SYLOW SUBGROUPS OF SYMMETRIC AND ALTERNATING GROUPS AND THE VERTEX OF S^{(kp−p,1^{p})} IN CHARACTERISTIC p

EUGENIO GIANNELLI, KAY JIN LIM, AND MARK WILDON

Abstract. We show that the Sylow p-subgroups of a symmetric group, respectively an alternating group, are characterized as the p-subgroups containing all elementary abelian p-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition (kp − p, 1^{p}) under the assumption that k ≡ 1 mod p and k ≢ 1 mod p^{2}.

1. Introduction

In this paper we prove two main theorems. The first gives a new characterization of the Sylow p-subgroups of the symmetric groups 𝔖_{n} and alternating groups 𝔄_{n}.

Theorem 1.1. Suppose that G is either 𝔄_{n} or 𝔖_{n}. Let p be a prime and let Q ≤ G be a p-subgroup. Then Q is a Sylow p-subgroup of G if and only if Q contains a G-conjugate of every elementary abelian p-subgroup of G.

It is easily seen that this characterization of Sylow subgroups does not hold in a general finite group. When n is large compared to p, the Sylow p-subgroups of 𝔖_{n} and 𝔄_{n} are much larger than the elementary abelian p-groups they contain. For example, if n is a power of 2 then a Sylow 2subgroup of 𝔖_{n} has order 2^{n−1}, whereas the order of the largest elementary abelian 2-subgroup of 𝔖_{n} is 2^{n/2}. This gives another indication that Theorem 1.1 is a non-trivial result. We state some open problems suggested by this theorem in the final section of the paper.

Date: November 13, 2014.

2010 Mathematics Subject Classification. Primary 20C30, Secondary 20B30, 20C20, 20D06, 20D20.

Key words and phrases. Specht module, vertex, symmetric group, alternating group, Sylow subgroup.

Part of this work was done while the third author was visiting the Institute for Mathematical Sciences, National University of Singapore in 2013. The visit was supported by the Institute.

The second author is supported by Singapore Ministry of Education AcRF Tier 1 grant RG13/14.

Theorem 1.1 is motivated by an application to vertices of indecomposable modules for the symmetric group. We recall the key definitions. Let F be a field of prime characteristic p, let G be a finite group and let M be an indecomposable F G-module. A subgroup Q of G is said to be a vertex of M if there exists a F Q-module U such that V is a direct summand of U ↑^{G} and Q is minimal with this property. This definition was introduced by Green in [11], where he showed that the vertices of M form a single conjugacy class of p-subgroups of G. We note that the vertices of M do not change under field extensions of F [15, Theorem 1.21]. Vertices play a key role in the Green correspondence [12], and in Alperin’s Weight Conjecture [3]. But despite the importance of vertices to local representation theory, there are relatively few cases where the vertices of particular families of modules have been computed. Our second main theorem is as follows.

Theorem 1.2. Let p be an odd prime. Suppose that k ≡ 1 mod p and that k ≢ 1 mod p^{2}. The Specht module S^{(kp−p,1^{p})}, defined over a field of prime characteristic p, has a Sylow p-subgroup of 𝔖_{kp} as a vertex.

We prove Theorem 1.2 as a corollary of the following result, which is of independent interest.

Theorem 1.3. Let p be an odd prime and let E be an elementary abelian p-subgroup of 𝔖_{kp}. If either

(i) E has at least two orbits of size p on {1, …, kp}, or (ii) E has at least one orbit of size p^{2} on {1, …, kp},

then E is contained in a vertex of the Specht module S^{(kp−p,1^{p})}, defined over an algebraically closed field of characteristic p.

The proof of Theorem 1.3 uses two key techniques: generic Jordan type for modules over elementary abelian p-groups (see [9, 24]), and the Brauer correspondence for modules, as developed by Broué in [5]. The isomorphism S^{(kp−p,1^{p})} ≅ ⋀^{p} S^{(kp−1,1)} (see Section 5) is used throughout the calculations.

Background. We now survey the existing results on vertices of Specht modules, emphasising the different techniques that have been used. We begin with results that, like Theorem 1.2, apply to Specht modules labelled by partitions of a special form. Note that, by [16, Corollary 13.18], Specht modules are always indecomposable in odd characteristic.

The simplest case is a Specht module S^{λ} in a block of abelian defect. In this case S^{λ} has dimension coprime to p and by [11, Corollary 1], its vertices are the defect groups of its block; these are elementary abelian.

THE VERTEX OF S^{(kp−p,1^{p})} 3

In [25, Theorem 2] the third author used the Brauer correspondence to show that when p does not divide n and S^{(n−r,1^{r})} is indecomposable, its vertex is a Sylow p-subgroup of 𝔖_{n−r−1} × 𝔖_{r}. When p = 2 and n is even it was shown by Murphy and Peel in [21, Theorem 4.5] that S^{(n−r,1^{r})} is indecomposable and has a Sylow 2-subgroup of 𝔖_{n} as a vertex. A key step in their proof is their Theorem 4.3, which states that if Q is an elementary abelian 2-subgroup of 𝔖_{n} of rank n/2 then S^{(n−r,1^{r})}↓_{Q} is indecomposable. The analogue of this result for odd primes is in general false: for example, when p = 5, calculations with the computer algebra package Magma [4] show that if Q = ⟨(1, 2, 3, 4, 5), (6, 7, 8, 9, 10)⟩ then S^{(8,1,1)}↓_{Q} is decomposable. It therefore seems impossible to extend the methods of [21] to the case when p is odd and p divides n.

The dimension of S^{(kp−r,1^{r})} is \binom{kp−1}{r}, which is coprime to p whenever r < p. In these cases S^{(kp−r,1^{r})} has a full Sylow p-subgroup of 𝔖_{kp} as a vertex again by [11, Corollary 1]. When r = p the dimension is divisible by p if and only if k ≡ 1 mod p, and divisible by p^{2} if and only if k ≡ 1 mod p^{2}. Theorem 1.2 therefore deals with the smallest case that cannot be decided solely by dimension arguments.

When p = 2 the Specht module S^{(n−2,2)} is always indecomposable. When n is even or n ≡ 1 mod 4, it was shown in [6, Theorem 1] that S^{(n−2,2)} has a Sylow 2-subgroup of 𝔖_{n} as a vertex, except when n = 4, when the unique vertex of S^{(2,2)} is the normal Sylow 2-subgroup of 𝔄_{4}. When n ≡ 3 mod 4, any Sylow 2-subgroup of 𝔖_{n−5} × 𝔖_{2} × 𝔖_{2} is a vertex. When p ≥ 3 the third author showed in [26, Theorem 8.1] that S^{(n−2,2)} has a Sylow p-subgroup of the defect group of its block as a vertex.

In the main result of [26] the third author used the Brauer correspondence to determine a subgroup contained in the vertex of each indecomposable Specht module. This result was later improved by the first author in [10]. In [19] the second author used the complexity theory of modules to prove a number of classification theorems on Specht modules with abelian vertices. In particular it was shown that if p ≥ 3 and S^{λ} has an abelian vertex of p-rank m then λ is a p^{2}-core of p-weight m, and that S^{(p^{p})} has a Sylow p-subgroup of 𝔖_{p^{2}} as a vertex.

In [14] Hemmer proved that, when p ≥ 3, a Specht module is simple if and only if it is a signed Young module, as defined by Donkin [8, page 651]. By [8, (3) page 675], it follows that the vertices of simple Specht modules in odd characteristic are Sylow p-subgroups of Young subgroups of symmetric groups. For example, when n is not divisible by p, the Specht module S^{(n−r,1^{r})} is isomorphic to the signed Young module Y ((n − r, 1^{r−ps})|p(s)) where ⌊r/p⌋ = s, and so its vertex may also be found using Donkin’s result. In general it is not possible to apply Donkin’s result in this way because Hemmer’s proof does not explicitly determine a pair of partitions labelling a signed Young module isomorphic to a given simple Specht module.

Finally we note that there has also been considerable work on the vertices of simple modules for symmetric groups. We refer the reader to [7] for a survey.

Outline. In Section 2 we prove Theorem 1.1 for symmetric groups. Then in Section 3 we use this result and further arguments to prove Theorem 1.1 for alternating groups. In Section 4 we give a short proof, using Theorem 1.1, that Theorem 1.3 implies Theorem 1.2.

We then turn to the proof of Theorem 1.3. In Section 5 we collect the necessary background results on hook-Specht modules. We review the Brauer correspondence in Section 6 and give a useful general result on the Brauer homomorphism for monomial modules. The notion of generic Jordan type is presented in Section 7. Then in Sections 8 and 9 we prove Theorem 1.3, under hypotheses (i) and (ii), respectively.

We end in Section 10 by stating some open problems related to our three main theorems.

2. Proof of Theorem 1.1 for symmetric groups

For background on Sylow subgroups of symmetric groups and their construction as product of iterated wreath products, we refer the reader to [17, (4.1.20)].

We need the following notation. Let 𝔖_{A} denote the symmetric group on a finite set A. For each A ⊂ ℕ such that A has size p^{d} where d ∈ ℕ, let E_{A} denote a fixed elementary abelian subgroup of 𝔖_{A} of order p^{d}. All such subgroups are permutation isomorphic to the regular representation of an elementary abelian group of order p^{d}, and so are conjugate in 𝔖_{A}. We highlight that the precise choice of E_{A} is irrelevant.

The main work in the proof of Theorem 1.1 comes in the case when n = mp for some m ∈ ℕ. Suppose that

(1)  n = m_{1}p + m_{2}p^{2} + ⋯ + m_{r}p^{r}

where m_{i} ∈ ℕ_{0} for each i and m_{r} ≠ 0. Note that this is not necessarily the p-adic expansion of n. For i ∈ {1, …, r} and each j such that 1 ≤ j ≤ m_{i}, let

z_{j}^{i} = (j − 1)p^{i} + ∑_{s=1}^{i−1} m_{s}p^{s}

and let B_{j}^{i} = {z_{j}^{i} + 1, …, z_{j}^{i} + p^{i}}. Note that |B_{j}^{i}| = p^{i} for all j such that 1 ≤ j ≤ m_{i}. If m_{i} = 0 then B_{j}^{i} = ∅ and E_{B_{j}^{i}} is the trivial group. For each expression for n as in (1), define an elementary abelian p-subgroup of 𝔖_{n} by

E(m_{1}, …, m_{r}) = ∏_{i=1}^{r} ∏_{j=1}^{m_{i}} E_{B_{j}^{i}}.

Proposition 2.1 ([2, 1.3 Chapter VI]). The p-subgroups E(m_{1}, …, m_{r}) where m_{1}, …, m_{r} satisfy (1) form a complete irredundant set of maximal elementary abelian p-subgroups of 𝔖_{mp} up to 𝔖_{mp}-conjugacy.

In fact each E(m_{1}, …, m_{r}) is maximal even as an abelian subgroup of 𝔖_{mp}; we leave this remark as an exercise to the reader.

The subgroup E(m) generated by m disjoint p-cycles is critical to our argument. To emphasise the role played by this subgroup, we denote it by C(m) below. Informally stated, the next lemma says that C(m) is the normal base group in any Sylow p-subgroup that contains it.

Lemma 2.2. If Q is a p-subgroup of 𝔖_{mp} containing C(m) then C(m)⊴Q. Moreover Q/C(m) acts faithfully on the orbits of C(m).

Proof. Let P be a Sylow p-subgroup of 𝔖_{mp} containing Q. By the construction of Sylow p-subgroups in [17, (4.1.20)], P has a normal base group B generated by m disjoint p-cycles τ_{1}, …, τ_{m}. If σ ∈ P then σ permutes the orbits of the τ_{k} as blocks for its action. It follows that if σ ∈ C(m) is a pcycle then σ ∈ ⟨τ_{k}⟩ for some k. Hence C(m) = B and C(m) ⊴ P. Moreover, if σ ∈ P fixes each orbit of C(m) setwise then σ is a product of disjoint p-cycles each contained in C(m). Therefore P/C(m) acts faithfully on the orbits of C(m). □

In particular, Lemma 2.2 determines how an elementary abelian p-subgroup that normalizes C(m) can act on the orbits of C(m).

Lemma 2.3. Let d ∈ ℕ and let σ ∈ 𝔖_{p^{d}}. Let C(p^{d−1}) be as defined when n = p^{d} and let X be the set of orbits of C(p^{d−1}). Let A = {1, 2, …, p^{d}}. Suppose that a p-subgroup P of 𝔖_{p^{d}} contains both ^{σ}E_{A} and C(p^{d−1}). Then σE_{A}C(p^{d−1})/C(p^{d−1}) acts regularly on X.

Proof. By Lemma 2.2 in the case n = p^{d}, the subgroup C(p^{d−1}) is normal in P and ^{σ}E_{A}C(p^{d−1})/C(p^{d−1}) acts faithfully on X. Since E_{A} acts transitively on {1, …, p^{d}}, the action of ^{σ}E_{A}C(p^{d−1})/C(p^{d−1}) on X is transitive. Since E_{A} is abelian, it follows that ^{σ}E_{A}C(p^{d−1})/C(p^{d−1}) ≅ ^{σ}E_{A}/(C(p^{d−1}) ∩^{σ}E_{A}) is also abelian and hence it acts regularly on X. □

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1 when G = 𝔖_{n}. We prove the theorem by induction on n. Let Q ≤ 𝔖_{n} be a p-subgroup containing an 𝔖_{n}-conjugate of every elementary abelian p-subgroup of 𝔖_{n}. Let n = mp + c where m ∈ ℕ and 0 ≤ c < p. If c ≥ 1 then by conjugating Q by an element of 𝔖_{n} we may assume that Q ≤ 𝔖_{mp}. By induction we get that Q is a Sylow p-subgroup of 𝔖_{mp}, and hence a Sylow p-subgroup of 𝔖_{n}. We may therefore assume that n = mp, and, by induction, that the theorem holds true for 𝔖_{m}.

By replacing Q with a 𝔖_{mp}-conjugate, we may assume that C(m) is contained in Q. Let X = {B_{1}^{1}, …, B_{m}^{1}} be the set of orbits of C(m). By Lemma 2.2 we have that C(m) ⊴ Q and Q/C(m) acts faithfully on X.

Let \overline{H} ≤ 𝔖_{X} denote the image of a subgroup H of Q under the quotient map Q → Q/C(m), regarding Q/C(m) as a subgroup of 𝔖_{X}. We aim to prove that \overline{Q} contains a 𝔖_{X}-conjugate of every elementary abelian psubgroup of 𝔖_{X}.

Let m = ℓ_{0} + ∑_{i=1}^{t} ℓ_{i}p^{i} such that ℓ_{t} ≠ 0. By assumption there exists σ ∈ 𝔖_{mp} such that Q contains ^{σ}E(ℓ_{0}, …, ℓ_{t}). (Note that because of the index shift, this subgroup has exactly ℓ_{i−1} orbits of size p^{i} on {1, …, mp}.) We have

σE(ℓ_{0}, …, ℓ_{t}) = ∏_{i=1}^{t+1} ∏_{j=1}^{ℓ_{i−1}} E_{σB_{j}^{i}}.

Fix i ∈ {1, …, t + 1} and j ∈ {1, …, ℓ_{i−1}}. Let Ω_{j}^{i} be the set of all k ∈ {1, …, m} such that the orbit B_{k}^{1} of C(m) is contained in the support of E_{σB_{j}^{i}}. Let X_{j}^{i} = {B_{k}^{1} : k ∈ Ω_{j}^{i}}. Thus X is the disjoint union of the X_{j}^{i}, the support of E_{σB_{j}^{i}} is ⋃_{k∈Ω_{j}^{i}} B_{k}^{1} and \overline{E_{σB_{j}^{i}}} acts on X_{j}^{i}. By Lemma 2.3 this action is regular. It follows that the elementary abelian p-subgroup

\overline{^{σ}E(ℓ_{0}, …, ℓ_{t})} ≤ 𝔖_{X}

has exactly ℓ_{i} orbits on X of size p^{i} for each i ∈ {1, …, t + 1}. Since the ℓ_{i} were freely chosen, we see that \overline{Q} contains a 𝔖_{X}-conjugate of every elementary abelian p-subgroup of 𝔖_{X}. It now follows by induction that \overline{Q} acts on X as a Sylow p-subgroup of 𝔖_{X}.

If p^{a} is the order of a Sylow p-subgroup of 𝔖_{m} then

|Q| = |C(m)| |\overline{Q}| = p^{m+a},

which is the order of a Sylow p-subgroup of 𝔖_{mp}. The theorem follows. □

3. Proof of Theorem 1.1 for alternating groups

Case p ≥ 3. In this case any p-subgroup of 𝔖_{n} is a p-subgroup of 𝔄_{n}. Hence if Q ≤ 𝔄_{n} contains an 𝔄_{n}-conjugate of every elementary abelian psubgroup of 𝔄_{n} then Q contains an 𝔖_{n}-conjugate of every elementary abelian p-subgroup of 𝔖_{n}, and so by Theorem 1.1 for 𝔖_{n}, we get that Q is a Sylow p-subgroup of 𝔖_{n}.

Case p = 2. Let n = 2m. We need the classification of all maximal elementary abelian 2-subgroups of 𝔄_{n} up to 𝔖_{n}-conjugacy. Suppose that

(2)  n = 2m_{1} + 2^{2}m_{2} + ⋯ + 2^{r}m_{r}

where m_{i} ∈ ℕ_{0} for each i, m_{r} ≠ 0 and m_{1} ≠ 2. For each such expression define

F (m_{1}, m_{2}, …, m_{r}) = E(m_{1}, m_{2}, …, m_{r}) ∩ 𝔄_{n}.

The reason for excluding the case m_{1} = 2 is that E(2, m_{2}, …, m_{r}) ∩ 𝔄_{n} is properly contained in F (0, m_{2}+1, m_{3}, …, m_{r}). Note also that F (m_{1}, m_{2}, …, m_{r}) has exactly two fixed points on {1, …, n} if m_{1} = 1.

Proposition 3.1 ([1, Proposition 5.2]). The subgroups F (m_{1}, …, m_{r}) where m_{1}, …, m_{r} satisfy (2) form a complete irredundant set of maximal elementary abelian 2-subgroups of 𝔄_{2m} up to 𝔖_{2m}-conjugacy.

In [1] the authors remark that the proposition follows from the analogous result for 𝔖_{n}. We take this opportunity to provide a complete proof. The following lemma is required.

Lemma 3.2. Suppose that m_{1}, …, m_{r} are as in (2) and m′_{1}, …, m′_{s} are as in (1). If F (m_{1}, …, m_{r}) ≤ ^{σ}E(m′_{1}, …, m′_{s}) for some σ ∈ 𝔖_{2m} then r = s and m_{i} = m′_{i} for all i ∈ {1, …, r}.

Proof. If m_{1} = 0 then F (0, m_{2}, …, m_{r}) = E(0, m_{2}, …, m_{r}) is maximal and by Proposition 2.1 we have m_{i} = m′_{i} for all i ∈ {2, …, r}. Hence m′_{1} = 0 and the lemma holds in this case. If m_{1} = 1 then F (1, m_{2}, …, m_{r}) is equal to the subgroup E(0, m_{2}, …, m_{r}). Since E(0, m_{2}, …, m_{r}) has exactly two fixed points and no orbit of size 2 on {1, …, 2m}, we have m′_{1} = 1. It follows that σ fixes the set {1, 2} and hence ^{σ}E(1, m_{2}, …, m_{r}) ≤ E(m′_{1}, …, m′_{s}). We may again apply Proposition 2.1.

Suppose that m_{1} ≥ 3. Let τ_{k} = (2k − 1, 2k)(2k + 1, 2k + 2) for k ∈ {1, …, m_{1} − 1}. Let B_{j}^{2} for j ∈ {1, …, m′_{2}} be as defined after (1) for the subgroup E(m′_{1}, …, m′_{s}). If there exist j and k such that τ_{k} ∈ ^{σ}E_{B_{j}^{2}} then σ(B_{j}^{2}) = {2k − 1, 2k, 2k + 1, 2k + 2} and an element of ^{σ}E(m′_{1}, …, m′_{s}) moves 2k − 1 if and only if it moves 2k + 1. But if k > 1 then τ_{k−1} ∈ F (m_{1}, …, m_{r}) and otherwise k = 1 and τ_{2} ∈ F (m_{1}, …, m_{r}), so this is a contradiction. Hence all the transpositions (2k − 1, 2k) for 1 ≤ k ≤ m_{1} lie in E(m′_{1}, …, m′_{s}) and so m′_{1} ≥ m_{1}. On the other hand, F (m_{1}, …, m_{r}) has m_{1} orbits of size 2 on {1, …, 2m} and ^{σ}E(m′_{1}, …, m′_{r}) has m′_{1} orbits of size 2 on {1, …, 2m}; since F (m_{1}, …, m_{r}) ≤ ^{σ}E(m′_{1}, …, m′_{r}) we have m_{1} ≥ m′_{1}. Therefore m_{1} = m′_{1}. Hence there exist τ, τ′ ∈ 𝔖_{2m} such that

F (m_{1}, m_{2}, …, m_{r}) = F (m_{1}) ×^{τ}F (0, m_{2}, …, m_{r})
E(m′_{1}, m′_{2}, …, m′_{s}) = ^{σ}E(m_{1}) ×^{τ′}E(0, m′_{2}, …, m′_{s})

and ^{τ}F (0, m_{2}, …, m_{r}) ≤ ^{τ′}E(0, m′_{2}, …, m′_{s}). Both groups have support of size 2m−2m_{1}. Hence, by the special case m_{1} = 0 we get r = s and m_{i} = m′_{i} for all i ∈ {2, …, r}. □

Proof of Proposition 3.1. Let F be an elementary abelian p-subgroup of 𝔄_{2m}. By Proposition 2.1 there exist m′_{1}, …, m′_{s} and σ ∈ 𝔖_{2m} such that

F ≤ ^{σ}E(m′_{1}, …, m′_{s}) ∩ 𝔄_{2m} = ^{σ}(E(m′_{1}, …, m′_{s}) ∩ 𝔄_{2m}).

If m′_{1} ≠ 2 then E(m′_{1}, …, m′_{s}) ∩ 𝔄_{2m} = F (m′_{1}, …, m′_{s}). In the remaining case we have E(2, m′_{2}, …, m′_{s}) ∩ 𝔄_{2m} ≤ F (0, m′_{2} + 1, …, m′_{s}). Suppose further that F = F (m_{1}, …, m_{r}). By Lemma 3.2, we have r = s and m_{i} = m′_{i} for all i ∈ {1, …, r}. Therefore the subgroups F (m_{1}, …, m_{r}) for m_{1}, …, m_{r} satisfying (2) form a complete set of maximal elementary abelian 2-subgroups of A_{2m} up to 𝔖_{2m}-conjugacy. It is clear from the orbits of these subgroups on {1, …, 2m} that no two of them are conjugate. □

We are now ready to prove Theorem 1.1 in the alternating group case.

Proof of Theorem 1.1 when G = 𝔄_{n} and p = 2. As in the proof of Theorem 1.1 for 𝔖_{n} we reduce to the case when n = 2m. The cases m = 1 and m = 2 are easily checked, so we may assume that m ≥ 3.

Let Q ≤ 𝔄_{2m} be a 2-subgroup containing an 𝔄_{2m}-conjugate of every elementary abelian 2-subgroup of 𝔄_{2m}. Without loss of generality we may assume that F (m) is contained in Q. Let P be a Sylow 2-subgroup of 𝔖_{2m} containing Q. As in the proof of Lemma 3.2, let τ_{k} = (2k − 1, 2k)(2k + 1, 2k + 2) ∈ Q for 1 ≤ k < m. Let (x, y) be a transposition in P with x < y. If y ≠ x + 1 then there exists k such that exactly one of x, y lies in {2k − 1, 2k, 2k + 1, 2k + 2}. It is easily checked that in this case [(x, y), τ_{k}] is a 3-cycle. So y = x + 1. Moreover, if x = 2k and y = 2k + 1 for some k then either k > 1 and (2k, 2k + 1)τ_{k−1} contains a 3-cycle, or k = 1 and (2, 3)τ_{2} contains a 3-cycle. It follows that the transpositions in P are of the form (2k − 1, 2k) for k ∈ {1, …, m}. Therefore F (m) ≤ C(m) ≤ P.

By Lemma 2.2, C(m) ⊴ P. In particular, Q ≤ N_{𝔖_{2m}}(C(m)). Since QC(m)/C(m) ≅ C(m)/(C(m) ∩ Q) = C(m)/F (m), we see that QC(m) = ⟨Q, (1, 2)⟩ is a 2-group having Q as a subgroup of index 2.

Let E = E(m_{1}, m_{2}, …, m_{r}) be a maximal elementary abelian 2-subgroup of 𝔖_{2m}. Let F = E(m_{1}, m_{2}, …, m_{r}) ∩ 𝔄_{2m}. By hypothesis there exists σ ∈ 𝔖_{2m} such that ^{σ}F ≤ Q. If m_{1} = 0 then ^{σ}E = ^{σ}F ≤ Q. Suppose that m_{1} ≥ 1. Let Ω = {σ(2m_{1} + 1), …, σ(2m)} and let 2i − 1 ≤ σ(1) ≤ 2i. Since σ(1) ∉ Ω, similar arguments to those used earlier in the proof show that {2i − 1, 2i} ∩ Ω = ∅. Again similar arguments show that either

(a) (σ(1), σ(2)) = (2i − 1, 2i), or

(b) m_{1} = 2 and (2i − 1, 2i) is either (σ(1), σ(3)) or (σ(1), σ(4)). So

⟨(2i − 1, 2i),^{σ}F ⟩ = \{ \begin{matrix}^{σ}E & in case (a), \\ ^{σ}(R × E(0, m_{2}, …, m_{r})) & in case (b), \end{matrix}

for some Sylow 2-subgroup R of 𝔖_{4}. In either case, ⟨(2i−1, 2i),^{σ}F ⟩ contains a 𝔖_{2m}-conjugate of E. Clearly, ⟨(2i − 1, 2i),^{σ}F ⟩ ≤ QC(m). It follows that QC(m) contains all maximal elementary abelian 2-subgroups of 𝔖_{2m} up to 𝔖_{2m}-conjugacy. By Theorem 1.1 for 𝔖_{n}, we see that QC(m) is a Sylow 2-subgroup of 𝔖_{2m}. Hence QC(m) ∩ 𝔄_{2m} = Q is a Sylow 2-subgroup of 𝔄_{2m}. □

We note that this proof only used the weaker hypothesis that Q contains an 𝔖_{2m}-conjugate of every elementary abelian 2-subgroup of 𝔄_{2m}. As remarked in [1], the 2-subgroups E(0, 0, m_{3}, …, m_{r}) are not normalized by any odd elements in 𝔄_{2m}, and so there are two 𝔄_{2m}-conjugacy classes of such subgroups; in all other cases the 𝔄_{2m}- and the 𝔖_{2m}-classes agree. This may be proved as follows:^{1} if A = {1, …, 2^{d}} then N_{S_{2^{d}}} (E_{A}) is permutation isomorphic to the affine general linear group AGL_{d}(𝔽_{2}); this group is generated by translations and transvections, both of which are even permutations of 𝔽_{2}^{d} when d ≥ 3. The normalizer of E(0, 0, m_{3}, …, m_{r}) factors as a direct product of wreath products N_{S_{2^{d}}} (E_{A_{i}}) ≀ S_{m_{i}} where A_{i} has size 2^{i}. The permutations in a copy of the top group 𝔖_{m_{i}} act on blocks of size 2^{i} and so are even. Hence N_{S_{2^{d}}} (E(0, 0, m_{3}, …, m_{r})) ≤ 𝔄_{2m}. It is easily seen that ⟨(12)(34)⟩ and ⟨(12)(34), (13)(24)⟩ are normalized by odd permutations in 𝔖_{4}, so these are the only exceptional cases.

4. Proof of Theorem 1.2 from Theorem 1.3

In this section, we deduce Theorem 1.2 from Theorem 1.3 using Theorem 1.1.

Let \overline{F} be the algebraic closure of F. By [15, Theorem 1.21], for any F G-module M and H ≤ G, we have that \overline{F} ⊗_{F} M is a direct summand of \overline{F} ⊗_{F} (M↓_{H}↑^{G}) ≅ (\overline{F} ⊗_{F} M)↓_{H}↑^{G} if and only if M is a direct summand of M↓_{H}↑^{G}. Thus the vertices of an indecomposable F G-module do not change under field extensions of F.

Without loss of generality, we may assume that F is an algebraically closed field. Let Q be a vertex of the Specht module S^{(kp−p,1^{p})} where k ≡ 1 mod p and k ≢ 1 mod p^{2}. By Theorem 1.1, it is sufficient to prove that if

kp = m_{1}p + m_{2}p^{2} + ⋯ + m_{r}p^{r}

where m_{i} ∈ ℕ_{0} for each i ∈ {1, …, r} and m_{r} ≠ 0 then Q contains a conjugate of E(m_{1}, m_{2}, …, m_{r}). If m_{2} ≠ 0 then Theorem 1.3(ii) applies. If m_{2} = 0 then k = m_{1} + m_{3}p^{2} + ⋯ + m_{r}p^{r−1} and so k ≡ m_{1} mod p^{2}. Hence m_{1} ≥ 2 and Theorem 1.3(i) applies. This completes the proof.

5. Hook-Specht modules

For maximal generality we work in this section with modules over an arbitrary commutative ring K. The definition of Specht modules given in [16, Definition 4.3] extends easily to this setting. Fix n ∈ ℕ. Let N = ⟨e_{1}, …, e_{n}⟩ be the K-free natural permutation module for KS_{n} of rank n. Then, by definition, S^{(n−1,1)} is the K-free submodule of N with K-basis {e_{2} − e_{1}, …, e_{n} − e_{1}}.

1The proof in [1] is indicated very briefly, but seems to incorrectly assume that if a subgroup of 𝔄_{n} is normalized by an odd element then it is normalized by a transposition.

We begin by establishing the isomorphism of K𝔖_{n}-modules S^{(n−r,1^{r})} ≅ ⋀^{r} S^{(n−1,1)}. This isomorphism can be found, most obviously in the case when K is a finite field of characteristic 2, in Peel’s papers [22, Section 6] and [23]. When K is a field of prime characteristic p and n = p it was proved by Hamernik [13]; it is easily seen that Hamernik’s proof also works in the general case. When K is a field it was proved by Müller and Zimmermann in [20, Proposition 23(a)]. The proof given here combines ideas from both [13] and [20].

Proposition 5.1. Let 1 ≤ r ≤ n − 1.

(i) The set

{(e_{i_{1}} − e_{1}) ∧ ⋯ ∧ (e_{i_{r}} − e_{1}) : 1 < i_{1} < ⋯ < i_{r} ≤ n}

is a K-basis of ⋀^{r} S^{(n−1,1)}. (ii) The map sending (e_{i_{1}} − e_{1}) ∧ ⋯ ∧ (e_{i_{r}} − e_{1}) ∈ ⋀^{r} S^{(n−1,1)} to e_{𝔱} ∈

S^{(n−r,1^{r})}, where 1 < i_{1} < ⋯ < i_{r} ≤ n and 𝔱 is the unique standard tableau of shape (n − r, 1^{r}) having i_{1}, …, i_{r} in its rows of length one, is an isomorphism of K𝔖_{n}-modules.

Proof. Part (i) is obvious from the basis of S^{(n−1,1)} above. By (i) and the Standard Basis Theorem for Specht modules (see [16, Lemma 8.2 and Corollary 8.9]), the map defined in (ii) is a K-linear isomorphism. So all we have to check is that it commutes with the action of 𝔖_{n}. For the subgroup of 𝔖_{n} fixing 1 this is obvious. So it suffices to check the action of the permutation (12). Let w = (e_{i_{1}} − e_{1}) ∧ ⋯ ∧ (e_{i_{r}} − e_{1}) where 1 < i_{1} < ⋯ < i_{r} ≤ n. If i_{1} = 2 then it is clear that (12)e_{𝔱} = −e_{𝔱} and (12)w = −w. Suppose that i_{1} ≠ 2. For each 1 ≤ a ≤ r, let 𝔱_{a} be the standard tableau having 2, i_{1}, …, î_{a}, …, i_{r} in its rows of length 1, where the hat over i_{a} indicates this entry is omitted. By the Garnir relation (see [16, Theorem 7.2]) involving all entries in the first column of 𝔱, and the single box in the second column of 𝔱, we have

(12)e_{𝔱} = e_{𝔱} − e_{𝔱_{1}} + e_{𝔱_{2}} − ⋯ + (−1)^{r}e_{𝔱_{r}}.

On the other hand, the action on the wedge product is given by

(12)w = ((e_{i_{1}} − e_{1}) − (e_{2} − e_{1})) ∧ ⋯ ∧ ((e_{i_{r}} − e_{1}) − (e_{2} − e_{1}))

= (e_{i_{1}} − e_{1}) ∧ ⋯ ∧ (e_{i_{r}} − e_{1})

+ ∑_{a=1}^{r}(−1)^{a}(e_{2} − e_{1}) ∧ (e_{i_{1}} − e_{1}) ∧ ⋯ ∧ { \atop (e_{i_{a}} − e_{1})} ∧ ⋯ ∧ (e_{i_{r}} − e_{1}).

The proof is now complete. □

We now introduce some further ideas from [13] and basic simplicial homology. For r ∈ ℕ define δ_{r} : ⋀^{r} N → ⋀^{r−1} N by

δ_{r}(e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}}) = ∑_{a=1}^{r}(−1)^{a−1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{a}} ∧ ⋯ e_{i_{r}}.

The subscript r in the map δ_{r} will be omitted when it is clear from the context. By definition ⋀^{0} N = K. We leave it to the reader to verify the relation

(3)  (e_{i_{1}} − e_{j}) ∧ ⋯ ∧ (e_{i_{r}} − e_{j}) = δ(e_{j} ∧ e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}})

for 1 ≤ j, i_{1}, …, i_{r} ≤ n.

Proposition 5.2 (Long exact sequence). The sequence

0 ⟶ ⋀^{n} N ⟶^{δ_{n}} ⋯ ⟶^{δ_{r+1}} ⋀^{r} N ⟶^{δ_{r}} ⋀^{r−1} N ⟶^{δ_{r−1}} ⋯ → N ⟶^{δ_{1}} K ⟶ 0

is exact. Moreover if 1 ≤ r < n then ker δ_{r} = im δ_{r+1} = ⋀^{r} S^{(n−1,1)}.

Proof. The K-linear map defined by e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} ↦ e_{1} ∧ e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} defines a homotopy equivalence between the identity map on the sequence above and the zero map. This shows that the sequence is null-homotopic and hence exact. Proposition 5.1(i) and Equation (3) imply that im δ_{r+1} is equal to ⋀^{r} S^{(n−1,1)}. □

We remark that there is an important homological interpretation of Proposition 5.2. Let e_{1}, …, e_{n} be the canonical basis of ℝ^{n} and fix an (n − 1)- simplex in ℝ^{n} with geometric vertices e_{1}, …, e_{n}. The wedge product e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} can then be identified with the oriented (r − 1)-simplex with vertices, in order, e_{i_{1}}, …, e_{i_{r}}. For each r ≥ 2, the map δ_{r} : ⋀^{r} N → ⋀^{r−1} N is the boundary map from simplicial homology, acting on oriented (r − 1)- simplices. Replace the map δ_{1} : N → K with the zero map N → 0. Then Proposition 5.2 is equivalent to the fundamental result that the solid (n−1)- simplex has trivial homology (with coefficients in K) in all non-zero degrees, and its zero homology group is N/ ker δ_{1} ≅ K.

We end this section with some further notation and results that are used in Section 9. Let r ∈ ℕ. Let

I^{(r)} = {(i_{1}, …, i_{r}) : 1 ≤ i_{1} < ⋯ < i_{r} ≤ n}.

We call the elements of I^{(r)} multi-indices. Let

J^{(r)} = {i ∈ I^{(r)} : i_{1} > 1}.

For i ∈ I^{(r)}, let e_{i} = e_{i_{1}} ∧⋯∧e_{i_{r}}. We say that {e_{i} : i ∈ I^{(r)}} is the monomial basis of ⋀^{r} N. By (3), the set

(4)  {δ(e_{1} ∧ e_{j}) : j ∈ J^{(r)}}

is a K-basis for ⋀^{r} S^{(n−1,1)}, corresponding under the isomorphism in Proposition 5.1 to the standard basis of S^{(n−r,1^{r})}. The following lemma gives a very useful way to express elements of ⋀^{r} S^{(n−1,1)} in this basis.

Lemma 5.3 (Rewriting Lemma). Let u = ∑_{i∈I^{(r)}} μ_{i}e_{i} ∈ ⋀^{r} S^{(n−1,1)}. Then

u = ∑_{j∈J^{(r)}} μ_{j}δ(e_{1} ∧ e_{j}).

Proof. By (4) we may write

u = ∑_{j∈J^{(r)}} ν_{j}δ(e_{1} ∧ e_{j})

for some ν_{j} ∈ K. Each of the monomial summands in δ(e_{1} ∧ e_{j}) involves e_{1}, with the sole exception of the first summand e_{j}. Therefore the coefficient of the monomial e_{j} in the right-hand side of the above equation is ν_{j}. Hence ν_{j} = μ_{j}, as required. □

Finally we note that if u ∈ ⋀^{r} N and v ∈ ⋀^{s} N, where r, s ∈ ℕ, then

(5)  δ(u ∧ v) = δ(u) ∧ v + (−1)^{r}u ∧ δ(v).

6. The Brauer homomorphism and monomial modules

Throughout this section let F be a field of prime characteristic p and let G be a finite group.

6.1. The Brauer homomorphism. Let V be an F G-module. For Q ≤ G, define

V^{Q} = {v ∈ V : σv = v for all σ ∈ Q}.

For R ≤ Q ≤ G, the relative trace map Tr_{R}^{Q} : V^{R} → V^{Q} is the linear map defined by

Tr_{R}^{Q}(v) = ∑_{σ} σv

where the sum is over a set of coset representatives for Q/R. The Brauer kernel of V with respect to Q is the subspace ∑_{R<Q} Tr_{R}^{Q} V^{R} of V^{Q}. The Brauer quotient of V with respect to Q is

V (Q) = V^{Q}/ ∑_{R<Q} Tr_{R}^{Q} V^{R}.

It is easy to show that V^{Q} and ∑_{R<Q} Tr_{R}^{Q} V^{R} are both N_{G}(Q)-invariant, and so V (Q) is a module for F N_{G}(Q)/Q.

The following proposition is proved in [5, (1.3)].

Proposition 6.1. Let V be an indecomposable F G-module. Let Q be a p-subgroup of G. If V (Q) ≠ 0 then V has a vertex containing Q.

The converse to Proposition 6.1 is true when V has the trivial module as its source, or, equivalently, if V is a direct summand of a permutation module. In general it is false. For example, calculations using Magma show that if p = 2 and P is a Sylow 2-subgroup of 𝔖_{6} then S^{(4,1,1)}(P) = 0. But by [21, Theorems 4.3, 4.5], S^{(4,1,1)} is indecomposable with vertex P.

6.2. Monomial modules. Let H be a subgroup of G and let θ : H → F^{×} be a representation of H such that θ(h) = 1 whenever h ∈ H is a p-element. Let ⟨v⟩ be the corresponding F H-module. Let V = F G ⊗_{F H} ⟨v⟩ be the monomial module induced from ⟨v⟩. If P is a Sylow p-subgroup of H then ⟨v⟩ has P as a vertex, and so ⟨v⟩ is a direct summand of ⟨v⟩↓_{P} ↑^{H}= F↑_{P}^{H}. Hence V is a p-permutation module. The following proposition establishes a close connection between V and the permutation module F↑_{H}^{G}.

Proposition 6.2. Let W = ⟨e_{γH} : γ ∈ G⟩ be the permutation module of G acting on the cosets of H. Let Q be a p-subgroup of G. Let γ_{1}H, …, γ_{L}H be representatives for the orbits of Q on G/H. For each ℓ let s_{ℓ} = ∑_{τ} τγ_{ℓ} ⊗ v where the sum is over a set of coset representatives τ for Q/ Stab_{Q}(γ_{ℓ}H). Then

(i) the map defined by αγ_{ℓ} ⊗ v ↦ αe_{γ_{ℓ}H} for α ∈ Q is an isomorphism V ↓_{Q} ≅ W↓_{Q},

(ii) s_{ℓ} = Tr_{Stab_{Q}(γ_{ℓ}H)}^{Q}(γ_{ℓ} ⊗ v), (iii) {s_{ℓ} : 1 ≤ ℓ ≤ L} is a basis for V^{Q}, (iv) if R ≤ Q then {s_{ℓ} : 1 ≤ ℓ ≤ L, Stab_{Q}(γ_{ℓ}H) = Stab_{R}(γ_{ℓ}H)} is a

basis for Tr_{R}^{Q} V^{R}.

Proof. Part (i) follows from Mackey’s induction/restriction formula on noting that if γ ∈ G then the restriction of^{γ}θ to^{γ}H∩Q is trivial. The orbit sums of Q acting on the canonical permutation basis of W are a basis for W^{Q}. Since the orbit sum containing e_{γ_{ℓ}H} is Tr_{Stab_{Q}(γ_{ℓ}H)}^{Q} e_{γ_{ℓ}H}, parts (ii) and (iii) follow from (i). For (iv), let γ ∈ G and note that by (iii) the orbit sum of R containing e_{γH} is w = Tr_{Stab_{R}(γH)}^{R} e_{γH}. If Stab_{R}(γH) = Stab_{Q}(γH) then Tr_{R}^{Q} w = Tr_{Stab_{Q}(γH)}^{Q} e_{γH} is the orbit sum of Q containing e_{γH}. On the other hand, if Stab_{R}(γH) < Stab_{Q}(γH) then

Tr_{R}^{Q} w = Tr_{Stab_{Q}(γH)}^{Q} Tr_{Stab_{R}(γH)}^{Stab_{Q}(γH)} e_{γH} = Tr_{Stab_{Q}(γH)}^{Q} 0 = 0.

Part (iv) now follows from the isomorphism in (i). □

7. Generic Jordan types of modules

Let E = ⟨g_{1}, …, g_{n}⟩ be an elementary abelian p-group of p-rank n and let F be an algebraically closed field of characteristic p. Let M be a finitedimensional F E-module. Let K = F (α_{1}, …, α_{n}) where α_{1}, …, α_{n} are indeterminates. With respect to a basis for M, the matrix representing the action of the element

1 + α_{1}(g_{1} − 1) + ⋯ + α_{n}(g_{n} − 1)

on M has order p. If [r] denotes a unipotent Jordan block of dimension r then the Jordan type of this matrix is [1]^{s_{1}} ⋯ [p]^{s_{p}} for some s_{i} ∈ ℕ_{0}. By Wheeler [24], this Jordan type is independent of the choice of the generators of E. It is called the generic Jordan type of the F E-module M. The stable generic Jordan type of M is [1]^{s_{1}} ⋯ [p−1]^{s_{p−1}}. The module M is generically free if s_{1} = ⋯ = s_{p−1} = 0. For further background on generic Jordan type, we refer the reader to [9].

We summarize below the properties which we need.

Proposition 7.1. Suppose that E is an elementary abelian p-group of finite order.

(i) The generic Jordan type of a direct sum of modules is the direct sum of the generic Jordan types of the modules.

(ii) Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence of F Emodules.

(a) If M_{2} is generically free then M_{1} has stable generic Jordan type [1]^{s_{1}} ⋯ [p − 1]^{s_{p−1}} if and only if M_{3} has stable generic Jordan type [1]^{s_{p−1}} ⋯ [p − 1]^{s_{1}}.

(b) If M_{3} is generically free then M_{1} and M_{2} have the same stable generic Jordan type.

(c) If M_{1} is generically free then M_{3} and M_{2} have the same stable

generic Jordan type.

(iii) Let D be a proper subgroup of E and let U be a F D-module. Then

the induced module U↑^{E} is generically free. (iv) Let G be a finite, not necessarily elementary abelian, supergroup of E

and let V be an indecomposable F G-module. If V ↓_{E} is not generically free then V has a vertex containing the subgroup E.

Proof. Part (i) follows from [9, Proposition 4.7] and will be used in the proofs of (ii), (iii) and (iv). Since we do not require these ideas elsewhere in our paper, we refer the reader to [9] for details. We write [α_{K}](M) and [α_{K}]^{⋆}(M) for the generic and stable generic Jordan type of a module M respectively.

For part (ii), we note that the short exact sequence induces the short exact sequence

0 → [α_{K}](M_{1}) → [α_{K}](M_{2}) → [α_{K}](M_{3}) → 0.

For (ii)(a) we have

[α_{K}]^{⋆}(M_{1}) ≃ Ω(Ω^{−1}[α_{K}]^{⋆}(M_{1})) ≃ Ω([α_{K}]^{⋆}(M_{3})).

Hence the stable generic Jordan types of M_{1} and M_{3} are complementary. For (ii)(b) we note that since M_{3} is generically free, the short exact sequence generically splits. Hence

[α_{K}]^{⋆}(M_{2}) ≃ [α_{K}]^{⋆}(M_{1}) ⊕ [α_{K}]^{∗}(M_{3}) = [α_{K}]^{∗}(M_{1}).

The proof of (ii)(c) is similar.

There exists a non-trivial subgroup C of E such that E = C × D. We have U↑^{E}≅ F E ⊗_{F D} U ≅ F (E/D) ⊗_{F} U′ where U′ has the same underlying vector space as U and the action of E is given by (hk)v = kv for h ∈ C, k ∈ D and v ∈ U′. Since F (E/D) is generically free as an F E-module, it follows that U↑^{E} is generically free, as required in (iii).

For (iv), suppose that V has Q as a vertex. Then there is an F Qmodule M such that V is a direct summand of M ↑^{G}. By the Mackey decomposition formula (M↑^{G})↓_{E} is a summand of a direct sum of the modules M_{g} = (^{g}M↓_{E∩gQg^{−1}})↑^{E} for some suitable g ∈ G. If E ∩ gQg^{−1} is a proper subgroup of E then M_{g} is generically free by (iii). Since V ↓_{E} is direct summand of (M ↑^{G}) ↓_{E} and, by hypothesis V ↓_{E} is not generically free, there exists g ∈ G such that E ∩ gQg^{−1} = E. For this g we have E ⊆ gQg^{−1}. □

Of independent interest, we mention a direct consequence of Proposition 7.1(iv) about hook Specht modules. Let n ∈ ℕ and let d = ⌊\frac{n}{p}⌋. Let

C(d) = ⟨(1, 2, …, p), …, ((d − 1)p + 1, (d − 1)p + 2, …, dp)⟩

be as defined in Section 2. The second author computed the generic Jordan type of S^{(n−r,1^{r})}↓_{C(d)} in [18, Corollary 4.2, Theorem 4.5]. In particular, the restricted module is not generically free. Hence by Proposition 7.1(iv), we obtain the following result.

Proposition 7.2. The hook Specht module S^{(n−r,1^{r})} has a vertex containing the maximal elementary abelian p-subgroup C(d) where d = ⌊\frac{n}{p}⌋.

We end this section with a result on the generic Jordan type of a monomial module.

Proposition 7.3. Let G be a finite group, let H ≤ G and let V = ⟨v⟩↑_{H}^{G} be a monomial F G-module. If E is an elementary abelian p-subgroup of G then V ↓_{E} has generic Jordan type [1]^{s} where s is the number of orbits of size 1 of E acting on G/H.

Proof. We have

V ↓_{E} = ⨁_{g}(^{g}⟨v⟩↓_{E ∩gHg^{−1}})↑^{E}

where the sum is over a set of representatives for the orbits of E on G/H. Orbits of size 1 correspond to representatives g such that E ⊆ gHg^{−1}. In this case,

(^{g}⟨v⟩↓_{E ∩gHg^{−1}})↑^{E} = ^{g}⟨v⟩↓_{E}.

Since ^{g}⟨v⟩↓_{E} is 1-dimensional, its generic Jordan type is clearly [1]. The result now follows from parts (i) and (iii) of Proposition 7.1. □

8. Proof of Theorem 1.3 under hypothesis (i)

Let F be an algebraically closed field of prime characteristic p. As in Section 5 we let N = ⟨v_{1}, …, v_{kp}⟩ be the kp-dimensional natural permutation module for 𝔖_{kp}. Recall from the end of Section 5 that ⋀^{r} N has as a basis the elements e_{i} = e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} for i ∈ I^{(r)}. Considering the action of H = 𝔖_{{1,...,r}} × 𝔖_{{r+1,...,kp}} on the generator e_{1} ∧ ⋯ ∧ e_{r} of ⋀^{r} N, we see that

(6)  ⋀^{r} N ≅ ⟨v⟩↑^{𝔖_{n}}

where ⟨v⟩ affords the representation sgn ⊠F of H. Hence ⋀^{r} N is a monomial module.

Corollary 8.1. Suppose that n = kp. Let E be an elementary abelian psubgroup of 𝔖_{kp}. Then the stable generic Jordan type of (⋀^{r} N)↓_{E} is [1]^{s} where s is the number of r-subsets of {1, …, kp} fixed by E in its action on all r-subsets of {1, …, kp}. In particular, (⋀^{r} N)↓_{E} is generically free if r ≢ 0 mod p.

Proof. Since ⋀^{r} N is a monomial module by (6), we may apply Proposition 7.3. Identifying 𝔖_{n}/H with the set of r-subsets of {1, …, kp} we see that orbits of E on 𝔖_{n}/H of size 1 correspond to r-subsets of {1, …, kp} fixed by E. Hence the generic Jordan type of (⋀^{r} N)↓_{E} is [1]^{s} where s is the number of r-subsets of {1, …, kp} which are fixed under the action of E. Notice that all the orbits of E on {1, …, kp} have sizes p^{j} for some j ≥ 1. Thus s = 0 if r is not divisible by p. □

We are now ready to prove Theorem 1.3 under hypothesis (i). Let E = E(m_{1}, m_{2}, …, m_{t}) where m_{1} ≥ 2. By Proposition 7.1(iv), it suffices to show that S^{(kp−p,1^{p})}↓_{E} ≅ ⋀^{p} S^{(kp−1,1)}↓_{E} is not generically free. We truncate the long exact sequence in Proposition 5.2 to obtain

0 ⟶ ⋀^{p} S^{(kp−1,1)} ⟶ ⋀^{p} N ⟶^{δ_{p}} ⋀^{p−1} N ⟶^{δ_{p−1}} ⋯ → N ⟶^{δ_{1}} F ⟶ 0.

Restrict each term in the long exact sequence to E. For each 1 ≤ i ≤ p, we have a short exact sequence

0 ⟶ ⋀^{i} S^{(kp−1,1)}↓_{E} ⟶ ⋀^{i} N↓_{E} ⟶^{δ_{i}} ⋀^{i−1} S^{(kp−1,1)}↓_{E} ⟶ 0.

By Corollary 8.1, ⋀^{i} N ↓_{E} is generically free if 1 ≤ i ≤ p − 1. Since F↓_{E} has generic Jordan type [1], it follows from Proposition 7.1(ii)(a) that S^{(kp−1,1)}↓_{E} has stable generic Jordan type [p−1]. Repeating this argument, with ⋀^{2} S^{(kp−1,1)}↓_{E} up to ⋀^{p−1} S^{(kp−1,1)}↓_{E}, we find that ⋀^{p−1} S^{(kp−1,1)} has stable generic Jordan type [1].

Suppose for a contradiction that ⋀^{p} S^{(kp−1,1)}↓_{E} is generically free. Then, by Proposition 7.1(ii)(c), ⋀^{p} N↓_{E} has stable generic Jordan type [1]. On the other hand, since E = E(m_{1}, m_{2}, …, m_{t}) where m_{1} ≥ 2, there are exactly m_{1} orbits of E on {1, …, kp} of size p. Each such orbit corresponds to a p-subset of {1, …, kp} fixed by E so, by Corollary 8.1, ⋀^{p} N↓_{E} has stable generic Jordan type [1]^{m_{1}}. The contradiction shows that S^{(kp−p,1^{p})}↓_{E} is not generically free. This completes the proof.

9. Proof of Theorem 1.3 under hypothesis (ii)

Let F be a field of prime characteristic p. Under the hypothesis (ii) of Theorem 1.3, we have that k ≥ p. As in Section 5, we let N = ⟨v_{1}, …, v_{kp}⟩ be the kp-dimensional natural permutation module for 𝔖_{kp} and define S^{(kp−1,1)} to be the submodule with F -basis {e_{2}−e_{1}, …, e_{kp}−e_{1}}. Let W = ⋀^{p} S^{(kp−1,1)}. By Proposition 5.1 we have W ≅ S^{(kp−p,1^{p})}.

Let T be a Sylow p-subgroup of the symmetric group on {p^{2} + 1, …, kp}. Let Q be the subgroup of 𝔖_{kp} generated by T together with

α = (1, 2, …, p) ⋯ (p^{2} − p + 1, p^{2} − p + 2, …, p^{2}),
β = (1, p + 1, …, p^{2} − p + 1) ⋯ (p, 2p, …, p^{2}),

and let

w = (e_{1} + e_{p+1} + ⋯ + e_{(p−1)p+1}) ∧ ⋯ ∧ (e_{p} + e_{2p} + ⋯ + e_{p^{2}}).

Since each factor in the wedge product lies in S^{(kp−1,1)}, and w is fixed by α, β and T, we have w ∈ W^{Q}. Observe that the coefficient of e_{1} ∧e_{2} ∧⋯∧e_{p} in w is 1.

The main result of this section is Proposition 9.1 below. It gives us the statement of Theorem 1.3(ii) almost immediately.

Proposition 9.1. If R < Q then no element of Tr_{R}^{Q} W^{R} has a non-zero coefficient of e_{1} ∧ e_{2} ⋯ ∧ e_{p} when expressed in the monomial basis of ⋀^{p} N.

Proof of Theorem 1.3(ii) assuming Proposition 9.1. Let E be an elementary abelian p-subgroup of 𝔖_{kp} having at least one orbit of size p^{2}. There exists σ ∈ 𝔖_{kp} such that

σEσ^{−1} ≤ ⟨α, β⟩ × T = Q.

By Proposition 9.1, we have W (Q) ≠ 0. Theorem 1.3(ii) now follows from Proposition 6.1. □

The proof of Proposition 9.1 occupies the remainder of this section. It is divided into two steps: in the first we reduce the proposition to a question about a specific relative trace map on the smaller module ⋀^{p} S^{(p^{2}−1,1)}. In the second step we answer this question (see Proposition 9.5) using a carefully chosen filtration of ⋀^{p} S^{(p^{2}−1,1)}.

Step 1: Reduction. We need the following lemma.

Lemma 9.2. Let R ≤ Q, let B^{(1)}, …, B^{(m)} be representatives for the orbits of R on the set of p-subsets of {1, …, kp} and let i^{(1)}, …, i^{(m)} be the corresponding multi-indices. Then a basis for (⋀^{p} N)^{R} is

B = {Tr_{Stab_{R}(B^{(k)})}^{R} e_{i^{(k)}} : 1 ≤ k ≤ m}.

Proof. By (6) at the beginning of Section 8, the module ⋀^{p} N is induced from the representation sgn ⊠F of H = 𝔖_{{1,...,p}} × 𝔖_{{p+1,...,kp}}. Since sgn(g) = 1 for all p-elements g ∈ 𝔖_{p}, the hypotheses of Proposition 6.2 are satisfied. The lemma now follows from parts (ii) and (iii) of this proposition, noting as in the proof of Corollary 8.1 that the coset space 𝔖_{kp}/H is isomorphic as an 𝔖_{kp}-set to the set of all p-subsets of {1, …, kp}. □

We now show that only one subgroup R needs to be considered in Proposition 9.1.

Proposition 9.3. If R is a proper subgroup of Q such that e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} appears with a non-zero coefficient in an element of Tr_{R}^{Q} W^{R}, then R = ⟨α, T ⟩.

Proof. Since W^{R} is a submodule of (⋀^{p} N)^{R}, it is sufficient to prove the proposition with W replaced by ⋀^{p} N. Let B be the basis of (⋀^{p} N)^{Q} given by Lemma 9.2. The unique element of B containing e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} with a non-zero coefficient is

Tr_{⟨α,T ⟩}^{Q}(e_{1}∧e_{2}∧⋯∧e_{p}) = e_{1}∧e_{2}∧⋯∧e_{p}+⋯+e_{(p−1)p+1}∧e_{(p−1)p+2}∧⋯∧e_{p^{2}}.

By Proposition 6.2(iv), Tr_{R}^{Q}(⋀^{p} N)^{R} has, as a basis, a subset B′ of B such that B′ contains Tr_{⟨α,T ⟩}^{Q}(e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p}) if and only if

Stab_{Q}({1, 2, …, p}) = Stab_{R}({1, 2, …, p}).

Since Stab_{Q}({1, 2, …, p}) = ⟨α, T ⟩ is a maximal subgroup of Q, this condition holds if and only if R = ⟨α, T ⟩. □

The next proposition completes the reduction step.

Proposition 9.4. If e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} appears with a non-zero coefficient in Tr_{⟨α,T ⟩}^{Q} v for some v ∈ (⋀^{p} S^{(kp−1,1)})^{⟨α,T ⟩} then e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} appears with a non-zero coefficient in Tr_{⟨α⟩}^{⟨α,β⟩} v′ for some v′ ∈ (⋀^{p} S^{(p^{2}−1,1)})^{⟨α⟩}.

Proof. Let B^{(1)}, …, B^{(L)} be representatives for the orbits of ⟨α⟩ on the set of p-subsets of {1, …, p^{2}} and let D^{(1)}, …, D^{(M)} be representatives for the orbits of ⟨α, T ⟩ on the set of p-subsets of {1, …, kp} that have non-empty intersection with {p^{2} +1, …, kp}. Let i^{(1)}, …, i^{(L)} and k^{(1)}, …, k^{(M)} be the corresponding multi-indices, and let

s_{ℓ} = Tr_{Stab_{⟨α,T ⟩}(B^{(ℓ)})}^{⟨α,T ⟩} e_{i^{(ℓ)}} for 1 ≤ ℓ ≤ L,
t_{m} = Tr_{Stab_{⟨α,T ⟩}(D^{(m)})}^{⟨α,T ⟩} e_{k^{(m)}} for 1 ≤ m ≤ M.

By Lemma 9.2, {s_{ℓ} : 1 ≤ ℓ ≤ L} is a basis for (⋀^{p}⟨e_{1}, … e_{p^{2}}⟩)^{⟨α⟩} and {s_{ℓ} : 1 ≤ ℓ ≤ L} ∪ {t_{m} : 1 ≤ m ≤ M} is a basis for (⋀^{p} N)^{⟨α,T ⟩}.

Let v = ∑_{ℓ=1}^{L} λ_{ℓ}s_{ℓ} + ∑_{m=1}^{M} μ_{m}t_{m} where the coefficients λ_{ℓ} and μ_{m} are in F. By Proposition 5.2, ⋀^{p} S^{(kp−1,1)} is the kernel of the map δ : ⋀^{p} N → ⋀^{p−1} N. There is a vector space decomposition ⋀^{p−1} N = U ⊕ U′ where

U = ⟨e_{i} : i ∈ I^{(p−1)}, 1 ≤ i_{1} < … < i_{p−1} ≤ p^{2}⟩,
U′ = ⟨e_{i} : i ∈ I^{(p−1)}, i_{p−1} > p^{2}⟩.

For each s_{ℓ} we have δ(s_{ℓ}) ∈ U. Let 1 ≤ m ≤ M. If |D^{(m)}∩{p^{2}+1, …, kp}| ≥ 2 then every monomial summand of t_{m} involves two or more of e_{p^{2}+1}, …, e_{kp}, and so δ(t_{m}) lies in U′. Suppose that D^{(m)} = {i_{1}, …, i_{p−1}, i_{p}} where 1 ≤ i_{1} ≤ … ≤ i_{p−1} ≤ p^{2} and i_{p} > p^{2}. Since Stab_{⟨α,T ⟩}({i_{1}, …, i_{p−1}}) = T, we have

t_{m} = Tr_{Stab_{T} ({i_{p}})}^{⟨α,T ⟩}((e_{i_{1}} ∧ ⋯ ∧ e_{i_{p−1}}) ∧ e_{i_{p}})
= (Tr_{1}^{⟨α⟩}(e_{i_{1}} ∧ ⋯ ∧ e_{i_{p−1}})) ∧ Tr_{Stab_{T} ({i_{p}})}^{T} e_{i_{p}}.

Since the orbit of T containing i_{p} has size a multiple of p, it now follows from (5) at the end of Section 5 that

δ(t_{m}) = δ(Tr_{1}^{⟨α⟩}(e_{i_{1}} ∧ ⋯ ∧ e_{i_{p−1}})) ∧ Tr_{Stab_{T} ({i_{p}})}^{T} e_{i_{p}}.

Hence δ(t_{m}) ∈ U′. Thus δ(∑_{ℓ=1}^{L} λ_{ℓ}s_{ℓ}) ∈ U, whereas δ(∑_{m=1}^{M} μ_{m}t_{m}) ∈ U′. Since δ(v) = 0 and U ∩ U′ = {0}, it follows that δ(∑_{ℓ=1}^{L} λ_{ℓ}s_{ℓ}) = 0. By Proposition 5.2, ∑_{ℓ=1}^{L} λ_{ℓ}s_{ℓ} ∈ ⋀^{p}(S^{(p^{2}−1,1)}). It is clear that no element of Tr_{⟨α,T ⟩}^{Q}⟨t_{1}, …, t_{M}⟩ contains the monomial e_{1} ∧ e_{2} ∧ … ∧ e_{p} with a non-zero coefficient. Therefore v′ = ∑_{ℓ=1}^{L} λ_{ℓ}s_{ℓ} has the properties required in the proposition. □

Step 2: Proof of the reduced version of Proposition 9.1. Fix u ∈ (⋀^{p} S^{(p^{2}−1,1)})^{⟨α⟩}. By Propositions 9.3 and 9.4, to prove Proposition 9.1, it suffices to show that the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} u is zero. We do this by analysing the behaviour of this trace map on a filtration of (⋀^{p} S^{(p^{2}−1,1)})^{⟨α⟩}. From now on we work inside ⋀^{p}⟨e_{1}, …, e_{p^{2}}⟩. To simplify the notation, let V = ⋀^{p} S^{(p^{2}−1,1)}, and let M = ⋀^{p}⟨e_{1}, …, e_{p^{2}}⟩. Let I = I^{(p)} and let J = J^{(p)}, where these sets of multi-indices are as defined in Section 5 when n = p^{2}. (Thus I = {(i_{1}, …, i_{p}) : 1 ≤ i_{1} < ⋯ < i_{p} ≤ p^{2}} and J = {i ∈ I : i_{1} > 1}.)

Let Δ = {1, 2, …, p}. For 0 ≤ c ≤ p let

I_{c} = {i ∈ I : i_{1}, …, i_{c} ∈ Δ, i_{c+1}, …, i_{p} ∉ Δ}

and let M_{c} = ⟨e_{i} : i ∈ I_{c}⟩. Since Δ is an orbit of α, the subspaces M_{c} are invariant under α, so as an F ⟨α⟩-module we have M = M_{0} ⊕ M_{1} ⊕ ⋯ ⊕ M_{p}. For 0 ≤ c ≤ p let

V_{c} = V ∩ (M_{c} ⊕ ⋯ ⊕ M_{p}).

Since M_{p} is spanned by e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} and δ(e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p}) ≠ 0, we have V_{p} = {0}. The required result that the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} u is zero therefore follows immediately from the case c = p − 1 of the next proposition.

Proposition 9.5. Let 0 ≤ c ≤ p − 1. There exists v_{0} ∈ V_{0}^{⟨α⟩}, … v_{c} ∈ V_{c}^{⟨α⟩} such that

(i) the coefficient of e_{1} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩}(v_{0} + ⋯ + v_{c}) is zero, (ii) u ∈ v_{0} + ⋯ + v_{c} + V_{c+1}.

We prove Proposition 9.5 by induction on c, using the following two lemmas.

Lemma 9.6. Fix 2 ≤ m ≤ p. For each X ⊆ {1, …, p} we define a p-tuple i(X) with elements in {1, …, p} ∪ {(m − 1)p + 1, …, (m − 1)p + p} by

i(X)_{a} = \begin{cases}a & if a ∈ X, \\ (m − 1)p + a & if a ∉ X. \end{cases}

Define

z_{m} = ∑_{X⊆{1,...,p}}(−1)^{|X|}e_{i(X)}.

Then (i) z_{m} ∈ V^{⟨α⟩}, (ii) z_{m} ∈ e_{(m−1)p+1} ∧ ⋯ ∧ e_{(m−1)p+p} + M_{1} ⊕ ⋯ ⊕ M_{p}, (iii) the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} z_{m} is zero.

Proof. The monomial summands in

δ(z_{m}) = ∑_{X⊆\{1,...,p\}}(−1)^{|X|} ∑_{a=1}^{p}(−1)^{a−1} e_{i(X)_{1}} ∧ … ∧ { \atop e_{i(X)_{a}}} ∧ … ∧ e_{i(X)_{p}}

are indexed by pairs (X, a) where X ⊆ {1, …, p} and 1 ≤ a ≤ p. We define an involution ⋆ on the set of such pairs by

(X, a)^{⋆} = (X △{a}, a).

where △ denotes symmetric difference. The summands for (X, a) and (X, a)^{⋆} cancel. Hence δ(z_{m}) = 0 and z_{m} ∈ V. Since αe_{i(X)} = e_{i(αX)} and obviously |αX| = |X|, we have z_{m} ∈ V^{⟨α⟩}. The unique monomial summand of z_{m} in M_{0} is e_{i(∅)} = e_{(m−1)p+1} ∧ ⋯ ∧ e_{(m−1)p+p}, so we have (ii). The two summands of z_{m} that contribute to the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} z_{m} are e_{i(∅)}, which appears with coefficient +1, and e_{i({1,2,...,p})} = e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p}, which appears with coefficient −1. Their contributions cancel, hence (iii). □

Lemma 9.7. Let 0 ≤ c ≤ p − 1 and let j = (j_{1}, …, j_{p}) ∈ J ∩ I_{c}. Define

w(j) = ∑_{ℓ=0}^{p−1} α^{ℓ}δ(e_{1} ∧ e_{j}).

Then (i) w(j) ∈ V_{c}^{⟨α⟩}, (ii) w(j) ∈ ∑_{ℓ=0}^{p−1} α^{ℓ}(δ(e_{1} ∧ e_{d}) ∧ e_{k}) + M_{c+1}, where d = (j_{1}, …, j_{c}) and

k = (j_{c+1}, …, j_{p}). (iii) the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} w(j) is zero.

Proof. Clearly w(j) ∈ V^{⟨α⟩}. The monomial summands of

α^{ℓ}δ(e_{1} ∧ e_{j}) = δ(e_{1+ℓ} ∧ e_{α^{ℓ}(j_{1})} ∧ ⋯ ∧ e_{α^{ℓ}(j_{c})} ∧ e_{α^{ℓ}(j_{c+1})} ∧ ⋯ ∧ e_{α^{ℓ}(j_{p})})

are e_{α^{ℓ}(j_{1})} ∧ ⋯ ∧ e_{α^{ℓ}(j_{p})} and e_{1+ℓ} ∧ e_{α^{ℓ}(j_{1})} ∧ ⋯ ∧ { \atop e_{α^{ℓ}(j_{a})}} ∧ ⋯ ∧ e_{α^{ℓ}(j_{p})} for 1 ≤

a ≤ p. The first summand and the summands for a such that 1 ≤ a ≤ c are in M_{c}, and the rest are in M_{c+1}, hence (i) and (ii). If e_{i} ∈ M_{b} where 1 ≤ b ≤ p − 1 then the coefficient of e_{1} ∧ … ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} e_{i} is zero. This implies that (iii) holds when 1 ≤ c ≤ p − 2.

Suppose that c = 0. Then the only contributions to the coefficient of e_{1}∧e_{2}∧⋯∧e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} w(j) can come from ∑_{ℓ=0}^{p−1} α^{ℓ}(e_{j}) when {j_{1}, …, j_{p}} = β^{k}(Δ) for some k. But then α(e_{j}) = e_{j}, and so w(j) ∈ M_{1}.

Suppose that c = p − 1. Then j = (2, 3, …, p, j_{p}) where j_{p} ∈ {(m − 1)p + 1, …, (m − 1)p + p} for some m such that 2 ≤ m ≤ p and

w(j) = δ(e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} ∧ (e_{(m−1)p+1} + ⋯ + e_{(m−1)p+p})).

It now follows from (5) at the end of Section 5 that w(j) = δ(e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p}) ∧ (e_{(m−1)p+1} + ⋯ + e_{(m−1)p+p}) ∈ M_{p−1}. This completes the proof of (iii). □

In the proof of Proposition 9.5 below we use the bilinear form on M defined on the monomial basis of M by

(e_{i}, e_{i′}) = \begin{cases}1 & if i = i′, \\ 0 & if i ≠ i′, \end{cases}

where i, i′ ∈ I.

Proof of Proposition 9.5. We work by induction on c. By induction we may assume that u ∈ V_{c}^{⟨α⟩}. For i ∈ I let μ_{i} = (u, e_{i}), so we have

u = ∑_{i∈I_{c}∪···∪I_{p}} μ_{i}e_{i}.

It suffices to find v ∈ V_{c}^{⟨α⟩} such that the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} v is zero and u − v ∈ V_{c+1}.

Case c = 0. Let

v′ = ∑_{m=2}^{p} μ_{((m−1)p+1,...,(m−1)p+p)}z_{m}

where z_{m} is as defined in Lemma 9.6. Note that the sets {(m − 1)p + 1, …, (m − 1)p + p} where 2 ≤ m ≤ p are the singleton orbits of ⟨α⟩ on the set of all p-subsets of {1, …, p^{2}}\Δ. Let B^{(1)}, …, B^{(S)} be representatives for the remaining orbits of size p, and let j^{(1)}, …, j^{(S)} ∈ J be the corresponding multi-indices. Let

v′′ = ∑_{s=1}^{S} μ_{j^{(s)}}w(j^{(s)})

where w(j) is as defined in Lemma 9.7 and let v = v′ + v′′. It follows from Lemmas 9.6 and 9.7 that v ∈ V^{⟨α⟩} and that the coefficient of e_{1} ∧e_{2} ∧⋯∧e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} v is zero.

Let j ∈ J ∩ I_{0}. Suppose that {j_{1}, …, j_{p}} is not fixed by α. There exists a unique set B^{(t)} and a unique q such that 0 ≤ q < p and α^{q}({j_{1}, …, j_{p}}) = B^{(t)}. Note that α^{q}e_{j} = ±e_{j^{(t)}} where the sign is determined by the permutation that puts (α^{q}(j_{1}), …, α^{q}(j_{p})) in increasing order. By Lemma 9.2 there is a basis of M^{⟨α⟩} in which Tr_{1}^{⟨α⟩} e_{j} is the unique basis element involving e_{j}. Since αu = u, it follows that (u, e_{j}) = (u, α^{q}e_{j}) = (u, ±e_{j^{(t)}}). We also have (v, e_{j^{(t)}}) = (v, α^{−q}e_{j^{(t)}}) = ±(v, e_{j}) where the sign is as before. By Lemmas 9.6(ii) and 9.7(ii), we have (u, e_{j^{(t)}}) = μ_{j^{(t)}} = (v, e_{j^{(t)}}). It follows that (u, e_{j}) = (v, e_{j}). Therefore u − v ∈ V_{1}.

Case 1 ≤ c ≤ p − 1. Let X be the set of all (p − c)-subsets of {1, …, p^{2}}\Δ. Each orbit of ⟨α⟩ on X has size p. Let B^{(1)}, …, B^{(S)} be representatives for these orbits and let k^{(1)}, …, k^{(S)} ∈ I^{(p−c)} be the corresponding multiindices. Define

D = {(i_{1}, …, i_{c}) ∈ I^{(c)} : i_{1}, …, i_{c} ∈ Δ\{1}}.

For each k^{(s)} define

v_{k^{(s)}} = ∑_{d∈D} μ_{d:k^{(s)}}w(d : k^{(s)})

where d : k^{(s)} ∈ I is the multi-index obtained by concatenating d and k^{(s)}. Let

v = ∑_{s=1}^{S} v_{k^{(s)}}.

It follows from parts (i) and (iii) of Lemma 9.7 that v ∈ V_{c}^{⟨α⟩} and that the coefficient of e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{p} in Tr_{⟨α⟩}^{⟨α,β⟩} v is zero. It remains to show that u − v ∈ V_{c+1}.

Let i ∈ I_{c}. There exists a unique set B^{(t)} and a unique q such that 0 ≤ q ≤ p − 1 and α^{q}({i_{c+1}, …, i_{p}}) = B^{(t)}. Let g = (g_{1}, …, g_{c}) be the multi-index corresponding to the set α^{q}({i_{1}, …, i_{c}}) ⊆ Δ. We have α^{q}e_{i} = ±e_{g:k^{(t)}}, where the sign is determined by the permutation putting (α^{q}(i_{1}), …, α^{q}(i_{p})) into increasing order. As in the previous case we have (u, e_{i}) = (u, α^{q}e_{i}) = ±(u, e_{g:k^{(t)}}) and (v, e_{g:k^{(t)}}) = (v, α^{−q}e_{g:k^{(t)}}) = ±(v, e_{i}) where the signs agree. Therefore (u, e_{i}) = (v, e_{i}), if and only if (u, e_{g:k^{(t)}}) = (v, e_{g:k^{(t)}}). By Lemma 9.7(ii) we have

(v, e_{g:k^{(t)}}) = ∑_{d∈D} μ_{d:k^{(t)}} ( ∑_{ℓ=0}^{p−1} α^{ℓ}(δ(e_{1} ∧ e_{d}) ∧ e_{k^{(t)}}), e_{g:k^{(t)}})

= ∑_{d∈D} μ_{d:k^{(t)}}(δ(e_{1} ∧ e_{d}) ∧ e_{k^{(t)}}, e_{g:k^{(t)}})

= ∑_{d∈D} μ_{d:k^{(t)}}(δ(e_{1} ∧ e_{d}), e_{g}). (7)

On the other hand, by (4) in Section 5 and Lemma 5.3 we have

u = ∑_{d∈D} μ_{d:k^{(t)}}δ(e_{1} ∧ e_{d} ∧ e_{k^{(t)}}) + u′

where u′ is an F -linear combination of elements δ(e_{1} ∧ e_{j}) of the standard basis for j ∈ J′, where

J′ = {j ∈ (I_{c} ∪ ⋯ ∪ I_{p}) ∩ J : j ≠ d : k^{(t)} for any d ∈ D}.

Note that (e_{g:k^{(t)}}, δ(e_{1} ∧ e_{j})) is zero for all j ∈ J′. (This is clear if g_{1} ≠ 1, since then (e_{g:k^{(t)}}, δ(e_{1} ∧ e_{j})) = (e_{g:k^{(t)}}, e_{j}) = 0. If g_{1} = 1 then k^{(t)} must be a subsequence of j; since j ∈ I_{c} ∪ ⋯ ∪ I_{p}, it follows that j ∈ I_{c}, all the entries of j not in k^{(t)} lie in Δ and so j = d : k^{(t)} where d ∈ D. But then j ∉ J′.) Hence (e_{g:k^{(t)}}, u′) = 0. It follows from (5) at the end of Section 5 that

u ∈ ∑_{d∈D} μ_{d:k^{(t)}}δ(e_{1} ∧ e_{d}) ∧ e_{k^{(t)}} + u′ + M_{c+1}

and so

(u, e_{g:k^{(t)}}) = ∑_{d∈D} μ_{d:k^{(t)}}(δ(e_{1} ∧ e_{d}), e_{g}).

Comparing this with (7) we get (v, e_{g:k^{(t)}}) = (u, e_{g:k^{(t)}}), as required. Hence u − v ∈ V_{c+1}. □

The proof of Proposition 9.1 is now complete.

10. Open problems

We end with some open problems suggested by our three main theorems.

Sylow subgroups. Say that a group G is p-elementarily large if it has the property in Theorem 1.1 that if Q is a p-subgroup of G containing a G-conjugate of every elementary abelian p-subgroup of G then Q is a Sylow p-subgroup of G. As remarked in the introduction, not every group is pelementarily large. For example, if G has a quaternionic Sylow 2-subgroup then any non-trivial 2-subgroup of G contains the unique elementary abelian 2-subgroup of G up to G-conjugacy. An abelian group is p-elementarily large if and only if its p-Sylow subgroup is elementary abelian.

Problem 10.1. Let p be a prime. Find sufficient conditions for a finite group G to be p-elementarily large. In particular, which finite simple groups are p-elementarily large?

Vertices. By Theorem 1.2, the following conjecture holds when r = p, k ≡ 1 mod p and k ≢ 1 mod p^{2}.

Conjecture 10.2. Let p be an odd prime. The hook Specht module S^{(kp−r,1^{r})} has a Sylow p-subgroup of 𝔖_{kp} as a vertex.

We hope that this conjecture will motivate new methods for computing vertices. The following two examples show some of the limitations of the main methods in this paper. Take p = 3. Let E = E(0, 0, 1); thus E is an elementary abelian 3-subgroup of 𝔖_{27} acting regularly on the set {1, …, 27}. Since the dimension of ⋀^{3} S^{(26,1)} is \binom{26}{3}, which is coprime to 3, ⋀^{3} S^{(26,1)} has a Sylow 3-subgroup of 𝔖_{27} as a vertex. However calculations using Magma show that (⋀^{3} S^{(26,1)})(E) = 0, so the Brauer correspondence fails to detect that E is contained in a vertex of S^{(24,1^{3})}. Similarly if E = E(1, 1) ≤ 𝔖_{12} then the F E(1, 1)-module S^{(9,1^{3})} ↓_{E(1,1)} is generically free, but by Theorem 1.2, S^{(9,1^{3})} has a Sylow 3-subgroup of 𝔖_{12} as a vertex.

Sources. If V is an F G-module with vertex Q and V is a direct summand of U↑^{G} where U is an indecomposable F Q-module, then U is said to be a source of V; the module U is well-defined up to conjugacy by N_{G}(Q). If n is not divisible by p then S^{(n−1,1)} is a direct summand of the permutation module ⟨e_{1}, …, e_{n}⟩ and it easily follows that ⋀^{r} S^{(n−1,1)} has trivial source whenever it is indecomposable. When p = 2 and n is even it follows from the result of Murphy and Peel mentioned in the introduction that the source of S^{(n−r,1^{r})} is S^{(n−r,1^{r})}↓_{P} where P is a Sylow 2-subgroup of 𝔖_{n}. The indirect methods used in this paper give no information about the source of S^{(kp−p,1^{p})} when p is odd.

Problem 10.3. Determine the source of S^{(kp−p,1^{p})} when p is odd.

Problem 10.4. Is there an indecomposable Specht module with a vertex properly contained in the defect group of its p-block that does not have trivial source?

References

[1] A. Adem, D. Karagueuzian, R. J. Milgram and K. Umland, The cohomology of the Lyons group and double covers of alternating groups, J. Algebra 208 (1998), no. 2, 452–479.

[2] A. Adem and R. J. Milgram, Cohomology of Finite Groups, Volume 309, Springer-Verlag 2004.

[3] J. L. Alperin, Weights for finite groups, Proc. Sympos. Pure Math. 47 (1987) 369–379.

[4] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993).

[5] M. Broué, On Scott modules and p-permutation modules: an approach through the Brauer homomorphism, Proc. Amer. Math. Soc. 93, no. 3 (1985), 401–408.

[6] S. Danz and K. Erdmann, The vertices of a class of Specht modules and simple modules for symmetric groups in characteristic 2, Algebra Colloq. 19 (2012), no. Special Issue No.1, 987–1016.

[7] S. Danz and B. Külshammer, Vertices of simple modules for symmetric groups: a survey, Proceedings of the International Conference on Modules and Representation Theory, Presa Univ. Clujeană, Cluj-Napoca, 2009, pp. 61–77.

[8] S. Donkin, Symmetric and exterior powers, linear source modules and representations of Schur superalgebras, Proc. London Math. Soc. 83 (2001), 647–680.

[9] E. Friedlander, J. Pevtsova and A. Suslin, Generic and maximal Jordan types, Invent. Math. 168 (2007), no. 3, 485–522.

[10] E. Giannelli, A lower bound on the vertices of Specht modules for symmetric groups, Arch. Math. (Basel) 103 (2014), no. 1, 1–9.

[11] J. A. Green, On the indecomposable representations of a finite group, Math. Zeitschrift 70 (1958/59), 430–445.

[12] J. A. Green, A transfer theorem for modular representations, J. Algebra 1 (1964), 73–84.

[13] W. Hamernik, Specht modules and the radical of the group ring over the symmetric group γ_{p}, Comm. Algebra 4, (1976), 435–475.

[14] D. J. Hemmer, Irreducible Specht modules are signed Young modules, J. Algebra 305 (2006), no. 1, 433–441.

[15] B. Huppert and N. Blackburn, Finite Groups II, Springer, Berlin, 1982.

[16] G. D. James. The representation theory of the symmetric groups, volume 682 of Lecture Notes in Mathematics. Springer, Berlin, 1978.

[17] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.

[18] K. J. Lim, The complexity of the Specht modules corresponding to hook partitions, Arch. Math. (Basel) 93 (2009), no. 1, 11–22.

[19] K. J. Lim, Specht modules with abelian vertices, J. Algebraic Combin. 35 (2012), no. 1, 157–171.

[20] J. Müller and R. Zimmermann, Green vertices and sources of simple modules of the symmetric group labelled by hook partitions, Arch. Math. (Basel) 89 (2007), no. 2, 97–108.

[21] G. M. Murphy and M. H. Peel, Vertices of Specht modules, J. Algebra 86 (1984), 85–97.

[22] M. H. Peel, Hook representations of the symmetric groups, Glasgow Math. J. 12 (1971), 136–149.

[23] M. H. Peel, Specht modules and symmetric groups, J. Algebra 36 (1975), 88–97.

[24] W. W. Wheeler, Generic module theory, J. Algebra 185 (1996), 205–228.

[25] M. Wildon, Two theorems on the vertices of Specht modules, Arch. Math. (Basel) 81 (2003), no. 5, 505–511.

[26] M. Wildon, Vertices of Specht modules and blocks of the symmetric group, J. Algebra 323 (2010), no. 8, 2243–2256.

(E. Giannelli) Department of Mathematics, Royal Holloway, University of London, United Kingdom.

   E-mail address: Eugenio.Giannelli.2011@live.rhul.ac.uk

(K. J. Lim) Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-03-01, 21 Nanyang Link, Singapore 637371.

   E-mail address: limkj@ntu.edu.sg

(M. Wildon) Department of Mathematics, Royal Holloway, University of London, United Kingdom.

   E-mail address: mark.wildon@rhul.ac.uk