COHOMOLOGY AND GENERIC COHOMOLOGY OF SPECHT MODULES FOR THE SYMMETRIC GROUP

DAVID J. HEMMER

Abstract. Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_{d}(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_{d}(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^{i}(Σ_{d}, S^{λ}). We recover work of James in the case i = 0. Then we prove two stability theorems, one of which is a “generic cohomology” result for Specht modules equating cohomology of S^{pλ} with S^{p^{2}λ}. This is the first theorem we know relating Specht modules S^{λ} and S^{pλ}. The second result equates cohomology of S^{λ} with S^{λ+p^{a}μ} for large a.

1. Introduction

1.1. For an algebraic group G and a G module M, the Frobenius morphism on G lets one define a new G module M^{(1)}, the Frobenius twist of M. If G = GL_{n}(k) and L(λ) is an irreducible module then L(λ)^{(1)} ≅ L(pλ). Thus theorems about G modules which involve multiplying partitions by p arise quite naturally and can be explained in terms of the Frobenius map.

In contrast, suppose S^{λ} is a Specht module for the symmetric group Σ_{d}, where λ ⊢ d. Then pλ is a partition of pd, so S^{pλ} is a Specht module for an entirely different group, Σ_{pd}, and there is no evident relation between the two modules. Indeed we know of no theorems involving both S^{λ} and S^{pλ}. This paper will prove the first such theorem, although an explanation involving only the symmetric group eludes us!

1.2. Let k be an algebraically closed field of characteristic p ≥ 3. For a partition λ ⊢ d let S^{λ} denote the corresponding Specht module for the symmetric group Σ_{d}, and let S_{λ} be its dual. In characteristic two every Specht module is also a dual Specht module, and the problem of calculating cohomology is quite different and seems more difficult. Proposition 3.1 below, which equates symmetric group cohomology with that of the general linear group, only holds through degree 2p − 4, so does not apply even to H^{1}(Σ_{d}, S^{λ}) when p = 2. Thus we will restrict to the odd characteristic case for this paper.

Much is known about the low degree cohomology groups H^{i}(Σ_{d}, S_{λ}). In [3] (where again only odd characteristic is considered), it is shown that this cohomology vanishes in degrees 1 ≤ i ≤ p−3. For p = 3 a complete description of the nonzero cohomology was given for i = 1, 2.

Date: March 2008.

2000 Mathematics Subject Classification. Primary 20C30, Secondary 20G05, 20G10.

Research of the author was supported in part by NSF grant DMS-0556260.

In contrast little is known about the cohomology H^{i}(Σ_{d}, S^{λ}) of Specht modules. For i = 0 this was computed by James [11, Thm. 24.4]. It is at most one-dimensional, and explicit conditions are given on λ for it to be nonzero, see Theorem 4.1 below. In [10] the cohomology H^{i}(Σ_{d}, S^{λ}) for 0 ≤ i ≤ p − 2 is shown to agree with certain B cohomology in degree i + \binom{d}{2}, where B is the Borel subgroup of GL_{d}(k). However this high degree B cohomology has not been computed, so no new symmetric group results were obtained.

In our approach the relevant cohomology for B and its Frobenius kernels can be computed, and so new results on the cohomology H^{1}(Σ_{d}, S^{λ}) of Specht modules are obtained. We prove some stability theorems that generalize the results of James for H^{0} and are inspired by the known results for two-part partitions. In particular we prove a “generic cohomology” theorem that H^{1}(Σ_{p^{a}d}, S^{p^{a}λ}) stabilizes for a ≥ 1. There is a corresponding result for algebraic groups where one twists by the Frobenius automorphism. However for the symmetric group there is no Frobenius automorphism, and it is quite mysterious why any theorem should relate Specht modules S^{λ} and S^{pλ}, which are apparently unrelated modules for two different groups!

2. Notation and preliminaries

2.1. Although our application is to symmetric group representation theory, the actual work will be done within the general linear group theory. A basic reference for key results and also for our notation is [13]. For information on the Schur algebra see [9]. We will also draw extensively from the paper [1] of Andersen.

Recall k is an algebraically closed field of characteristic p ≥ 3 and let G = GL_{n}(k) be the general linear group. Let B (resp. B_{+}) be the Borel subgroup of lower (resp. upper) triangular matrices and T the torus of diagonal matrices. Let R denote the root system with respect to T. Let X(T) ≅ ℤ^{n} denote the weight lattice and S = {α_{1}, α_{2}, …, α_{n−1}} a set of simple roots such that B corresponds to the negative roots. For α ∈ S the corresponding coroot is α^{∨} and the corresponding reflection on X(T) is s_{α}. Let ρ denote half the sum of the positive roots.

The set of dominant weights is X(T)_{+} = {λ ∈ X(T) | ⟨α^{∨}, λ⟩ ≥ 0 for all α ∈ S.} Let ≤ denote the ordering on X(T) given by λ ≤ μ if and only if μ − λ is a linear combination of positive roots with nonnegative coefficients. The set of p^{r}-restricted weights is denoted X_{r}(T) = {λ ∈ X(T) | 0 ≤ ⟨α^{∨}, λ⟩ < p^{r} for all α ∈ S}.

For a module M and a simple module S, the composition multiplicity of S in M will be denoted [M : S]. For a GL_{n}(k) module M, the module M^{(r)} will denote the rth Frobenius twist of M. The rth Frobenius kernel of B is denoted B_{r}, see [13] for definitions. We will often use the fact that B/B_{r} is isomorphic to B and the modules can be identified using the rth power of the Frobenius twist.

2.2. The Schur algebra S(n, d) is the finite-dimensional associative k-algebra End_{kΣ_{d}}(V^{⊗d}) where V ≅ k^{n} 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).

Simple S(n, d) modules are in bijective correspondence with the set Λ^{+}(n, d) of partitions of d with at most n parts, and are denoted by L(λ). Note that one can also identify Λ^{+}(n, d) as the set of dominant polynomial weights of G of degree d. For λ ∈ Λ^{+}(n, d), let H^{0}(λ) = ind_{B}^{G}λ be the induced module and let V (λ) be the Weyl module. For λ, μ ∈ Λ^{+}(n, d) let ⊵ denote the usual dominance order.

2.3. We recall two results about Ext groups for GL_{n}(k) modules. The first is that for two S(n, d) modules M, N, the Ext^{i} groups are the same whether we work in the category of all G modules or of S(n, d) modules. The second is that for n > d the category of modules for S(n, d) is equivalent to that of S(d, d), where the equivalence “preserves” L(λ), H^{0}(λ) and V (λ) for λ ⊢ d. (Although the modules certainly have different dimensions, for example V (1, 1, 1) is a one-dimensional S(3, 3) module but a four-dimensional S(4, 3) module.) Both results are special cases of a more general result regarding truncated categories (cf. [13, Chpt. A]).

Proposition 2.1. [13, A.10, A.18] Let M and N be S(n, d) modules, and hence also G modules. Then for all i ≥ 0

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

The next proposition will let us equate Σ_{d} cohomology with that of GL_{n}(k) for various n ≥ d.

Proposition 2.2. [9, 6.5g] Suppose n > d. Then there is an idempotent e ∈ S(n, d) such that eS(n, d)e ≅ S(d, d) and the functor taking M to eM is an equivalence of categories from mod-S(n, d) to mod-S(d, d) mapping L(λ) to L(λ) and similarly for H^{0}(λ) and V (λ).

3. Specht module cohomology and symmetric powers

3.1. Although we will be studying symmetric group cohomology, we first transfer the problem to the general linear group. We recall below the result that allows this. For additional such results and explanations of how they are obtained using the Schur functor and the higher derived functors of its adjoint, see [14]. We use only the following.

Proposition 3.1. [14, Corollary 6.3(b)(iii)] For 0 ≤ i ≤ 2p − 4 we have

H^{i}(Σ_{d}, S^{λ}) ≅ Ext_{GL_{d}(k)}^{i}(H^{0}(d), H^{0}(λ)).

Remark. Proposition 3.1 is stated in [14] for 0 ≤ i ≤ 2p−3 but actually only holds up to i = 2p−4, cf. [10, 2.4] for an explanation.

We will actually do our computations using B cohomology, so the following is crucial, where part (b) is immediate from part (a) and Proposition 3.1.

Proposition 3.2.

(a) [13, 4.7a] Let V be a G-module and λ ⊢ d. Then for all i ≥ 0:

Ext_{G}^{i}(V, H^{0}(λ)) ≅ Ext_{B}^{i}(V, λ)

where λ also denotes the one-dimensional B module of weight λ.

(b) If 0 ≤ i ≤ 2p − 4 and G = GL_{n}(k) for n ≥ d then

H^{i}(Σ_{d}, S^{λ}) ≅ Ext_{B}^{i}(H^{0}(d), λ).

3.2. Structure of symmetric powers. Proposition 3.2 suggests one must understand H^{0}(d) in order to compute low degree Specht module cohomology. This module is isomorphic to S^{d}(V), the d-th symmetric power of the natural module V ≅ k^{n}. In [7], Doty gave a complete description of the composition factors and submodule lattice of H^{0}(d), which we recall below.

Let

B(d) := {β = (β_{1}, β_{2}, …, β_{n}) | β_{i} ≥ 0 and ∑ β_{i} = d}.

Then H^{0}(d) has a basis of weight vectors indexed naturally by B(d) [7, 2.1]. For β ∈ B(d), associate a sequence of nonnegative integers c_{i}(β) as follows. First write each β_{i} out in base p. Then add them base p to get d. For i ≥ 1, let c_{i}(β) be the number that is “carried” to the top of the p^{i} column during the addition. For example if p = 3 and β = (5, 5, 2) then the addition 5 + 5 + 2=12 base three looks like:

1 2 1 2 1 2 + 0 2 1 1 0

so c(β) = (2, 1). Doty calls this the carry pattern of β.

Let C(d) be the set of all carry patterns which occur for some β ∈ B(d). Define a partial order on C(d) by declaring (c_{1}, c_{2}, …, c_{m}) ≤ (c′_{1}, c′_{2}, …, c′_{m}) if c_{i} ≤ c′_{i} for all i. Given a subset B ⊆ C(d), let T_{B} be the subspace of H^{0}(d) spanned by all β with c(β) ∈ B. We can now state Doty’s result:

Theorem 3.3. [6] The correspondence B → T_{B} defines a lattice isomorphism between the lattice of ideals in the poset C(d) and the lattice of submodules of H^{0}(d). In particular the composition factors of H^{0}(d) are in one-to-one correspondence with the subspaces T_{c}, for c ∈ C(d). If L(λ) corresponds to T_{c} then λ is the maximal partition in Λ^{+}(n, d) with carry pattern c.

Given λ ∈ Λ^{+}(n, d), it is not difficult to determine if [H^{0}(d) : L(λ)] ≠ 0. The next lemma is an easy exercise working with carry patterns.

Lemma 3.4. Let λ = (λ_{1}, λ_{2}, …, λ_{n}) ∈ Λ^{+}(n, d). Define numbers a_{ij} with 0 ≤ a_{ij} < p by λ_{i} = ∑ a_{ij}p^{j}. Then:

(a) [H^{0}(d) : L(λ)] ≠ 0 if and only if a_{ij} ≠ p − 1 implies a_{lj} = 0 for all l > i.

(b) [H^{0}(pd) : L(pλ)] = [H^{0}(d) : L(λ)].

We will later use a strengthened version of Lemma 3.4(b), namely that H^{0}(d)^{(1)} is a submodule of H^{0}(pd) in a particularly nice way, cf. Proposition 5.6.

4. Previously known Specht module cohomology

4.1. Computing Hom_{Σ_{d}}(k, S^{λ}). The degree zero cohomology H^{0}(Σ_{d}, S^{λ}) ≅ Hom_{Σ_{d}}(k, S^{λ}) was determined by James. For an integer t let l_{p}(t) be the least nonnegative integer satisfying t < p^{l_{p}(t)}. James proved:

Theorem 4.1. [11, 24.4] Over a field of characteristic p, H^{0}(Σ_{d}, S^{λ}) is zero unless λ_{i} ≡ −1 mod p^{l_{p}(λ_{i+1})} for all i, in which case it is one-dimensional.

Remark. James’ result is proved entirely using symmetric group theory, applying the famous kernel intersection theorem. We observe that Doty’s description of H^{0}(d) allows one to calculate precisely when Hom_{B}(H^{0}(d), λ) is nonzero, and thus rederive James’ result in an entirely different way. The space of homomorphisms can only be nonzero for λ ⊢ d with [H^{0}(d) : L(λ)] = 1, so we must determine for each L(λ) a constituent of H^{0}(d), if there is a nonzero B module homomorphism from H^{0}(d) to λ. There is such a map precisely when there does not exist a μ ⊳ λ such that [H^{0}(d) : L(μ)] = 1 and c(μ) > c(λ). This can easily be seen to be equivalent to the condition in Theorem 4.1. Since the result is already known, we leave the details to the reader. One must show that given such λ and μ (i.e. μ ⊳ λ and c(μ) > c(λ)) the λ weight vector always lies in the B submodule generated by the μ weight vector, so there is no homomorphism in this case. Doty’s paper explicitly describes the B action, so this can be done.

Example 4.2. In characteristic five, [H^{0}(25) : L(20, 5)] = 1 but H^{0}(Σ_{25}, S^{(20,5)}) = 0. This is because (24, 1) ⊳ (20, 5) and c(24, 1) = (1, 1) > (1, 0) = c(20, 5). The (20, 5) weight vector lies in the B submodule generated by the (24, 1) weight vector, and so Hom_{B}(H^{0}(25), (20, 5)) = 0.

4.2. Two-part partitions. The other known example of H^{1}(Σ_{d}, S^{λ}) is for λ = (λ_{1}, λ_{2}) a twopart partition, where the answer can be deduced from work of Erdmann [8] on Ext^{1} between Weyl modules for SL_{2}(k), as we describe below.

For r > 0 write r = ∑_{i≥0} r_{i}p^{i} with 0 ≤ r_{i} ≤ p−1. For 0 ≤ a ≤ p−1 define \overline{a} so that 0 ≤ \overline{a} ≤ p−1 and a + \overline{a} ≡ p − 1 (mod p). Erdmann defined a set of integers Ψ_{p}(r) by

(4.2.1)  Ψ_{p}(r) = \{∑_{i=0}^{u−1} \overline{r_{i}}p^{i} + p^{u+a} : r_{u} ≠ p − 1, a ≥ 1, u ≥ 0\} ∪ \{∑_{i=0}^{u} \overline{r_{i}}p^{i} : r_{u} ≠ p − 1, u ≥ 0\}.

From Lemma 3.2.1 in [10] we deduce H^{1}(Σ_{d}, S^{(λ_{1},λ_{2})}) is at most one-dimensional, and it is nonzero precisely when λ_{2} ∈ Ψ_{p}(λ_{1} − λ_{2}). The conditions on λ for this to occur were stated incorrectly in [10], so we correct it here.

First recall from Theorem 4.1 that H^{0}(Σ_{d}, S^{(λ_{1},λ_{2})}) ≠ 0, precisely when λ_{1} ≡ −1 mod p^{l_{p}(λ_{2})}. This is the same as λ_{2} lying in the second subset in the definition of Ψ_{p}(λ_{1} − λ_{2}) in (4.2.1). Thus we consider when λ_{2} lies in the first subset. In this case we will have λ_{1} ≡ −1 mod p^{u} but (since r_{u} ≠ p − 1) λ ≢ −1 mod p^{u+1}. In this case λ_{2} is just p^{u+a} plus an arbitrary number less than p^{u}. We obtain:

Proposition 4.3. Let λ = (λ_{1}, λ_{2}) ⊢ d with λ_{2} ≠ 0. Then H^{1}(Σ_{d}, S^{λ}) is zero except in the two cases below, when it is one-dimensional:

(i) If Hom_{Σ_{d}}(k, S^{λ}) ≠ 0 or (ii) If λ_{1} ≡ −1 mod p^{u} but λ_{1} ≢ −1 mod p^{u+1} for some u ≥ 0 such that λ_{2} = c + p^{b} for c < p^{u}

and b > u.

Example 4.4. In characteristic five, H^{1}(Σ_{54}, S^{(29,25)}) ≅ k. Here choose u = 1, a = 2, c = 0 in part (ii) above.

Example 4.5. In characteristic p suppose a, b > 0 and pa ≥ p^{b}. Then choosing u = 0 we get H^{1}(Σ_{d}, S^{(pa,p^{b})}) ≠ 0.

Case (i) in Proposition 4.3 is a special case of a more general result conjectured in [10] which can be proved using a suggestion of Andersen.

Proposition 4.6 (Andersen). Suppose H^{0}(Σ_{d}, S^{λ}) ≠ 0 and λ ≠ (d). Then H^{1}(Σ_{d}, S^{λ}) ≠ 0.

Proof. This follows from the universal coefficient theorem. The key observation is that the Specht module is defined over the integers, but when λ ≠ (d) then Hom_{ℤ}(ℤ, S_{ℤ}^{λ}) = 0. □

4.3. The following corollary is immediate from Proposition 4.3, and is the motivation for the generalizations proved in the next two sections.

Corollary 4.7. Let λ = (λ_{1}, λ_{2}) ⊢ d < p^{a}. Then

(a) H^{1}(Σ_{pd}, S^{pλ}) ≅ H^{1}(Σ_{p^{2}d}, S^{p^{2}λ}).

(b) H^{1}(Σ_{d}, S^{λ}) ≅ H^{1}(Σ_{d+p^{a}}, S^{(λ_{1}+p^{a},λ_{2})})

5. Generic cohomology for Specht modules?

5.1. For any G-module M there is a series of injections:

(5.1.1)  H^{i}(G, M) → H^{i}(G, M^{(1)}) → H^{i}(G, M^{(2)}) → · · · .

This sequence is known [4] to stabilize, and the limit is called the generic cohomology of M. For example we get injections H^{i}(G, L(λ)) → H^{i}(G, L(pλ)) → H^{i}(G, L(p^{2}λ)). It is natural then to have theorems for G which involve multiplying a partition by p, as this reflects what happens to the weights after a Frobenius twist.

For the symmetric group, we know of no theorems involving multiplying a partition by p. There is nothing that seems to play the role of the Frobenius twist. Moreover, the modules S^{λ} and S^{pλ} are not even modules for the same symmetric group. However, for two-part partitions we observed in Corollary 4.7 that H^{1}(Σ_{pd}, S^{pμ}) ≅ H^{1}(Σ_{p^{2}d}, S^{p^{2}μ}). In this section we generalize this stability result to arbitrary partitions, and show there is a generic cohomology for Specht modules in degree one. The main result is an isomorphism H^{1}(Σ_{pd}, S^{pλ}) ≅ H^{1}(Σ_{p^{2}d}, S^{p^{2}λ}).

5.2. Relating cohomology and Frobenius twists. There is a spectral sequence which relates Ext_{B}^{•}(M_{1}, M_{2}) and Ext_{B}^{•}(M_{1}^{(r)}, M_{2}^{(r)}).

Proposition 5.1. [13, I.6.10, II.10.14] Let M_{1}, M_{2} be B modules and B_{r} be the rth Frobenius kernel. There is a spectral sequence:

(5.2.1)  E_{2}^{i,j} = Ext_{B}^{i}(M_{1}, M_{2} ⊗ H^{j}(B_{r}, k)^{(−r)}) ⇒ Ext_{B}^{i+j}(M_{1}^{(r)}, M_{2}^{(r)}).

We can use this spectral sequence as our first step in relating cohomology of S^{λ} and S^{pλ}, namely to show for M = μ ⊗ H^{0}(d)^{∗}, the sequence in (5.1.1) stabilizes immediately.

Lemma 5.2. Let μ ⊢ d. Then

Ext_{B}^{1}(H^{0}(d)^{(1)}, pμ) ≅ Ext_{B}^{1}(H^{0}(d), μ).

SPECHT MODULE COHOMOLOGY 7

Proof. Take r = 1 and consider the five-term exact sequence for the spectral sequence (5.2.1)

0 → Ext_{B}^{1}(H^{0}(d), μ) → Ext_{B}^{1}(H^{0}(d)^{(1)}, pμ) → Hom_{B}(H^{0}(d), H^{1}(B_{1}, k)^{(−1)} ⊗ μ) → · · · . Now observe that H^{1}(B_{1}, k) = 0 by [12], so we have the desired isomorphism. □ Remark. If the left side in Lemma 5.2 had H^{0}(pd) instead of H^{0}(d)^{(1)} we would have an isomorphism between H^{1}(Σ_{d}, S^{λ}) and H^{1}(Σ_{pd}, S^{pλ}), however this is false in general.

Next we prove a technical lemma we will need later. Lemma 5.3. Suppose λ ⊢ p^{2}d such that λ ≠ pτ for any τ ⊢ pd. Suppose further that μ ⊢ d with λ ⊳ p^{2}μ. Then:

⟨λ − p^{2}μ, α_{i}^{∨}⟩ ≥ p^{2} − p

for some i. Proof. Let λ = (λ_{1}, λ_{2}, …, λ_{n}) and denote the p-adic expansion of λ_{i} by λ_{i} = a_{i0} + a_{i1}p + · · · . If a_{11} = p − 1 then ⟨λ − p^{2}μ, α_{1}^{∨}⟩ ≥ (p − 1)p and we are done. If not then, by Lemma 3.4, for each i > 1 we have a_{i1} = 0. Since λ_{1} > μ_{1} and λ ⊳ p^{2}μ, there must be some j with λ_{j} > p^{2}μ_{j} and λ_{j+1} ≤ p^{2}μ_{j+1}. The p-adic expansion of λ_{j} and λ_{j+1} ensures that ⟨λ−p^{2}μ, α_{j}^{∨}⟩ ≥ p^{2}−(p−1) > p^{2}−p as desired. □ Corollary 5.4. Let λ and p^{2}μ be as above. Then λ−p^{2}μ−(p−1)ρ is not a weight in the Steinberg module St_{1} = L((p − 1)ρ). Proof. Let γ = λ − p^{2}μ − (p − 1)ρ and choose i as in Lemma 5.3. Then

⟨(p − 1)ρ − γ, α_{i}^{∨}⟩ = ⟨2(p − 1)ρ − (λ − p^{2}μ), α_{i}^{∨}⟩
= 2p − 2 − ⟨(λ − p^{2}μ), α_{i}^{∨}⟩
≤ 2p − 2 − (p^{2} − p) by Lemma 5.3
= −p^{2} + 3p − 2 < 0.

Thus (p − 1)ρ ≱ γ so γ is not a weight in St_{1}. □ Remark. We observe that Lemma 5.3 requires p^{2}μ and the corresponding statement for pμ is false. For example if p = 5, λ = (9, 1) and 5μ = (5, 5) then λ − 5μ − 4ρ is a weight in the Steinberg module. This is why our stability theorem requires comparing pμ and p^{2}μ. 5.3. Corollary 5.4 lets us prove a key vanishing result for B-cohomology, which we state next. Proposition 5.5. Let λ ⊢ p^{2}d with [H^{0}(p^{2}d) : L(λ)] ≠ 0. Suppose λ is not of the form pτ and let μ ⊢ d. Then:

Ext_{B}^{1}(L(λ), p^{2}μ) = 0.

Proof. We can assume that λ ⊳ p^{2}μ without loss by [13, II.4.14]. Write λ = λ_{(0)} + pτ with ∅ ≠ λ_{(0)} ∈ X_{1}(T) and write:

Ext_{B}^{1}(L(λ), p^{2}μ) ≅ Ext_{B}^{1}(L(λ_{(0)} ⊗ L(pτ) ⊗ (−p^{2}μ), k).

In this case the LHS spectral sequence [13, I.6.6(1)] with B_{1} ⊲ B looks like:

(5.3.1)  E_{2}^{i,j} = Ext_{B/B_{1}}^{i}(L(pτ) ⊗ (−p^{2}μ), Ext_{B_{1}}^{j}(L(λ_{(0)}), k)) ⇒ Ext_{B}^{i+j}(L(λ), p^{2}μ).

By [1, (3.1)] we know Hom_{B_{1}}(L(λ_{(0)}), k) = 0, and thus the E^{1,0} term is zero. Now consider the E^{0,1} term. For any ξ ∈ X(T) we have:

Hom_{B/B_{1}}(L(τ)^{(1)} ⊗ (−p^{2}μ), pξ) ≅ Hom_{B}(L(τ) ⊗ (−pμ), ξ) ≅ Hom_{B}(L(τ), ξ + pμ).

Applying this to the E^{0,1} term we find it is zero unless pτ − p^{2}μ is a weight in Ext_{B_{1}}^{1}(L(λ_{(0)}), k). By Proposition 3.2 in [1], this can only occur if λ_{(0)} + pτ − p^{2}μ − (p − 1)ρ is a weight in St_{1}. But this is ruled out by Corollary 5.4. Thus the E^{0,1} term vanishes as well, and so E^{1} = 0 as desired.

□

5.4. Twisted symmetric powers. The next observation is that the twisted symmetric powers H^{0}(d)^{(1)} embed nicely in H^{0}(pd) with cokernel containing no simple modules of the form L(pτ).

Proposition 5.6. There is a short exact sequence

(5.4.1)  0 → H^{0}(d)^{(1)} → H^{0}(pd) → Q → 0

where ∀τ ⊢ d, [Q : L(pτ)] = 0.

Proof. Apply Hom_{G}(−, H^{0}(pd)) to the sequence 0 → L(pd) → H^{0}(d)^{(1)} → U → 0 to obtain:

0 → Hom_{G}(H^{0}(d)^{(1)}, H^{0}(pd)) → k → Ext_{G}^{1}(U, H^{0}(pd)).

But Ext_{G}^{1}(U, H^{0}(pd)) = 0 since (pd) is not dominated by any weight in U (see [13, p.210]). Thus Hom_{G}(H^{0}(d)^{(1)}, H^{0}(pd)) ≅ k and by comparing socles we see the map must be an injection. The statement about Q is immediate by Lemma 3.4(b) since H^{0}(pd) is multiplicity free. □

Now consider (5.4.1) with pd and p^{2}d, and suppose p^{2}μ ⊢ p^{2}d. Applying Hom_{B}(−, p^{2}μ) to (5.4.1) we obtain:

(5.4.2)

⋯ → Ext_{B}^{1}(Q, p^{2}μ) → Ext_{B}^{1}(H^{0}(p^{2}d), p^{2}μ) → Ext_{B}^{1}(H^{0}(pd)^{(1)}, p^{2}μ) → Ext_{B}^{2}(Q, p^{2}μ) → ⋯

We will show the first and last term in (5.4.2) are zero.

Lemma 5.7. Let Q be as in (5.4.2). Then Ext_{B}^{1}(Q, p^{2}μ) = 0.

Proof. Let [Q : L(λ)] ≠ 0. By Proposition 5.6 we know λ is not of the form pτ. The result then follows by Proposition 5.5.

□

5.5. Analyzing Ext_{B}^{2}(Q, p^{2}μ). It remains to prove the analogue of Proposition 5.5 for Ext^{2}, which will require even more intricate spectral sequence calculations. Let λ = λ_{(0)} + pτ be as above and consider the spectral sequence (5.3.1). We will prove the E^{2} term is zero by showing the terms E^{0,2}, E^{1,1} and E^{2,0} all vanish.

Lemma 5.8. The E^{2,0} term in (5.3.1) is zero.

Proof. This is immediate since λ_{(0)} ≠ ∅ implies Hom_{B_{1}}(L(λ_{(0)}), k) = 0 by [1, (3.1)]. □

To show the E^{1,1} term is zero we extend an argument of Andersen’s. For ν ∈ X_{1}(T) he defined ([1, p.495]) R(ν) by the exact sequence

(5.5.1)  0 → ν → St_{1} ⊗ [(p − 1)ρ + ν] → R(ν) → 0.

We will use this to prove:

Lemma 5.9. The E^{0,2} term in (5.3.1) is zero.

Proof. The E^{0,2} term is:

Hom_{B/B_{1}}(L(pτ) ⊗ (−p^{2}μ), Ext_{B_{1}}^{2}(L(λ_{(0)}), k).

We will showing that Ext_{B_{1}}^{2}(L(λ_{(0)}), k) = 0 cannot have pτ − p^{2}μ as a weight.

Apply Hom_{B_{1}}(L(λ_{(0)}, −) to (5.5.1) and use the fact that St_{1} is injective to obtain:

Ext_{B_{1}}^{2}(L(λ_{(0)}), k) ≅ Ext_{B_{1}}^{1}(L(λ_{(0)}), R(0)).

Suppose σ is a weight in R(0), so ν := σ −(p−1)ρ ∈ St_{1}. If Ext_{B_{1}}^{1}(L(λ_{(0)}), σ) has a weight pτ −p^{2}μ in it, then by [1, Prop. 3.2] we must have

λ_{(0)} − σ + pτ − p^{2}μ − (p − 1)ρ ∈ St_{1}

so adding ν ∈ St_{1} we must also have

ω := λ − p^{2}μ − 2(p − 1)ρ ∈ St_{1}.

But choosing i as in Lemma 5.3, we have:

⟨(p − 1)ρ − ω, α_{i}^{∨}⟩ = 3p − 3 − ⟨λ − p^{2}μ, α_{i}^{∨}⟩
≤ 3p − 3 − (p^{2} − p)
= −p^{2} + 4p − 3 ≤ 0

Unlike Corollary 5.4 we have a minor technicality here, the final inequality above is not strict precisely when p = 3. In this case we must rule out the possibility that λ − p^{2}μ = 6ρ separately. This can be done using Lemma 3.4(b) since λ_{(0)} ≠ ∅. Thus for any odd prime, (p − 1)ρ ≱ ω, so ω is not a weight in St_{1}. □

Before tackling the E^{1,1} term we prove a preliminary lemma:

Lemma 5.10. Suppose λ_{(0)} + pζ − (p − 1)ρ ∈ St_{1}. Then 0 ≤ ⟨ζ, α^{∨}⟩ ≤ 1 for all α ∈ S.

Proof. By Lemma 3.4 we know 0 ≤ ⟨λ_{(0)}, α^{∨}⟩ < p and we know the weights in St_{1} lie between −(p − 1)ρ and (p − 1)ρ. Now just check the corresponding inequalities for ⟨ζ, α^{∨}⟩. □

Lemma 5.11. The E^{1,1} term in (5.3.1) is zero.

Proof. Suppose pζ is a weight in Ext_{B_{1}}^{1}(L(λ_{(0)}), k) and consider

(5.5.2)  Ext_{B/B_{1}}^{1}(L(pτ) ⊗ (−p^{2}μ), pζ) ≅ Ext_{B}^{1}(L(τ), ζ + pμ).

It suffices to show that (5.5.2) is zero. Set τ = τ_{(0)} + pγ with τ_{(0)} ∈ X_{1}(T) and apply the LHS spectral sequence again:

(5.5.3)  E_{2}^{i,j} = Ext_{B/B_{1}}^{i}(L(pγ) ⊗ (−pμ), Ext_{B_{1}}^{j}(L(τ_{(0)}), ζ) ⇒ Ext_{B}^{i+j}(L(τ), ζ + pμ).

We are trying to show the E^{1} term in (5.5.3) is zero. Notice that E^{1,0} is immediately zero unless τ_{(0)} = ζ in which case it is

Ext_{B/B_{1}}^{1}(L(pγ), pμ) ≅ Ext_{B}^{1}(L(γ), μ)

which is also zero since γ ⋭μ. (In particular γ is a partition of a smaller integer than μ).

Finally consider Ext_{B_{1}}^{1}(L(τ_{(0)}), ζ) in the E^{0,1} term and let pχ be a weight, so 0 ≤ ⟨χ, α^{∨}⟩ ≤ 1 for all α ∈ S, by Lemma 5.10. Note that μ ⊢ d whereas λ_{(0)} ≠ ∅ implies γ ⊢ c < d. Thus we cannot have ν = χ + μ so Hom_{B}(L(ν), χ + μ) = 0 and we are done. □

From the preceding four lemmas we obtain:

Proposition 5.12. Let Q be as in (5.4.2). Then Ext_{B}^{1}(Q, p^{2}μ) = 0.

Now suppose G = GL_{p^{2}d}(k). By Propositions 2.1 and 2.2 and Lemma 5.7, we can use Propositions 5.5 and 5.12 to obtain the main result of this section.

Theorem 5.13. Let λ ⊢ d. Then there is a isomorphism:

H^{1}(Σ_{pd}, S^{pλ}) ≅ H^{1}(Σ_{p^{2}d}, S^{p^{2}λ}).

Notice we are using Proposition 2.2 here. We know H^{1}(Σ_{pd}, S^{pλ}) ≅ Ext_{GL_{pd}(k)}^{1}(H^{0}(pd), H^{0}(pλ)), but this is the same as taking Ext^{1} in the category of GL_{p^{2}d}(k) modules.

We immediately obtain a “generic cohomology” result.

Theorem 5.14. Let λ ⊢ d. Then for any c ≥ 1 we have

H^{1}(Σ_{p^{c}d}, S^{p^{c}λ}) ≅ H^{1}(Σ_{p^{c+1}d}, S^{p^{c+1}λ}).

Remark. A corresponding result for H^{0}(Σ_{d}, S^{λ}) follows from Theorem 4.1 since H^{0}(Σ_{pd}, S^{pλ}) = 0 unless λ = (d). For H^{1} the generic cohomology can definitely be nonzero, see Example 4.5 for instance.

6. A second stability result

6.1. In this section we prove a stability result for Specht module cohomology relating to adding a large power of p to the first part of the partition. This will generalize Corollary 4.7(b) for two-part partitions. The previous section used the fact that H^{0}(d)^{(1)} sits nicely as a submodule in H^{0}(pd). In this section we exploit the fact that H^{0}(d) ⊗ L(1)^{(r)} sits nicely inside H^{0}(d + p^{r}) for large r. For this section choose r so that p^{r} > d and let n = d + p^{r}. We wish to analyze the GL_{n}(k) module H^{0}(d) ⊗ L((p^{r})) = H^{0}(d) ⊗ L(1)^{(r)} Lemma 6.1. The module H^{0}(d) ⊗ L(1)^{(r)} has simple socle isomorphic to L((d + p^{r})).

Proof. Its constituents are all of the form L(λ) ⊗ L(1)^{(r)} where [H^{0}(d) : L(λ)] = 1. From the 0, 0 term in the LHS spectral sequence ([13, I.6.6]), we have:

(6.1.1) Hom_{G}(L(λ)⊗L(1)^{(r)}, H^{0}(d)⊗L(1)^{(r)}) ≅ Hom_{G/G_{r}}(L(1)^{(r)}, Hom_{G_{r}}(L(λ), H^{0}(d)⊗L(1)^{(r)})).

Since d < p^{r}, constituents of H^{0}(d) are of the form L(μ), μ ∈ X_{r}(T). So

Hom_{G_{r}}(L(λ), H^{0}(d)) ≅ Hom_{G}(L(λ), H^{0}(d))

which is zero unless λ = (d). Thus the homomorphism space in (6.1.1) is nonzero precisely when λ = (d) and the statement about the socle follows.

□

Lemma 6.2. There is an injection:

0 → H^{0}(d) ⊗ L(1)^{(r)} → H^{0}(d + p^{r})

Proof. We show equivalently that there is a surjection from V (d + p^{r}) onto V (d) ⊗ L(1)^{(r)}. Note that [13, Lemma I.2.13] implies

Hom_{G}(V (d + p^{r}), V (d) ⊗ L(1)^{(r)}) ≅ Hom_{B^{+}}(d + p^{r}, V (d) ⊗ L(1)^{(r)})

which is one-dimensional. However both modules have isomorphic simple heads and are multiplicity free, so the map must be onto.

□

For notational simplicity, if λ ⊢ d let λ + p^{r} denote (λ_{1} + p^{r}, λ_{2}, …, λ_{n}). Next we consider the cokernel of the map in Lemma 6.2.

Lemma 6.3. There is a short exact sequence

(6.1.2)  0 → H^{0}(d) ⊗ L(1)^{(r)} → H^{0}(d + p^{r}) → U → 0

where if [U : L(μ)] ≠ 0 then μ_{1} < p^{r}. In particular μ ⋭λ + p^{r}.

Proof. The sequence comes from Lemma 6.2. Now suppose [H^{0}(d + p^{r}) : L(μ)] ≠ 0 and μ_{1} ≥ p^{r}. Then the p-adic expansion of μ_{1} is μ_{1} = p^{r} +c_{r−1}p^{r−1}+⋯+c_{0}. Since d < p^{r} then μ_{2} < p^{r}. However μ is supposed to be maximal with its carry pattern, which can not happen unless μ_{1} − μ_{2} ≥ p^{r}. Thus μ is of the form μ̃ + p^{r}, where [H^{0}(d) : L(μ̃)] ≠ 0. Thus L(μ) occurs in H^{0}(d) ⊗ L(1)^{(r)} and, since H^{0}(d + p^{r}) is multiplicity free, not in U.

□

From [13, p.210] we immediately obtain:

Lemma 6.4. Let U be as in (6.1.2). Then Ext_{G}^{i}(U, H^{0}(λ + p^{r})) = 0∀i.

Applying Hom_{G}(−, H^{0}(λ + p^{r})) to (6.1.2) and using the Lemma 6.4, we obtain:

Lemma 6.5.

Ext_{G}^{1}(H^{0}(d + p^{r}), H^{0}(λ + p^{r})) ≅ Ext_{G}^{1}(H^{0}(d) ⊗ L(1)^{(r)}, H^{0}(λ + p^{r})).

By Lemma 6.5 and Proposition 3.2, we know that H^{1}(Σ_{d+p^{r}}, S^{λ+p^{r}}) is isomorphic to Ext_{B}^{1}(H^{0}(λ)⊗ L(1)^{(r)}, λ + p^{r}). The module L(1)^{(r)} is just the natural representation V, twisted r times. As a module for B it is uniserial, write

(6.1.3)  0 → Q → L(1)^{(r)} → (p^{r}, 0, …, 0) → 0

Lemma 6.6. Let Q be as in (6.1.3). Then

Ext_{B}^{1}(H^{0}(d) ⊗ Q, λ + p^{r}) = 0.

Proof. Consider τ = (0, 0, …, 0, p^{r}, 0 …) a weight in Q. Since p^{r} > d, it is clear that λ + p^{r} − τ is not dominant, nor is it of the form s_{α}.σ for α ∈ S and σ ∈ X_{+}(T). The result follow from [1, Prop. 2.3]. □

We can now obtain a result which will imply the stability theorem.

Theorem 6.7.

Ext_{B}^{1}(H^{0}(d) ⊗ L(1)^{(r)}, λ + p^{r}) ≅ Ext_{B}^{1}(H^{0}(d), λ).

Proof. Take (6.1.3) and tensor it by H^{0}(d) then apply Hom_{B}(−, λ + p^{r}) to obtain

(6.1.4)  ⋯ → Hom_{B}(H^{0}(d) ⊗ Q, λ + p^{r}) → Ext_{B}^{1}(H^{0}(d) ⊗ (p^{r}, 0, …, 0), λ + p^{r}) →
Ext_{B}^{1}(H^{0}(d) ⊗ L(1)^{(r)}, λ + p^{r}) → Ext_{B}^{1}(H^{0}(d) ⊗ Q, λ + p^{r}) → ⋯

The left hand term in (6.1.4) is zero just by considering weight spaces. The right hand term is zero by Lemma 6.6, so we have the result. □

By Proposition 3.1(b) we obtain the main result of this section.

Theorem 6.8. Let λ ⊢ d and p^{r} > d. Then:

H^{1}(Σ_{d}, S^{λ}) = H^{1}(Σ_{d+p^{r}}, S^{λ+p^{r}}).

7. Open problems and future directions

We suspect these results are the tip of a large iceberg of “generic cohomology” type theorems for the symmetric group. In a recent paper [5] of the author with Cohen and Nakano we proved generic cohomology results for Young modules in characteristic two, conjectured to hold for all primes.

One question is whether our stability result can be explicitly realized, i.e.

Problem 7.1. Given an element 0 → S^{pλ} → M → k → 0 in H^{1}(Σ_{pd}, S^{pλ}), can one explicitly construct an extension of S^{p^{2}λ} by k realizing the isomorphism in Theorem 5.14.

Another obvious question is whether there is generic cohomology in all degrees. With our evidence from degrees zero and one we conjecture:

Conjecture 7.2. Fix i > 0. There is constant c(i) such that for any d with λ ⊢ d and any a ≥ c(i) that

H^{i}(Σ_{p^{a}d}, S^{p^{a}λ}) ≅ H^{i}(Σ_{p^{a+1}d}, S^{p^{a+1}λ}).

So c(0) = c(1) = 1. One obstacle to generalizing our proof is that Proposition 3.1(ii) only holds through degree 2p − 4. There is certainly evidence that for larger i one must “twist” more times before stability for H^{i} begins. For example we have the following analogue of Lemma 5.2.

Proposition 7.3. Let μ ⊢ d. Then

Ext_{B}^{2}(H^{0}(d)^{(1)}, pμ) ≅ Ext_{B}^{2}(H^{0}(d)^{(2)}, p^{2}μ).

Proof. Consider the spectral sequence (5.2.1) for the case r = 1, M_{1} = H^{0}(d)^{(1)} and M_{2} = pμ. The E^{2,0} term is Ext_{B}^{2}(H^{0}(d)^{(2)}, p^{2}μ), which we must show is equal to E^{2}. It suffices to show the E^{1,1} and E^{0,2} terms are both zero. The E^{1,1} term is zero, as H^{1}(B_{1}, k) = 0 as mentioned in the proof of Lemma 5.2. Finally we have:

E^{0,2} ≅ Hom_{B}(H^{0}(d)^{(1)}, pμ ⊗ H^{2}(B_{1}, k)^{(−1)}).

The cohomology H^{2}(B_{1}, k) was computed in [2, Thm 5.3]. From this calculation one can see that the weights in H^{2}(B_{1}, k)^{(−1)} are all of the form −α for α ∈ S , in particular they are not of the form pσ. But every weight in H^{0}(d)^{(1)} ⊗ (−pμ) is of this form, so the E^{0,2} term is zero. □

Remark. The additional twist going from Lemma 5.2 to Proposition 7.3 is indeed necessary. Without it, in the proof above the E^{0,2} term may be nonzero. For example if p = 5 and μ = (50, 25) then Hom_{B}(H^{0}(75), μ ⊗ (−α_{1})) ≠ 0. Further, Proposition 7.3 required information about higher degree B_{1} cohomology.

7.1. It would be nice to have a proof of Proposition 4.6 using our approach, or perhaps more generally for any composition factor L(λ) in H^{0}(d). We conjecture:

Conjecture 7.4. Suppose [H^{0}(d) : L(λ)] ≠ 0 for λ ≠ (d). Then H^{1}(Σ_{d}, S^{λ}) ≠ 0.

Note that Conjecture 7.4 is a strengthening of Proposition 4.6. More generally the problem of computing H^{1} is still open.

Problem 7.5. For which λ ⊢ d is H^{1}(Σ_{d}, S^{λ}) nonzero? Is it at most one-dimensional?

There is another type of “stability” which occurs, at least for H^{0}(Σ_{d}, S^{λ}). Namely the following is an easy consequence of Theorem 4.1:

Lemma 7.6. Suppose λ = (λ_{1}, λ_{2}, …, λ_{s}) ⊢ d and suppose a ≡ −1 mod p^{l_{p}(λ_{1})}. Then

(7.1.1)  H^{0}(Σ_{d}, S^{λ}) ≅ H^{0}(Σ_{d+a}, S^{(a,λ_{1},λ_{2},...,λ_{s})}).

This leads to the following

Problem 7.7. Does the isomorphism in (7.1.1) hold for H^{i} for any other i > 0?

7.2. Finally we ask for stronger results like that of Theorem 6.8.

Problem 7.8. Let λ ⊢ d and μ ⊢ c. Can one find more results that related the cohomology H^{i}(Σ_{d}, S^{λ}) and H^{i}(Σ_{d+cp^{r}}, S^{λ+p^{r}μ})?

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Department of Mathematics, University at Buffalo, SUNY, 244 Mathematics Building, Buffalo, NY 14260, USA

E-mail address: dhemmer@math.buffalo.edu