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 (i.e. when k = \overline{𝔽_{p}}) the relationship between their representation theories is not as direct. However, James [Ja3] showed that the decomposition matrix of the symmetric group is a submatrix of the decomposition matrix of the general linear group, and Erdmann [E] 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 covariant exact functor from M(n, d) to modules for the group algebra kΣ_{d} ([Gr]). This functor is constructed by using V^{⊗d} (see Section 2.2). Consequently, it is no surprise that some of the aforementioned results can be proved by using this approach. In the same spirit, in order to relate the cohomology theories of GL_{n}(k) and Σ_{d}, Doty, Erdmann and the second author [DEN] employed the actions of GL_{n}(k) and kΣ_{d} on V^{⊗d}. The projectivity of V^{⊗d} in M(n, d) is used to construct a spectral sequence starting with extensions for GL_{n}(k) and converging to extensions for kΣ_{d}. With this spectral sequence, a precise relationship between the cohomology for both groups can be obtained by computing the higher right derived functors R^{•}G_{Hom}(−) ≅ Ext_{kΣ_{d}}^{•}(V^{⊗d}, −), where G_{Hom} is a right adjoint functor to F (see Section 2.2). These derived functors are used extensively throughout the paper.

We wish to address the following fundamental 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 [Do1, 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.

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

Research of the first author was supported in part by NSF grant DMS-9600124

Research of the second author was supported in part by NSF grant DMS-9800960

1.2 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}.

A simple GL_{n}(k)-module (resp. kΣ_{d}-module) is called completely splittable if its restriction to any Levi (resp. Young) subgroup is semisimple. For general linear (resp. symmetric) groups these modules were introduced in [MP, BKS] (resp. [K]). The class of completely splittable representations is very interesting and has appeared (in different forms) in various contexts, see [Wen, Mat, Ry, K, BaK, BKS, MP]. The completely splittable representations can be classified, see [K, BKS] and Section 3.2. It follows from this classification that

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

is precisely the set of completely splittable kΣ_{d}-modules.

1.3 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 [DEN] and state the elementary properties of the higher right derived functors R^{•}G_{Hom}(−). Section 3 begins with stating the fundamental results on completely splittable modules. With this information we look at the image of G := G_{Hom} on a non-trivial completely splittable module where G is a right adjoint functor to F. Our result in Theorem 3.4(a) says that

G(D_{λ}) = L(λ). (1.3.1)

if D_{λ} is completely splittable 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 [DEN, 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 (1.3.1) is quite remarkable. Using this result we show in Theorem 3.4(b) that if λ and μ are p-restricted, L(λ) is completely splittable and D_{λ} ≅ k̸ then

Ext_{GL_{n}(k)}^{1}(L(μ), L(λ)) ≅ Ext_{kΣ_{d}}^{1}(D_{μ}, D_{λ}). (1.3.2)

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

Section 4 contains vanishing results about the cohomology of symmetric groups with coefficients in completely splittable modules. This information is later used to prove the vanishing of some R^{•}G_{Hom}(D_{λ}). The results of Section 4 allow us to generalize the isomorphism (1.3.2) 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 5.1 and 5.2, which give other examples where (1.1.1) holds. For example, Corollary 5.1a 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 [CPS, (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 heavily 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 5.2 (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 5.1 can also be used to prove a conjecture on the cohomology of dual Specht modules made in [BKM]. In fact, we get an even stronger result that for any Specht module S^{λ} over kΣ_{d} and 1 ≤ i ≤ p − 3,

H^{i}(Σ_{d}, (S^{λ})^{∗}) = 0.

This result can be interpreted as an isomorphism in (1.1.1) with M = ∧^{d}(V) and N being any costandard module. Finally, in Section 5.4 we obtain some results on cohomology of the alternating groups.

1.4 Acknowledgements This work was initiated while the second author was visiting the University of Oregon during the spring of 1998. The second author would like to acknowledge the Department of Mathematics at the University of Oregon for their hospitality and support. This work was completed during the conference on Algebraic Representation Theory held at Aarhus University in August 1998. Both authors thank Henning H. Andersen and Jens C. Jantzen for organizing a productive meeting.

2 Comparing GL_{n}(k) and kΣ_{d}

2.1 Correspondences under the Schur functor: The basic references for this section are [Gr], [Ja2] (see also [Mar]). Let k be an algebraically closed field of characteristic p > 0. Moreover, let G = GL_{n}(k) be the general linear group over k and kΣ_{d} be the group algebra of the symmetric group on d letters Σ_{d}. 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.

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 [Gr, (6.1)]. Then eS(n, d)e ≅ kΣ_{d}. The Schur functor F is the covariant exact functor from Mod(S(n, d)) to Mod(kΣ_{d}) defined 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. For λ ∈ Λ, set ∇(λ) = ind_{B}^{G}λ where B is the lower triangular Borel subgroup of G. Also, let Δ(λ) be the Weyl module for G. 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 M ∈ mod(S(n, d)).

Set D_{λ} = eL(λ). As mentioned in Section 1.2, D_{λ} = (0) unless λ ∈ Λ is p-restricted, and {D_{λ} | λ ∈ Λ is p-restricted} is a complete set of (non-isomorphic) simple kΣ_{d}-modules. Thus we label the simple kΣ_{d}-modules by the p-restricted partitions λ ∈ Λ. Recall that a partition (λ_{1}, λ_{2}, …) is called p-restricted if λ_{i} − λ_{i+1} ≤ p − 1 for all i. We denote the set of the p-restricted partitions of d 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 [Ja2], 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 [Gr, § 6]:

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

The Specht (resp. Young) module corresponding to a partition λ ∈ Λ over kΣ_{d} is denoted by S^{λ} (resp. Y^{λ}). One has following correspondences between S(n, d)-modules and kΣ_{d}- modules under F (see [Gr, § 6] and [Do2, 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 both a Hom functor and a tensor functor:

F(M) ≅ eM ≅ Hom_{A}(Ae, M) ≅ eA ⊗_{A} M.

By using these identifications, F admits two adjoint functors, a right adjoint G_{Hom} and a left adjoint G_{⊗} defined by

G_{Hom}(N) = Hom_{eAe}(eA, N) and G_{⊗}(N) = Ae ⊗_{eAe} N for N ∈ Mod(eAe).

Furthermore, G_{Hom} and G_{⊗} are both left inverses to F. Since Ae ≅ V^{⊗d}, and G_{Hom} (resp. G_{⊗}) takes injective (resp. projective) eAe-modules to injective (resp. projective) A-modules, one can construct two spectral sequences:

Theorem. [DEN, Thm. 2.2] Let M ∈ Mod(S(n, d)), N ∈ Mod(kΣ_{d}) with n ≥ d. There exists two first-quadrant Grothendieck spectral sequences, with E_{2}-pages given by

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

2.3 Properties of R^{•}G_{Hom}: One of the main goals of this paper is compare Ext groups for S(n, d) and kΣ_{d}. The spectral sequence (2.2.1) will serve as a tool for this purpose. Set G = G_{Hom} for the remainder of this paper. 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 provide information about the composition factors of the higher right derived functors of G.

Theorem. [DEN] Let N ∈ mod(kΣ_{d}), and μ ∈ Λ.

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

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

3 Completely splittable representations

3.1 Definition. [K] 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}.

3.2 We now recall two basic results proved in [K] on completely splittable modules. 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 completely splittable.

Theorem (A). [K] 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 demonstrates how completely splittable representations decompose upon restriction to Σ_{d−1}.

Theorem (B). [K] 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.

3.3 The branching rule given in Theorem 3.2B, can be used to give conditions on more general restrictions from Σ_{m} to the naturally embedded Σ_{d} for m ≥ d.

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 3.2A, D^{λ} is completely splittable. If h(λ) < s then λ_{s} = 0, and χ(λ) < λ_{1} − λ_{s} + s ≤ p. So by Theorem 3.2A again, D^{λ} is completely splittable in this case, too. Now apply Theorem 3.2B. _{□} 3.4 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 completely splittable 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 2.3a, we have [G(D_{λ}) : L(ν)] = dim_{k} Hom_{kΣ_{d}}(Y^{ν}, D_{λ}). It is known that head Y^{ν} = D_{ν} if μ is p-restricted. 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}. Let M^{ν} = k↑_{Σ_{ν}}^{Σ_{d}} be the permutation module for ν. 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 3.3, D^{γ}↓_{kΣ_{ν_{1}}} does not contain a trivial component. This completes the proof of part (a).

(b) By [DEN, 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 completely splittable representations. It is a special case of [KS, 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 > 2 and D be any completely splittable module over kΣ_{d}. Then

Ext_{kΣ_{d}}^{1}(D, D) = 0.

Proof. First suppose that D_{λ} is completely splittable module which is non-trivial. Then by Theorem 3.4b,

Ext_{kΣ_{d}}^{1}(D_{λ}, D_{λ}) = Ext_{S(n,d)}^{1}(L(λ), L(λ)).

The algebra S(n, d) is quasi-hereditary and therefore Ext_{S(n,d)}^{1}(L(λ), L(λ)) = 0. Note that this also follows by [Jan, II 2.12], since Ext_{GL_{n}(k)}^{1}(−, −) = Ext_{S(n,d)}^{1}(−, −). On the other hand, if D_{λ} ≅ k, then by Ext_{kΣ_{d}}^{1}(k, k) ≅ H^{1}(Σ_{d}, k), which is clearly 0 as p > 2. _{□}

4 Vanishing results

Theorem 3.4 motivates us to study images of completely splittable modules under the higher derived functors of the functor G. A vanishing result depending on the height will be proved in Section 4.4. But first we need three lemmas on symmetric group cohomology.

4.1 We begin with a calculation of some cohomology groups over Σ_{p} with coefficients in simple modules. Note that all simple modules for kΣ_{p} are completely splittable.

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 self dual 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,

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. _{□}

4.2 We now prove a vanishing result on cohomology of Σ_{p^{m}} with coefficients in completely splittable modules, using Lemma 4.1 as an induction base.

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

(a) If D^{λ} is non-trivial with 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 4.1. 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}). (4.2.1)

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 3.3, 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 (4.2.1). 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). _{□}

4.3 In this subsection a general vanishing result for the cohomology of Σ_{d} with coefficients in a completely splittable module is given.

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

(a) If D^{λ} is non-trivial with 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}. Note that we have used the fact that the restriction is completely reducible. Furthermore, by Proposition 3.3, each μ(l) satisfies satisfies h(μ(l)) ≤ s and μ(l)_{1} − μ(l)_{s} + s ≤ p. Hence, by Lemma 4.2, H^{j}(G, D^{λ}) = 0 for 0 ≤ j ≤ s − 2 when D^{λ} is not trivial and H^{j}(G, k) = 0 for 1 ≤ j ≤ 2(p − 2). _{□}

4.4 The following theorem gives an information about vanishing of the higher derived functors of the functor G.

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

(a) If D^{λ} is non-trivial with 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) By 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 last two sums are over all compositions μ of d. For 1 ≤ j ≤ s − 2 and a composition μ = (μ_{1}, …, μ_{a}) of d consider 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 3.3, every partition λ(i) satisfies h(λ(i)) ≤ s and λ(i)_{1} − λ(i)_{s} + s ≤ p. Now by Lemma 4.3a and the Kunneth formula Ext_{kΣ_{μ}}^{j}(k, D^{λ(1)} ⊗ ⋯ ⊗ D^{λ(a)}) = 0. 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 4.3b. _{□}

Remark. The sign representation is of course completely splittable and non-trivial if p > 2. Moreover, it is known to be equal to D^{ε} with h(ε) = p − 1 if n ≥ p − 1 (this follows e.g. from [Ja1]). On the other hand, kΣ_{n} is semisimple if n < p, so the theorem above implies

R^{j}G(sgn) = 0 for 1 ≤ j ≤ p − 3. (4.4.1)

5 Applications

5.1 Stability of Extensions I: The algebra S(n, d) is a quasi hereditary algebra and thus has finite global dimension. On the other hand, kΣ_{d} is a finite-dimensional Hopf algebra (thus quasi-Frobenius), so kΣ_{d} has infinite global dimension. Consequently, there is no hope for the extension groups of these algebras to agree in all degrees. Nevertheless, under suitable assumptions one can actually 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 completely splittable module and let N ∈ M(n, d).

(a) For 0 ≤ i ≤ h(ν) − 2 we have Ext_{S(n,d)}^{i}(N, L(λ)) ≅ Ext_{kΣ_{d}}^{i}(eN, 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) Consider the spectral sequence given in (2.2.1),

E_{2}^{i,j} = Ext_{S(n,d)}^{i}(N, R^{j}G(D_{λ})) ⇒ Ext_{kΣ_{d}}^{i+j}(eN, D_{λ}) (5.1.1)

According to Theorem 4.4a, for 1 ≤ j ≤ h(ν) − 2, we have

R^{j}G(D_{λ}) = R^{j}G(D^{ν}) = (0).

Therefore, E_{2}^{i,j} = 0 for j > 0 and 1 ≤ i + j ≤ h(ν) − 2 and

E_{2}^{i,0} ≅ Ext_{S(n,d)}^{i}(N, G(D_{λ})) ≅ Ext_{S(n,d)}^{i}(N, L(λ)) (5.1.2)

by Theorem 3.4a. The spectral sequence has differentials d_{r} with bidegree (r, 1 − r). Therefore, E_{2}^{i,0} ≅ Ext_{kΣ_{d}}^{i}(eN, D_{λ}) for 0 ≤ i ≤ h(ν) − 2.

(b) Part (ii) (resp. (iii)) follows from (a) by simply setting N = ∇(μ) (resp. N = Δ(μ)) and using the results stated in Section 2.1. Part (i) also uses (a) with N = L(μ) and the fact that eL(λ) = 0 for λ ∉ Λ_{res}. To prove part (iv), set N = T (μ) and use the isomorphism given in (2.1.1). For part (v), set N = 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 for i > 0 or λ ≠ μ and isomorphic to k for i = 0 λ = μ. Now apply part (a). _{□}

Remark. (a) Note that even in the case i = 0 the theorem above is saying something new. Theorem 5.1b(v) implies that a non-trivial completely splittable 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.

The theorem above can be used to show how the cohomology groups for Σ_{d} are related to cohomology theory for GL_{d}(k).

Corollary. Let N ∈ mod(S(d, d)) and δ be the determinant representation for S(d, d). (a) For 0 ≤ i ≤ p − 3, H^{i}(Σ_{d}, eN ⊗ sgn) ≅ Ext_{S(d,d)}^{i}(δ, N).

(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 5.1a and (4.4.1) we have for 0 ≤ i ≤ p − 3,

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

COHOMOLOGY OF GENERAL LINEAR GROUPS AND SYMMETRIC GROUPS 11

(b) Note that D^{μ′} = D_{μ} ⊗ sgn, and (S^{μ′})^{∗} = S^{μ} ⊗ sgn so parts (i)-(iii) follow. Part (iv) follows from Theorem 5.1b(v). _{□} 5.2 Stability of Extensions II: The next two results indicate that one can improve the range of the stability of extension groups as long as one considers extensions of Δ(d), which is the same as the contravariant dual S^{d}(V)^{τ} of the dth symmetric power of the natural module V.

Theorem. Let N ∈ mod(S(n, d)) with n ≥ d.

(a) For 0 ≤ i ≤ 2(p − 2) we have Ext_{S(n,d)}^{i}(N, Δ(d)) ≅ Ext_{kΣ_{d}}^{i}(eN, 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) Once again consider the spectral sequence given in (2.2.1),

E_{2}^{i,j} = Ext_{S(n,d)}^{i}(N, R^{j}G(k)) ⇒ Ext_{kΣ_{d}}^{i+j}(eN, k) (5.2.1)

Theorem 4.4b indicates that for 1 ≤ j ≤ 2(p − 2), one has R^{j}G(k) = 0. Hence, E_{2}^{i,j} = 0 for j > 0 and 1 ≤ i + j ≤ 2(p − 2) and

E_{2}^{i,0} ≅ Ext_{S(n,d)}^{i}(N, G(k)) ≅ Ext_{S(n,d)}^{i}(N, Δ(d)) (5.2.2)

by [DEN, Thm. 5.5]. The spectral sequence has differentials d_{r} with bidegree (r, 1−r), thus E_{2}^{i,0} ≅ Ext_{kΣ_{d}}^{i}(eN, k) for 0 ≤ i ≤ 2(p − 2).

(b) Parts (i)-(v) follows from (a) by simply setting N = ∇(μ) (resp. Δ(μ), T (μ), P (μ)) in the same way as in the proof of Theorem 5.1b. For (v) we also use the fact that Δ(d) is multiplicity free. _{□}

Corollary. Let N ∈ mod(S(n, d)) with n ≥ d.

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

(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}(S^{d}(V), L(μ)) for μ ∈ Λ_{res}. (ii) H^{i}(Σ_{d}, (S^{μ})^{∗}) ≅ Ext_{S(n,d)}^{i}(S^{d}(V), Δ(μ)). (iii) H^{i}(Σ_{d}, S^{μ}) ≅ Ext_{S(n,d)}^{i}(S^{d}(V), ∇(μ)). (iv) H^{i}(Σ_{d}, Y^{μ′} ⊗ sgn) ≅ Ext_{S(n,d)}^{i}(S^{d}(V), T (μ)).

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

Proof. (a) By Theorem 5.2a, one has for 0 ≤ i ≤ 2(p − 2)

H^{i}(Σ_{d}, eN) ≅ Ext_{kΣ_{d}}^{i}(k, eN)
≅ Ext_{kΣ_{d}}^{i}((eN)^{∗}, k)
≅ Ext_{kΣ_{d}}^{i}(eN^{τ}, k)
≅ Ext_{S(n,d)}^{i}(N^{τ}, V (d))
≅ Ext_{S(n,d)}^{i}(S^{d}(V), N).

(b) For parts (i)-(v) one simply makes the appropriate substitutions for N. Note for part (v) one sets N = I(μ), the injective hull of L(μ), and eI(μ) = Y^{μ}. Alternatively, (v) follows from Theorem 5.2b(v) above by dualizing. _{□}

5.3 B-K-M conjecture: In this section we note that the conjecture made by Burichenko, Kleshchev and Martin in 1996 [BKM, Conjecture 6.2] (stated in the corollary below) is a rather special case of Corollary 5.1b(iii).

Corollary. For a fixed i > 0 there exists some N such that for p > N, H^{i}(Σ_{d}, (S^{λ})^{∗}) = 0 for all d and λ ∈ Λ.

Proof. Let i > 0. Set N = i+2. If p > i+2, or in other words i < p−2 then H^{i}(Σ_{d}, (S^{λ})^{∗}) = 0 by Corollary 5.1b(ii) for all d and λ. _{□}

5.4 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) [Nak]. For the prime p = 2, there is a method to compute H^{•}(A_{d}, k) by using the calculation of H^{•}(Σ_{d}, k) [AM]. 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 4.3 and Remark 4.4. _{□}

Consider the spectral sequence (2.2.1) with M = S^{d}(V) and N = sgn. Then the spectral sequence takes on the form

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_{S(n,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