SPECHT MODULES WITH ABELIAN VERTICES

KAY JIN LIM

Abstract. In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p^{2}- cores where p is the characteristic of the underlying field. Furthermore, in the case of p ≥ 3, or p = 2 and μ is 2-regular, we show that the complexity of the Specht module S^{μ} is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S^{(p^{p})} for p ≥ 3.

1. Introduction

The representation theory of symmetric groups has been one of the major research areas since early on in the last century. The theory has been well-developed but yet it seems that very little is known for the modular case. One way of understanding the structure of representations of finite groups is through the notion of relative projectivity, on which J. A. Green [13] defined the vertices of modules of finite groups about fifty years ago. The vertices of modules are, in some way, related to the complexity of the modules defined by J. Alperin and L. Evens [2], and the rank variety defined by J. Carlson [6].

Classically, Young modules, Specht modules and simple modules are the major objects extensively studied in the representation theory of symmetric groups. The computation of the vertices of Young modules has been done by J. Grabmeier [12]. The vertices of signed Young modules have been computed by S. Donkin [9]. However, the vertices of simple modules and Specht modules remain mostly unknown. The vertices of Specht modules of hook shape in p = 2 case were first considered by G. M. Murphy and M. H. Peel in [27], with a mistake which has been corrected by M. Wildon [31]. M. Wildon has made some progress on the computation of the vertices of Specht modules which he has computed the vertices of simple Specht modules of hook shape [31] and showed that, in general, a vertex of a Specht module contains some large p-subgroup [32]. Recently, there are some computation of the vertices of simple modules made by S. Danz, B. Külshammer, J. Müller and R. Zimmermann [8, 25].

It is well known that, in general, if the defect groups of an indecomposable module are abelian then its vertices are necessarily abelian too. In particular, for the representations of symmetric groups in the modular case, representations with abelian defect correspond to partitions of p-weights strictly less than the characteristic p of the underlying field. But there are examples of Specht modules of hook shape whose vertices are abelian but yet their defect groups are not abelian. For example, for p = 3 the Specht module S^{(7,1^{3})} has vertices the Sylow 3-subgroups of 𝔖_{6} × 𝔖_{3} (see [31, Theorem 2]) but with defect groups the Sylow 3-subgroups of 𝔖_{9}.

Date: January 2011.

2000 Mathematics Subject Classification. 20C20, 20C30.

Supported by MOE Academic Research Fund R-146-000-135-112.

In this article, we shall mainly be concerned with Specht modules with abelian vertices. Motivated by the question of classifying these Specht modules, we give some necessary conditions for such Specht modules. Indeed, for p ≥ 3, we show that no other abelian subgroups can be the vertices of Specht modules besides the elementary abelian ones. For p = 2, we could not give a definite answer as in the odd characteristic case unless the corresponding partitions are 2-regular. Under the hypothesis that the vertices of the Specht modules are abelian, a class of partitions arises naturally; namely the p^{2}-core partitions. In fact, for p = 2, we show that a 2-regular partition μ is a 4-core if and only if the Specht module S^{μ} has elementary abelian vertices. For all the Specht modules mentioned above, we conclude that their complexities are precisely the p-weights of their labeled partitions.

We organize the article in the following way. In §2, we lay down some basic knowledge about the representations of symmetric groups, the complexities and the rank varieties for modules. We state our main results in §3 and prove them in §4. In §5, we draw some consequences of the main results, in which we generalize a result of M. Wildon and show that the vertices of S^{(p^{p})} are the Sylow p-subgroups of 𝔖_{p^{2}} when p ≥ 3. This is an example where the partition (p^{p}) is a p^{2}-core but the Specht module S^{(p^{p})} does not have abelian vertices. In the last section §6, we post some questions which arise naturally from our results.

2. Preliminaries

We introduce the notation and background which we require. General references for this section are [4, 17, 19].

2.1. The representations. Let G be a finite group and F be a field of characteristic p. In this article, all F G-modules are finite dimensional vector spaces over F.

Let M be a F G-module and H be a subgroup of G. We regard the restriction Res_{H}^{G} M as the F H-module in the obvious way. If S is a F H-module we write Ind_{H}^{G} S for the induced module. Let N be another F G-module. We write N | M if N is isomorphic to a direct summand of M as F G-modules.

Suppose that p > 0. We say that the module M is relatively H-projective if M | Ind_{H}^{G} Res_{H}^{G} M. A vertex Q of an indecomposable F G-module M is a minimal subgroup of G subject to the condition that M is relatively Q-projective. Given a vertex Q of M, a source of M is an indecomposable F Q-module S such that M | Ind_{Q}^{G} S.

Let F G = I_{1} ⊕⋯⊕I_{m} be a decomposition of the F (G×G)-module F G, with the action given by (g, h)x = gxh^{−1}, into indecomposables. Suppose that 1 = ∑_{i=1}^{m} e_{i} with e_{i} ∈ I_{i} for each 1 ≤ i ≤ m. Indeed, I_{i} = e_{i}F G. The elements e_{i} are mutually orthogonal primitive central idempotents of the algebra F G and they are called the blocks of F G. For an indecomposable F G-module M, there is a unique block e_{j} such that e_{j}M ≠ 0 and e_{i}M = 0 for all i ≠ j. In this case, e_{j}M = M and we say that the module M lies in the block e_{j}. Note that F G ≅ Ind_{Δ(G)}^{G×G} F where Δ(G) is the diagonal embedding of G into G × G. Thus, for each 1 ≤ i ≤ m, a vertex of I_{i} is of the form Δ(D_{i}) for some subgroup D_{i} of G and the subgroup D_{i} is called a defect group of the block e_{i}.

Theorem 2.1 ([13]). Let F be a field of characteristic p > 0, G be a finite group and M be an indecomposable F G-module. Then

(i) any vertex of M is a p-subgroup of G, (ii) the vertices of M are conjugate to each other in G, and (iii) if M lies inside the block e_{j} of F G then M is relatively D_{j}-projective where

D_{j} is a defect group of e_{j}. In particular, a vertex of M is necessarily a subgroup of D_{j} up to conjugation.

Suppose that L is a field extension of F. We write L ⊗_{F} M for the LG-module upon field extension. In the latter part of this article, we are required to deal with vertices of modules over field extension. We include a little lemma here.

Lemma 2.2 ([11, §III Lemma 4.14]). Let L be a field extension of F and H be a p-subgroup of a finite group G. Then an F G-module M is relatively H-projective if and only if L ⊗_{F} M is relatively H-projective. In particular, if M is indecomposable and L ⊗_{F} M remains indecomposable as a LG-module then a p-subgroup Q of G is a vertex of M if and only if Q is a vertex of L ⊗_{F} M.

2.2. Rank varieties. Let p be a prime. The p-rank of a finite group G is the largest integer n subject to the condition that G contains an elementary abelian p-subgroup of order p^{n}. The p-rank of the abelian p-group ℤ_{p^{n_{1}}} ×⋯×ℤ_{p^{nm}} with n_{1}, …, n_{m} > 0 is m.

Let E be an elementary abelian p-group of p-rank n with generators g_{1}, …, g_{n}. For each non-zero point α = (α_{1}, …, α_{n}) of the space F^{n}, we write u_{α} = 1 + ∑_{i=1}^{n} α_{i}(g_{i} − 1) ∈ F E. Note that ⟨u_{α}⟩ is a cyclic p-group. Suppose that F is algebraically closed. The rank variety of a F E-module M is the set

V_{E}^{#}(M) = {0} ∪ {0 ≠ α ∈ F^{n} | M is not projective as F ⟨u_{α}⟩-module}.

The rank variety V_{E}^{#}(M) is a homogeneous and closed subvariety of the affine space 𝔸^{n}(F) [6, Theorem 4.3]. We shall write dim V_{E}^{#}(M) for the dimension of the algebraic variety V_{E}^{#}(M). The rank variety depends on the choice and order of the generators of E, but its dimension does not. In the case where p ∤ dim_{F} M, for each 0 ≠ α ∈ F^{n}, M necessarily has a summand of dimension coprime to p as F ⟨u_{α}⟩-modules. In this case, we have V_{E}^{#}(M) = V_{E}^{#}(F) and thus dim V_{E}^{#}(M) = n. Let N be another F E-module. It is clear from the definition that V_{E}^{#}(M ⊕ N) = V_{E}^{#}(M) ∪ V_{E}^{#}(N).

Let G be a finite group and M be a F G-module. Let

P : ⋯ → P_{2} → P_{1} → P_{0} → M

be a minimal projective resolution of M. Following [2], the complexity c of the module M is the smallest non-negative integer such that

lim_{n→∞} \frac{dim_{F} P_{n}}{n^{c}} = 0;

in other words, the polynomial rate growth of the dimensions of the terms of the sequence P. We write c_{G}(M) for the number c to indicate the dependence of c on M as F G-module.

Theorem 2.3.

(i) ([2, §1 Theorem], [3, Theorem 1.1] and [6, Theorem 7.6]) Let M be an F Gmodule, E be a set of representatives of elementary abelian p-subgroups of G up to conjugation and E^{max} be a set of representatives of maximal elementary abelian p-subgroups of G up to conjugation. Then

c_{G}(M) = max_{E∈E}{dim V_{E}^{#}(M)} = max_{E∈E^{max}}{dim V_{E}^{#}(M)}.

(ii) [2, Corollary 4,5] The complexity of an indecomposable F G-module M is bounded above by the p-rank of a defect group of the block in which M lies.

Following Theorem 2.3 (i), we have the conclusion that c_{G}(M) is the p-rank of G provided p ∤ dim_{F} M. We have a more refined statement of Theorem 2.3 (ii).

Proposition 2.4 ([2, Lemmas 5.2, 5.3]). Let M be a F G-module where F is a field of characteristic p > 0.

(i) Suppose that M is relatively H-projective for some subgroup H of G. Then c_{G}(M) = c_{H}(M).

(ii) Suppose that M is indecomposable. Let Q be a vertex of M with a source S. Then c_{G}(M) = c_{Q}(S).

Readers who are familiar with the variety theory for modules would have realized that the author has avoided taking cohomological variety into discussion, thanks to the Quillen stratification theorem and [3, Theorem 1.1].

2.3. The representations of symmetric groups. We now briefly go through the representations of symmetric groups. Let n be a non-negative integer. A partition μ of n is a sequence of positive integers (μ_{1}, μ_{2}, ⋯, μ_{s}) such that μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{s} and ∑_{i=1}^{s} μ_{i} = n. In this case, we write |μ| = n. Note that we allow the empty partition ∅ to be the unique partition of 0. The partition μ is called p-singular if there is some i ≥ 0 such that μ_{i+1} = μ_{i+2} = ⋯ = μ_{i+p} > 0; otherwise, it is called p-regular. Let Λ(n) be the set consisting of all partitions of n. There is a one-to-one correspondence between Λ(n) and the set of Young diagrams with n nodes in an obvious way. The Young diagram of μ is written as [μ]. Fix a positive integer m, not necessarily a prime. Each partition μ is associated to a partition μ̃ and an non-negative integer w such that |μ| = |μ̃| + mw. The partition μ̃ and the integer w is called the m-core and the m-weight of μ, respectively [17, §2.7]. The m-core of μ is obtained by successively removing w removable rim m-hooks. In the case where w = 0 or equivalently μ = μ̃, we say that the partition μ is an m-core.

We write 𝔖_{n} for the symmetric group acting on n letters. To each partition μ of n, we have the F 𝔖_{n}-module S_{F}^{μ}, the Specht module labeled by the partition μ. We usually write S^{μ} for S_{F}^{μ} if the underlying field is understood. The dimension of the Specht module S_{F}^{μ} is given by the hook formula

dim_{F} S_{F}^{μ} = \frac{n!}{∏_{(i,j)∈[μ]} h_{i,j}}

where h_{i,j} denotes the hook length of the node (i, j) of [μ] and n = |μ|. We note that the dimension of S_{F}^{μ} is independent of the field F.

Theorem 2.5. Let p be the characteristic of a field F and n be a positive integer.

(i) [17, 2.7.40] Let m be a positive integer. The number of hook lengths of a Young diagram [μ] which are divisible by m is precisely the m-weight of μ.

(ii) [19, Theorem 4.12] For p = 0, the set {S^{μ} | μ ∈ Λ(n)} is a complete set of

non-isomorphic simple F 𝔖_{n}-modules. (iii) [19, Corollary 13.18] Let μ be a partition. For p ≥ 3, or p = 2 and μ is

2-regular, the Specht module S^{μ} is indecomposable. (iv) [17, §6.1 and Theorem 6.2.45] The blocks of F 𝔖_{n} are parametrized by the

p-cores of the partitions of n such that the block eμ̃ labeled by the p-core μ̃ contains the Specht module S^{μ}. In particular, if μ, λ are partitions of n then the Specht modules S^{μ}, S^{λ} lie inside the same block of F 𝔖_{n} if and only if μ̃ = λ̃. A defect group of the block eμ̃ is conjugate to a Sylow p-subgroup of 𝔖_{wp} where w is the p-weight of μ.

We shall take a step further to discuss the vertices of a special class of modules of the symmetric groups, the Young modules [20]. Here, and hereafter, whenever we have subgroups H ⊆ 𝔖_{a}, K ⊆ 𝔖_{b} where a + b ≤ n for some positive integers a, b, n, we write H × K ↪ 𝔖_{a} × 𝔖_{b} ↪ 𝔖_{n} for the obvious inclusions.

Let μ = (μ_{1}, …, μ_{r}) be a partition of n. Let 𝔖_{μ} be the Young subgroup of 𝔖_{n}, i.e.

𝔖_{μ} = 𝔖_{μ_{1}} × 𝔖_{μ_{2}} × ⋯ × 𝔖_{μ_{r}}.

Let M^{μ} ≅ Ind_{𝔖_{μ}}^{𝔖_{n}} F be the associated permutation module. Suppose that M^{μ} = M_{1} ⊕ ⋯ ⊕ M_{k} is a decomposition of M^{μ} into indecomposable summands. Then there is a unique direct summand M_{j} of M^{μ} such that S^{μ} ⊆ M_{j}. We write Y^{μ} for M_{j} and call it a Young module. Clearly, the Young modules are well-defined up to isomorphism. Every indecomposable direct summand of M^{μ} is isomorphic to some Young module and Y^{μ} ≅ Y^{λ} if and only if μ = λ. The Young modules have trivial sources. It is well known that if μ is a p-regular partition and S^{μ} is a simple Specht module then S^{μ} ≅ Y^{μ}; see for example [7, §1].

Every partition μ can be written as its p-adic expansion as follows. Let μ_{(0)} be the partition obtained from μ by successively stripping off all horizontal p-hooks. Then we have the coordinate-wise summation μ = pμ(1) + μ_{(0)} for some partition μ(1). Inductively, we can write μ(k − 1) = pμ(k) + μ_{(k−1)} for k ≥ 2 and hence μ = p^{k}μ(k) + ⋯ + pμ_{(1)} + μ_{(0)}. The process ends if for some k, μ(k) has no removable horizontal p-hooks. Let μ_{(k)} = μ(k). Then μ = p^{k}μ_{(k)} + ⋯ + pμ_{(1)} + μ_{(0)} is the p-adic expansion of μ. Let ρ(μ) be the partition

(\underbrace{p^{k}, …, p^{k}}_{|μ_{(k)}| factors} , …, \underbrace{p, …, p}_{|μ_{(1)}| factors} , \underbrace{p^{0}, …, p^{0}}_{|μ_{(0)}| factors})

of n and 𝔖_{ρ(μ)} be the corresponding Young subgroup of 𝔖_{n}.

Theorem 2.6 ([12], [10, §4.1]). Any vertex of the Young module Y^{μ} is conjugate to a Sylow p-subgroup of 𝔖_{ρ(μ)}.

The main purpose of this article is to study the vertices and complexity of Specht modules. Thus, by Theorem 2.3 (i), it is crucial to understand the maximal elementary abelian p-subgroups of a symmetric group up to conjugation. Let m be a positive integer. Let (ℤ_{p})^{m} act on itself by the left regular action. This induces an injective group homomorphism (ℤ_{p})^{m} ↪ 𝔖_{p^{m}}. Since the action is faithful we write V_{m}(p) for the image of this homomorphism.

Theorem 2.7 ([1, §VI Theorem 1.3]). Let n be a positive integer. There is a oneto-one correspondence between the ways of writing n = i_{0} + i_{1}p + ⋯ + i_{r}p^{r} for some non-negative integer r such that all i_{0}, …, i_{r} are non-negative integers and 0 ≤ i_{0} ≤ p − 1, and the set of representatives of all maximal elementary abelian p-subgroups of 𝔖_{n} up to conjugation, given by

\underbrace{V_{1}(p) × ⋯ × V_{1}(p)}_{i_{1} times} × ⋯ × \underbrace{V_{r}(p) × ⋯ × V_{r}(p)}_{i_{r} times} ↪

\underbrace{𝔖_{p} × ⋯ × 𝔖_{p}}_{i_{1} times} × ⋯ × \underbrace{𝔖_{p^{r}} × ⋯ × 𝔖_{p^{r}}}_{i_{r} times} ↪ 𝔖_{n}.

Note that the p-rank of the maximal elementary abelian p-subgroup of 𝔖_{n} given in Theorem 2.7 is i_{1} + 2i_{2} + ⋯ + ri_{r}. We shall draw an easy conclusion, following Theorem 2.5 (iv) and Theorem 2.3 (ii), that the complexity of a Specht module S^{μ} is bounded above by the p-weight of μ. The statement remains valid if we replace the Specht module by the simple module D^{μ}, whenever μ is p-regular, of F 𝔖_{n} as there is an analogous statement to Theorem 2.5 (iv) for simple modules, though this is not of the main concern of this article.

3. Main results

We present our main results in this section.

Theorem 3.1. Let F be a field of characteristic p > 0 and S^{μ} be an indecomposable Specht module. Suppose that S^{μ} has an abelian vertex of p-rank m. Then μ is a p^{2}-core and the complexity of S^{μ} is m.

Theorem 3.2. Let F be a field of characteristic p > 0 and μ be a partition. Suppose that p ≥ 3, or p = 2 and μ is 2-regular. Suppose that the Specht module S^{μ} has an abelian vertex Q of p-rank m. Let c be the complexity of the Specht module S^{μ} and w be the p-weight of the partition μ. Then c = m = w and Q is conjugate to the elementary abelian p-subgroup

\underbrace{V_{1}(p) × ⋯ × V_{1}(p)}_{w factors} ↪ 𝔖_{p} × ⋯ × 𝔖_{p} ↪ 𝔖_{n}.

Theorem 3.3. Let p = 2, μ be a 2-regular partition and Q be a vertex of the Specht module S^{μ}. Let w be a non-negative integer. Then the following statements are equivalent.

(i) The vertex Q is abelian of 2-rank w.

(ii) The vertex Q is elementary abelian of 2-rank w.

(iii) The partition μ is a 4-core of 2-weight w. In any of these cases, the Specht module S^{μ} has trivial source, complexity w and is a simple Young module.

Remark 3.4. The hypothesis of Theorem 3.3 cannot be loosened in the following sense. For p = 2, it is known that the Specht modules S^{(n−r,1^{r})} are indecomposable when 2 | n [28, Theorem 4.1]. In this case, the vertices of S^{(n−r,1^{r})} are the Sylow 2-subgroups of 𝔖_{n} [27, Theorem 4.5]. Thus, for instance, the Specht module S^{(4,1,1)} has a non-abelian vertex but yet the partition (4, 1, 1) is a 4-core. So, in Theorem 3.3, we cannot replace the 2-regularity condition merely by the indecomposability condition.

4. Proof of the main results

In this section we shall prove our main theorems. We first prove Theorem 3.1. We then divide the latter part of this section into two subsections. The first part concerns the proof of Theorem 3.2 when p ≥ 3. The second part concerns the proofs of Theorem 3.2 when p = 2 and Theorem 3.3.

We state two known results upon which our proofs rely heavily.

Theorem 4.1 ([29], [21, Theorem 1]). Let n_{1}, …, n_{s} be positive integers. If there is an injective group homomorphism ℤ_{p^{n_{1}}} ×…×ℤ_{p^{ns}} ↪ 𝔖_{m} then p^{n_{1}} +…+p^{n_{s}} ≤ m.

For any positive integer n and prime number p, we write n_{p} for the prime power p^{a} such that p^{a} | n and gcd(p, n/p^{a}) = 1.

Theorem 4.2 ([5, Theorem 1.1]). Let M be a F G-module for some finite group G. Suppose that M is relatively H-projective for some subgroup H of G. Let Q ≅ ℤ_{p^{n_{1}}} × … × ℤ_{p^{nm}} be an abelian subgroup of H of p-rank m with n_{1} ≥ ⋯ ≥ n_{m} ≥ 1. Let c = c_{Q}(M) be the complexity of the F Q-module Res_{Q}^{G} M. Then

\frac{|G : H|_{p}|Q|_{p}}{p^{n_{1}} ⋯ p^{n_{c}}} | (dim_{F} M)_{p}.

We can now prove Theorem 3.1.

Proof of Theorem 3.1. We assume the notation as in Theorem 3.1. Let n = |μ| and w be the p-weight of μ. Let D be a defect group of the block eμ̃ and Q be a vertex of S^{μ}. Without loss of generality, we may assume that

Q ⊆ D ⊆ 𝔖_{pw}.

We apply Theorem 4.2 and Proposition 2.4 by taking M = S^{μ}, H = Q ≅ ℤ_{p^{n_{1}}} × … × ℤ_{p^{nm}} with n_{1} ≥ ⋯ ≥ n_{m} ≥ 1. Let c = c_{Q}(S^{μ}) = c_{𝔖_{n}}(S^{μ}). We have

p^{a−n_{1}−···−n_{m}} · p^{n_{c+1}} ⋯ p^{n_{m}} = p^{a−n_{1}−···−n_{c}} | (dim_{F} S^{μ})_{p}

where p^{a} = (n!)_{p}. By Theorem 2.5 (i) and the hook formula, we let s be the non-negative integer such that

p^{s} = \frac{p^{a}}{p^{w} · (dim_{F} S^{μ})_{p}} = \frac{∏_{(i,j)∈[μ]}(h_{i,j})_{p}}{p^{w}} .

Note that s = 0 if and only if μ is a p^{2}-core; namely every hook length of [μ] is not divisible by p^{2}. With this notation, we deduce that

s + w ≤ n_{1} + ⋯ + n_{c}.  (4.1)

On the other hand, we have the injection Q ↪ 𝔖_{pw}. As such, by Theorem 4.1, we get p^{n_{1}} + ⋯ + p^{n_{m}} ≤ pw; namely p^{n_{1}−1} + ⋯ + p^{n_{m}−1} ≤ w. Combining this inequality with (4.1), we have

(p^{n_{1}−1} − n_{1}) + ⋯ + (p^{n_{c}−1} − n_{c}) + p^{n_{c+1}−1} + ⋯ + p^{n_{m}−1} + s ≤ 0.  (4.2)

Since p^{k−1} − k ≥ 0 for every positive integer k we conclude that m = c, s = 0, p^{n_{i}−1} = n_{i} for all 1 ≤ i ≤ c and w = n_{1} + ⋯ + n_{c}. We have proved Theorem 3.1. □

4.1. The case where p ≥ 3. Assuming all the computations what we have just done in the proof of Theorem 3.1, we continue to prove Theorem 3.2 for the case of p ≥ 3.

Proof of Theorem 3.2 when p ≥ 3. If p is odd then, from p^{n_{i}−1} = n_{i}, necessarily n_{i} = 1 for all 1 ≤ i ≤ c; namely m = c = w and Q is an elementary abelian p-group of p-rank w. Since Q is also a subgroup of 𝔖_{pw}, by Theorem 2.7, we let

E = (V_{1}(p))^{i_{1}} × ⋯ × (V_{r}(p))^{i_{r}}

be an maximal elementary p-subgroup of 𝔖_{pw} containing Q. Comparing their pranks, we have w ≤ i_{1} + 2i_{2} + ⋯ + ri_{r}. On the other hand, we also have i_{1}p + i_{2}p^{2} + ⋯ + i_{r}p^{r} = pw. Thus we deduce that

i_{2}(p − 2) + ⋯ + i_{r}(p^{r−1} − r) ≤ 0

and hence i_{2} = ⋯ = i_{r} = 0, i_{1} = w. Namely, Q is necessarily conjugate to (V_{1}(p))^{w}. This completes the proof of Theorem 3.2 when p is odd. □

Remark 4.3. The author would like to point out that the proofs given above, for Theorem 3.1 and Theorem 3.2 when p ≥ 3, is very similar to the proof of [31, Theorem 1] in which Wildon dealt with the case where m = 1.

4.2. The case where p = 2. Note that the proof of Theorem 3.1 assumes nothing about the 2-regularity of the partition μ, in the case of p = 2. It works for all indecomposable Specht modules with abelian vertices. In view of Theorem 3.2, we still have a result for p = 2 case without assuming the 2-regularity condition.

Proposition 4.4. Let p = 2 and S^{μ} be indecomposable. Suppose that a vertex Q of S^{μ} is abelian of p-rank m. Let c be the complexity of S^{μ} and w be the 2-weight of μ. Then m = c, c ≤ w ≤ 2c and the direct factors of Q are either ℤ_{2} or ℤ_{4}.

Proof. By (4.2) with p = 2, we have m = c, 2^{n_{i}−1} = n_{i} for all 1 ≤ i ≤ c and w = n_{1} + ⋯ + n_{c}. Thus each of the n_{i} is either 1 or 2. □

In general, we do not know when a Specht module over an even characteristic field is decomposable. Since vertices are defined only for indecomposable modules, in view of Theorem 2.5 (iii), the 2-regularity assumption is technically required.

Note that the proof which we have just given in §4.1 for Theorem 3.2 fails when p = 2 because of the possibility that Q may have ℤ_{4} as a direct factor. Thus we need to take a closer look at these partitions μ which are 2-regular and the vertices of S^{μ} are abelian.

Lemma 4.5. Let μ = (μ_{1}, …, μ_{r}) be a partition. Then μ is both a 2-regular and 4-core partition if and only if exactly one of the following holds.

(i) μ_{r} = 1 and there exists a number 0 ≤ s ≤ r − 1 such that μ_{i} − μ_{i+1} = 3 for all 1 ≤ i ≤ s and μ_{i} − μ_{i+1} = 1 for all s + 1 ≤ i ≤ r − 1.

(ii) μ_{r} ∈ {2, 3} and μ_{i} − μ_{i+1} = 3 for all 1 ≤ i ≤ r − 1.

Proof. Suppose that μ is 2-regular and is a 4-core. It is easy to see that 1 ≤ μ_{i} − μ_{i+1} ≤ 3 for all 1 ≤ i ≤ r − 1 and μ_{i} − μ_{i+1} ≠ 2 for all 1 ≤ i ≤ r − 2. Suppose that μ_{j} − μ_{j+1} = 3 and μ_{j−1} − μ_{j} = 1 for some 2 ≤ j ≤ r − 1. Then the node (j − 1, μ_{j−1} − 2) has hook length 4. If μ_{r} is 2 or 3, then μ_{r−1} − μ_{r} ≠ 1; otherwise the node (r − 1, 1), respectively (r − 1, 2), has hook length 4. This proves one direction of the characterization. The converse is easy. □

Using the above characterization of 2-regular and 4-core partitions, and the characterization of simple Specht modules for p = 2 [18, Main Theorem], we obtain the following corollary.

Corollary 4.6 (of [18, Main Theorem]). Let p = 2 and μ be a 2-regular partition. If μ is a 4-core then S^{μ} is simple and D^{μ} ≅ S^{μ} ≅ Y^{μ}.

Corollary 4.7. Let p = 2 and μ be a 2-regular partition of n. Suppose that μ is a 4-core. Then any vertex Q of S^{μ} is conjugate to

\underbrace{V_{1}(2) × ⋯ × V_{1}(2)}_{w factors} ↪ (𝔖_{2})^{w} ↪ 𝔖_{n}

where w is the 2-weight of μ. Furthermore, the complexity of S^{μ} is w.

Proof. Let μ = 2^{k}μ_{(k)}+⋯+2μ_{(1)}+μ_{(0)} be the 2-adic expansion of μ with |μ_{(k)}| ≠ 0. We claim that 0 ≤ k ≤ 1 and μ_{(0)} is the 2-core of μ. Note that there are only two types of removable skew 2-hooks, the horizontal 2-hook and the vertical 2-hook. Since μ_{(0)} is obtained from μ by successively removing all horizontal 2-hooks it is then not a 2-core if there is some removable vertical 2-hook of μ_{(0)}. In this case, the original partition μ has two successive non-empty rows whose difference of their sizes is divisible by 2. However, this is not allowed by the characterization in Lemma 4.5. Thus μ_{(0)} = μ̃. It is clear that, in general, if μ is a p^{k}-core for some positive integer k then μ_{(l)} = ∅ for all l ≥ k.

By Corollary 4.6, the Specht module S^{μ} is isomorphic to the Young module Y^{μ}. By Theorem 2.6, we deduce that any vertex Q of the Specht module S^{μ} ≅ Y^{μ} is conjugate to a Sylow 2-subgroup of 𝔖_{ρ(μ)} = (𝔖_{2})^{|μ_{(1)}|} where

|μ_{(1)}| = (n − |μ_{(0)}|)/2 = (n − |μ̃|)/2 = w.

Since the Young module Y^{μ} has a trivial source; namely the F Q-module F, by Proposition 2.4 (ii), we have

c_{𝔖_{n}}(S^{μ}) = c_{Q}(F) = w.

The proof of our corollary is now complete. □

The proofs for Theorem 3.2 when p = 2 and Theorem 3.3 are now clear:

Proof of Theorem 3.2 when p = 2. We assume all the notation in Theorem 3.2. By Theorem 3.1, the partition μ is necessarily a 4-core. By Corollary 4.7, we get m = w = c and Q is conjugate to (V_{1}(2))^{w}. □

Proof of Theorem 3.3. Corollary 4.7 shows that (iii) implies (ii). The implication from (ii) to (i) is trivial. Theorem 3.1 shows that (i) implies (iii). The final statement comes from Corollaries 4.6 and 4.7. □

5. Some applications of the main results

In [31, Theorem 1], M. Wildon showed that the vertices of an indecomposable Specht module S^{μ} are non-trivial cyclic if and only if the p-weight of μ is 1. We have a generalization of this result.

Corollary 5.1. Let p be the characteristic of the field F and suppose that p ≥ 3. Let 1 ≤ m ≤ p − 1 be an integer and μ be a partition such that the Specht module S^{μ} is indecomposable. Then a vertex Q of S^{μ} is abelian of p-rank m if and only if the p-weight of μ is m. In this case, Q is necessarily elementary abelian.

Proof. Assume that Q is abelian. Theorem 3.2 shows that Q is elementary abelian and m = w where w is the p-weight of μ. Conversely, suppose that m = w. In this case, we have the abelian defect case. Thus Q is necessarily abelian as a subgroup of some defect group of S^{μ}. By Theorem 3.2, Q is has p-rank w = m. □

Remark 5.2. By virtue of [31, Theorem 1], Corollary 5.1 also holds for p = 2.

There is a special class of Specht modules S^{μ} whose partition μ is of the form (μ_{1}^{a_{1}}, μ_{2}^{a_{2}}, …, μ_{r}^{a_{r}}) where both μ_{i}, a_{i} are multiples of p for every 1 ≤ i ≤ r. We call them the (p × p)-partitions. As another application of our results, we study the vertices of the Specht modules S^{μ} where μ is a (p × p)-partition for p ≥ 3. We have seen in Lemma 2.2 that the vertices of an indecomposable module remain unchanged, should the module remains indecomposable upon field extension. Since all Specht modules in the case of p ≥ 3 are indecomposable, we may assume that F is algebraically closed for the rest of this section.

Let E_{p} = V_{1}(p) × ⋯ × V_{1}(p) (p factors) be the maximal elementary abelian psubgroup of 𝔖_{p^{2}} as in Theorem 2.7. We have c_{𝔖_{p^{2}}} (S^{(p^{p})}) = c_{E_{p}}(S^{(p^{p})}) = p − 1 [23, Theorem 3.1(i)]. Furthermore, for p ≥ 3, with respect to the generators

(1, …, p), (p + 1, …, 2p), …, (p^{2} − p + 1, …, p^{2})

of E_{p}, a description of the union of irreducible components W_{p−1} of dimension p − 1 of V_{E_{p}}^{#}(S^{(p^{p})}) and the vanishing ideal of W_{p−1} is given in [23, Theorem 3.1(ii)]. The vanishing ideal I of W_{p−1} is generated by the element

f(x_{1}, …, x_{p}) = (x_{1} ⋯ x_{p})^{p−1}f̃+ ∑_{i=1}^{p} x_{1}^{n(p−1)} ⋯ \begin{matrix} \\ x_{i} & n(p−1) \end{matrix} ⋯ x_{p}^{n(p−1)}

for some homogeneous polynomial f̃ and positive integer n, and where the term

x_{1}^{n(p−1)} ⋯ \begin{matrix} \\ x_{i} & n(p−1) \end{matrix} ⋯ x_{p}^{n(p−1)} is the product of all x_{j}^{n(p−1)}’s with 1 ≤ j ≤ p except

the term x_{i}^{n(p−1)}. In particular, we have the following lemma.

Lemma 5.3. Any irreducible component of W_{p−1} ⊆ V_{E_{p}}^{#}(S^{(p^{p})}) is not a hyperplane of 𝔸^{p}(F).

Proof. Suppose that W_{p−1} contains a hyperplane, i.e. f(x_{1}, …, x_{p}) ∈ I(W_{p−1}) ⊆ ⟨a_{1}x_{1} + ⋯ + a_{p}x_{p}⟩ for some non-zero vector v = (a_{1}, …, a_{p}) ∈ 𝔸^{p}(F). Let g ∈ F [x_{1}, …, x_{p}] be the polynomial such that

f(x_{1}, …, x_{p}) = (a_{1}x_{1} + ⋯ + a_{p}x_{p})g(x_{1}, …, x_{p}).  (5.1)

Suppose that at least two of the coordinates of v are non-zero; say a_{r} and a_{s}. Let 1 ≤ i ≤ p be such that i is different from either r or s. Then there exists elements b_{1}, …, b_{p} of F with b_{i} = 0, and b_{j} ≠ 0 for all 1 ≤ j ≤ p and j ≠ i, such that a_{1}b_{1} + ⋯ + a_{p}b_{p} = 0. Substitute the values b_{1}, …, b_{p} into Equation (5.1), we have the contradiction where

0 ≠ b_{1}^{n(p−1)} ⋯ \begin{matrix} \\ b_{i} & n(p−1) \end{matrix} ⋯ b_{p}^{n(p−1)} = 0 · g(b_{1}, …, b_{p}) = 0.

This shows that the coordinates of v are all zero except a_{j} ≠ 0 for some unique 1 ≤ j ≤ p. We deduce that f(x_{1}, …, x_{p}) = a_{j}x_{j}g(x_{1}, …, x_{p}) and hence x_{j} is a factor of f(x_{1}, …, x_{p}). However, it is clear from the description of f that x_{j} is not a factor. The proof is now complete. □

For p ≥ 3, Hemmer show that the complexity of the Specht module S^{μ} when μ is a (p × p)-partition is strictly less than the p-weight of μ [15, Corollary 1.4]. Thus the following corollary is immediate by applying Theorem 3.2.

Corollary 5.4. Suppose that p ≥ 3 and μ is a (p × p)-partition. Then the vertices of S^{μ} are non-abelian.

The next corollary describes the vertices of the Specht module S^{(p^{p})} for p ≥ 3.

Corollary 5.5. For p ≥ 3, any vertex of S^{(p^{p})} is a Sylow p-subgroup of 𝔖_{p^{2}}.

Proof. Let Q be a vertex of M = S^{(p^{p})} and S be a source of M. By Mackey Decomposition Theorem, we have

Res_{E_{p}}^{𝔖_{p^{2}}} M | Res_{E_{p}}^{𝔖_{p^{2}}} Ind_{Q}^{𝔖_{p^{2}}} S ≅ ⨁_{g∈E_{p}\𝔖_{p^{2}}/Q} Ind_{^{g}Q∩E_{p}}^{E_{p}} Res_{^{g}Q∩E_{p}}^{^{g}Q} ^{g}S

where E_{p}\𝔖_{p^{2}}/Q is a set of double coset representatives of (E_{p}, Q) in 𝔖_{p^{2}}. Suppose that ^{g}Q ∩ E_{p} ≠ E_{p} for all g; namely ^{g}Q ∩ E_{p} is a proper subgroup of E_{p} and hence has order at most p^{p−1}. Let

U_{g} = V_{E_{p}}^{#}(Ind_{^{g}Q∩E_{p}}^{E_{p}} Res_{^{g}Q∩E_{p}}^{^{g}Q} ^{g}S).

In fact, dim U_{g} = dim V_{^{g}Q∩E_{p}}^{#}(Res_{^{g}Q∩E_{p}}^{^{g}Q} ^{g}S). Since U_{g} is an induced module induced from a proper subgroup of E_{p} we have that the rank variety

U := V_{E_{p}}^{#}(Res_{E_{p}}^{𝔖_{p^{2}}} Ind_{Q}^{𝔖_{p^{2}}} S) = ⋃ U_{g} ⊆ 𝔸^{p}(F)

is a finite union of subvarieties of dimension at most p−1. If U_{g} has dimension p−1 for some g then it is necessarily that V_{^{g}Q∩E_{p}}^{#}(Res_{^{g}Q∩E_{p}}^{^{g}Q} ^{g}S) ≅ 𝔸^{p−1}(F) and hence U_{g} is a union of hyperplanes of 𝔸^{p}(F).

Let N be a kE_{p}-module such that N ⊕ Res_{E_{p}}^{𝔖_{p^{2}}} M = Res_{E_{p}}^{𝔖_{p^{2}}} Ind_{Q}^{𝔖_{p^{2}}} S. We have

V_{E_{p}}^{#}(M) ∪ V_{E_{p}}^{#}(N) = U. Let V be an irreducible component of V_{E_{p}}^{#}(M) of dimension p − 1. By the unique decomposition property of varieties into their irreducible varieties, see [14, Corollary 1.6], we conclude that V is a hyperplane of 𝔸^{p}(F). This contradicts to Lemma 5.3 and hence we conclude that ^{g}Q ∩ E_{p} = E_{p} for some g ∈ 𝔖_{p^{2}}; namely E_{p} ⊆ Q up to some conjugation.

Since S^{(p^{p})} has complexity p − 1, different from the p-weight of (p^{p}), by Theorem 3.2 or Corollary 5.4, the vertices cannot be abelian. Thus Q has order strictly larger than p^{p}, i.e. p^{p+1} and hence is a Sylow p-subgroup of 𝔖_{p^{2}}. □

Remark 5.6. The case of p = 2 is dealt with in [31, Lemma 6]: the Specht module S^{(2,2)} has the Klein four group V_{2}(2) as its vertex, which is different from the description in Corollary 5.5.

6. Some further questions

In this section, we include some further questions. The following question is natural following our Theorem 3.3.

Question 6.1. Given that μ is p-regular and p ≥ 3, are the vertices of the Specht module S^{μ} abelian if μ is a p^{2}-core?

To answer Question 6.1, our method used in §4.2 does not help so much. The main obstruction is that there are Specht modules S^{μ} which are not Young modules, with μ p-regular and a p^{2}-core (for example, the Specht module S^{(6,3)} with p = 3). Readers may have observed that, in order to achieve Theorem 3.3, the main idea is to prove the simplicity of the corresponding Specht modules as in Corollary 4.6. Thus we can partly answer Question 6.1 by imposing this crude assumption.

Proposition 6.2. Let p ≥ 3, μ be a p-regular partition and Q be a vertex of the Specht module S^{μ}. Suppose that S^{μ} is simple. Let w be a non-negative integer. Then the following statements are equivalent.

(i) The vertex Q is abelian of p-rank w.

(ii) The vertex Q is elementary abelian of p-rank w.

(iii) The partition μ is a p^{2}-core of p-weight w. In any of these cases, the Specht module S^{μ} has trivial source, complexity w and is a simple Young module.

Proof. We only need to show that (iii) implies (i). As we have mentioned in the proof of Corollary 4.7, the p-adic expansion of μ is μ_{(0)} + pμ_{(1)}, here we allow μ_{(1)} = ∅. By the simplicity condition, we deduce that S^{μ} ≅ Y^{μ} and hence Q is conjugate to a Sylow p-subgroup of 𝔖_{ρ(μ)} = (𝔖_{p})^{|μ_{(1)}|} by Theorem 2.6. So Q is abelian, necessarily of p-rank w and the Specht module S^{μ} has complexity w by Theorem 3.1. □

Of course, ultimately, we will be delighted to achieve the following goal, if the computation of the vertices of all Specht modules is difficult.

Question 6.3. Classify all indecomposable Specht modules with abelian vertices.

As we have pointed out in the beginning of §4.2, the technical assumption of 2-regularity is required in Theorem 3.2. The result will be nicer if it works for all indecomposable Specht modules with abelian vertices. The possibility of the existence of an example of a Specht module S^{μ} with abelian vertices Q which contain ℤ_{4} as a direct factor is the main obstruction.We do not have an example of a Specht module that support this possibility of Proposition 4.4.

Question 6.4. For p = 2, is there an indecomposable Specht module S^{μ} whose vertices are abelian and contain ℤ_{4} as a direct factor?

Acknowledgment. The results of this article have been partly achieved during my days in Aberdeen [22, Remark 4.14] and partly done in Singapore. I would like to take the chance to thank Dave Benson and Radha Kessar. I would like to thank Kai Meng Tan for his availability for discussion and careful reading of this article, and Mark Wildon for pointing out Theorem 4.1 and improving an earlier version of Theorem 3.1.

I would like to thank the referees for their valuable suggestions.

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Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076.

  E-mail address: matlkj@nus.edu.sg