

(ii) Ext_{S(n,d)}^{i}(∇(μ), T (λ)) ≅ Ext_{kΣ_{d}}^{i}(S^{μ} ⊗ sgn, Y^{λ′}) = 0.

(iii) Ext_{kΣ_{d}}^{i}(Y^{μ}, Y^{λ}) = 0.

(iv) Ext_{kΣ_{d}}^{i}(Y^{μ} ⊗ sgn, Y^{λ}) = 0.

Proof. (a) By Theorem 3.1, T (λ) = G(Y^{λ′} ⊗ sgn). Furthermore, if μ is a composition of d then the restriction of Y^{λ′} ⊗ sgn to Σ_{μ} admits a filtration with sections of the form (Y^{λ(1)} ⊗ sgn)⊗⋯⊗(Y^{λ(a)}⊗sgn). Now use the same argument as in Theorem 6.4 along with Corollary 6.1b(iv) to show that R^{j}G(Y^{λ′} ⊗ sgn) = 0 for 1 ≤ j ≤ p − 3. Finally apply Theorem 2.3.

(b) Substitute M = L(μ), Δ(μ), T (μ) and P (μ), respectively. _{□}

We remark that the results given in Theorem 6.4b(iv)-(v) and Theorem 6.5b(iii) can be viewed as natural generalizations involving higher extension groups of the Ext^{1}-results given in the work of Cline, Parshall and Scott [3, §3.8, §4.6].

6.6 Alternating group cohomology. Let A_{d} be the alternating group on d letters. Nakaoka computed the structure of the cohomology of the symmetric group H^{•}(Σ_{d}, k) [18]. For the prime p = 2, there is a method to compute H^{•}(A_{d}, k) by using the calculation of H^{•}(Σ_{d}, k) [1]. The first proposition shows that one can compute the cohomology of the alternating group by knowing the cohomology of the symmetric group with coefficients in the trivial and sign representations.

Proposition . Let p ≥ 3. Then H^{•}(A_{d}, k) ≅ H^{•}(Σ_{d}, k) ⊕ H^{•}(Σ_{d}, sgn) as k-vector spaces.

Proof. Note that the module induced from the trivial kA_{d}-module to kΣ_{d} is isomorphic to k ⊕ sgn. The result now follows from the Eckmann-Shapiro lemma. _{□}

Corollary . Let p ≥ 3. Then H^{i}(A_{d}, k) = 0 for 1 ≤ i ≤ p − 3.

Proof. This follows from the proposition above, Lemma 5.3 and Remark 5.4. _{□}

Remark . Consider the spectral sequence (2.2.1) with M = S^{d}(V) and N = sgn:

E_{2}^{i,j} = Ext_{S(n,d)}^{i}(S^{d}(V), R^{j}G(sgn)) ⇒ H^{i+j}(Σ_{d}, sgn).

This indicates that there may be some hope to compute the cohomology of the alternating group by determining R^{•}G(sgn) = Ext_{kΣ_{d}}^{•}(V^{⊗d}, sgn) as a GL_{n}(k)-module.

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Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

E-mail address: klesh@math.uoregon.edu

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA

E-mail address: nakano@sunfs.math.usu.edu