PERIODIC LIE MODULES

KAY JIN LIM AND KAI MENG TAN

Abstract. Let p be a prime number and k be a positive integer not divisible by p. We describe the Heller translates of the periodic Lie module Lie(pk) in characteristic p and show that it has period 2p − 2 when p is odd and 1 when p = 2.

1. Introduction

The main difficulty in the study of modular representation theory of finite groups is that the group algebras are not only non-semisimple but also are ‘almost surely’ of wild representation type. Various contraptions, such as vertices and complexities, have been invented as a measure of how non-projective the modules are and to group the modules into classes. One of the easiest and yet non-trivial classes to study are those having complexity one, which are precisely the periodic modules. Informally, a module is periodic if and only if the terms in its minimal projective resolution run over a repeating cycle, and the period of such a module is the length of the cycle.

The current article deals with the determination of the periods and various Heller translates of some Lie modules of the symmetric groups. The Lie modules are classical objects of study and has appeared in [We] and [Wi]. They have deep relations to algebraic topology too, as homology groups in spaces related to the braid groups [Co] as well as in the study of Goodwillie calculus [AM]. More recently, the Lie modules have been tied to the finest coalgebra decomposition of the tensor algebras [SW].

Throughout this article, 𝔽 denotes an algebraically closed field of prime characteristic p.

Let n ∈ ℤ^{+}. The Lie module Lie(n) of the symmetric group 𝔖_{n} of n letters may be defined as the left ideal of 𝔽𝔖_{n} generated by the Dynkin-Specht-Wever element

w_{n} = (1 − d_{2})(1 − d_{3}) ⋯ (1 − d_{n}),

where d_{i} is the descending i-cycle (i, i − 1, …, 1) and we compose the elements of 𝔖_{n} from right to left.

Let k ∈ ℤ^{+} with p ∤ k. In [ELT, Theorem 3.2], the complexity of Lie(p^{m}k) is shown to be bounded above by m. Since Lie(pk) is non-projective, this in particular shows that Lie(pk) has complexity 1 and hence is periodic. It is therefore natural to ask what its various Heller translates are and what its period is, and we answer these questions in this article. More precisely, we show the following:

Theorem 1.1. Let k ∈ ℤ^{+} with p ∤ k, and let 𝔽_{𝔖_{p}} denote the trivial 𝔽𝔖_{p}-module. (i) For any i ∈ ℤ, we have

Ω^{i}(Lie(pk)) ≅ Ω^{i+1} (Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}} (𝔽_{𝔖_{p}} ⊠ Lie(k))) ≅ Ω^{0} (Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}} (Ω^{i+1} (𝔽_{𝔖_{p}}) ⊠ Lie(k))) .

We thank Karin Erdmann for providing the arguments for Proposition 3.3. We are supported by Singapore Ministry of Education AcRF Tier 1 grants RG13/14 and R-146-000-172-112 respectively.

(ii) The period of Lie(pk) is equal to the period of 𝔽_{𝔖_{p}}, i.e.

\begin{cases}2p − 2, & if p is odd; \\ 1, & if p = 2. \end{cases}

Note that in part (i) of Theorem 1.1, the group 𝔖_{p} × 𝔖_{k} is identified with the subgroup Δ(𝔖_{p})𝔖_{k}^{[p]} of 𝔖_{pk} (see subsection 2.2 for the definition of these notations). In fact, we give a more detailed description of the Heller translates of Lie(pk) in Theorem 3.5.

Our approach builds on the recent work on Lie(pk) in [ES]. Following [ES, Theorem 10], we analyse the pth symmetrization of Lie(k), denoted as S^{p}(Lie(k)). More precisely, we compute the various Heller translates and period of S^{p}(Lie(k)) and these directly imply our desired results about the Lie module Lie(pk). This in particular provides an alternative proof that the complexity of Lie(pk) is 1, independent of the results in [ELT]. It is easier to deal with S^{p}(Lie(k)), instead of Lie(pk), because the former is isomorphic to Ind_{𝔖_{k}×𝔖_{p}}^{𝔖_{pk}}(Lie(k) ⊠ 𝔽) in the stable module category (see Proposition 3.3).

This article is organised as follows: In Section 2, we gather together some background knowledge and introduce the notations that we shall need. In Section 3, we describe the Heller translates Ω^{i}(Lie(pk)) in terms of the Specht and dual Specht modules in the principal block of 𝔽𝔖_{p} and the projective indecomposable 𝔽𝔖_{k}-modules. As a consequence, we obtain the vertices and sources of the non-projective indecomposable summands of Ω^{i}(Lie(pk)). The period of Lie(pk) is computed in Section 4.

2. Preliminaries

In this section, we provide the necessary background, introduce the notations and prove some preliminary results that we shall need.

2.1. Vertices and sources. Let G be a finite group, and let H be a subgroup of G. An 𝔽G-module M is relatively H-projective if and only if M is a direct summand of Ind_{H}^{G} N for some 𝔽H-module N. Every 𝔽G-module is relatively P -projective for any Sylow psubgroup P of G, and is projective if and only if it is relatively {1}-projective.

A vertex Q of an indecomposable 𝔽G-module M is a subgroup of G that is minimal subject to M being relatively Q-projective. The vertices of M form a single conjugacy class of p-subgroups of G. If M has vertex Q, with M being a direct summand of Ind_{Q}^{G} N for some indecomposable 𝔽Q-module N, then we call N a Q-source of M.

Let Q be a p-subgroup of G. Green’s correspondence provides a bijection between the set of isomorphism classes of the indecomposable 𝔽G-modules with vertex Q and and that of the indecomposable 𝔽N_{G}(Q)-modules with vertex Q. In particular, if Q has order p, and M is an indecomposable 𝔽G-module with vertex Q, and N is its Green correspondent, then

Res_{N_{G}(Q)}^{G} M ≅ N ⊕ P, Ind_{N_{G}(Q)}^{G} N ≅ M ⊕ R

for some projective modules P and R respectively.

2.2. Periodic modules. Let G be a finite group. Traditionally (see [B, §5.10]) an 𝔽Gmodule M is periodic if and only if Ω^{n}(M) ≅ M for some n ∈ ℤ^{+}, and the least such n for which this is true is the period of M. However, this definition excludes modules which have non-zero projective direct summands. As the Lie modules have large non-zero projective direct summands (see [BLT] for details), this definition does not suit our purpose here.

In this paper, we say that an 𝔽G-module M is periodic if and only if Ω^{n}(M) ≅ Ω^{0}(M) ≠ 0 for some n ∈ ℤ^{+}, and the least such n for which this is true is the period of M. Equivalently, M is periodic if and only if M is non-projective and there exists a long exact sequence of 𝔽G-modules

0 → Ω^{0}(M) → P_{m} → ⋯ → P_{1} → M → 0

where P_{1}, …, P_{m} are projectives, and the least such m ∈ ℤ^{+} for which such a long exact sequence exists is the period of M.

We gather together some elementary results about periodic modules.

Lemma 2.1. Let G be a finite group and let M be a periodic 𝔽G-module of period n. (i) Any finite external direct sum of the module M is periodic of period n. (ii) Ω^{a}M is periodic of period n for all a ∈ ℤ. (iii) Let H be a subgroup of G. Then Res_{H}^{G} M is either projective or periodic of period

dividing n. (iv) Let N be a group of which G is a subgroup. Then Ind_{G}^{N} M is periodic of period

dividing n.

Proof. Parts (i) and (ii) are clear. For (iii) and (iv), they follow from the fact that Res_{H}^{G} Ω^{n}(M) ≅ Ω^{n}(Res_{H}^{G} M) ⊕ P and Ind_{G}^{N} Ω^{n}(M) ≅ Ω^{n}(Ind_{G}^{N} M) ⊕ Q, where P and Q are some projective 𝔽H- and 𝔽N-modules respectively [Ca, Proposition 4.4(vii,viii)]. □

2.3. Symmetric groups. In this subsection, we set up the notations of some elements and subgroups of symmetric groups that shall be used in this paper.

For n ∈ ℤ^{+}, the symmetric group of n letters will be denoted by 𝔖_{n}.

Let a, b ∈ ℤ^{+}.

For τ ∈ 𝔖_{a}, define τ^{[b]} ∈ 𝔖_{ab} by

τ^{[b]}((i − 1)b + j) = (τ(i) − 1)b + j

for all 1 ≤ i ≤ a and 1 ≤ j ≤ b, so that τ^{[b]} permutes the a successive blocks of size b in {1, …, ab} according to τ. Clearly, the map τ ↦ τ^{[b]} is an injective group homomorphism.

For σ ∈ 𝔖_{b} and 1 ≤ r ≤ a, define σ[r] ∈ 𝔖_{ab} by

(σ[r])(i + (j − 1)a) = \begin{cases}σ(i) + (r − 1)a & if j = r, \\ i + (j − 1)a & otherwise, \end{cases}

for all 1 ≤ i ≤ a and 1 ≤ j ≤ b, so that σ[r] acts on the rth successive block of size b in {1, …, ab} according to σ, and fixes everything else. In addition, define Δ_{a}(σ) ∈ 𝔖_{ab} by Δ_{a}(σ) = ∏_{r=1}^{a} σ[r], so that Δ_{a}(σ) permutes each of the a successive blocks of size b in {1, …, ab} simultaneously according to σ. Clearly, the maps σ ↦ σ[r] and σ ↦ Δ_{a}(σ) are again injective group homomorphisms.

If H ⊆ 𝔖_{a}, L ⊆ 𝔖_{b} and r ∈ {1, …, a}, we write

H^{[b]} = {h^{[b]} : h ∈ H},
L[r] = {ℓ[r] : ℓ ∈ L},
Δ_{a}(L) = {Δ_{a}(ℓ) : ℓ ∈ L}.

Note that Δ_{a}(𝔖_{b}) and 𝔖_{a}^{[b]} centralises each other.

2.4. Some miscellaneous results. In this subsection, we state and prove three results which we shall require. These results are most probably known and their proofs are provided here for the sake of completeness.

For any finite group G, we write 𝔽_{G} for the trivial 𝔽G-module and subsequently drop the subscript G if it is clear from the context.

Lemma 2.2. Let H be a subgroup of a group G of index ℓ with p ∤ ℓ. Denote the trivial 𝔽Gand 𝔽H-module by 𝔽_{G} and 𝔽_{H} respectively. Then 𝔽_{G} is a direct summand of Ind_{H}^{G} 𝔽_{H} = 𝔽G ⊗_{𝔽H} 𝔽_{H} as an 𝔽G-module.

Proof. Let T be a complete set of left coset representatives of H in G. It is straightforward to check that the maps

ϕ : 𝔽_{G} → Ind_{H}^{G} 𝔽_{H}  ψ : Ind_{H}^{G} 𝔽_{H} → 𝔽_{G}
ϕ(λ) = \frac{1}{ℓ} ∑_{t∈T} t ⊗ λ  ψ (∑_{t∈T} t ⊗ μ_{t}) = ∑_{t∈T} μ_{t}

are 𝔽G-module homomorphisms satisfying ψ∘ϕ = id_{𝔽_{G}}. Thus ϕ is injective, and Ind_{H}^{G} 𝔽_{H} = ker(ψ) ⊕ ϕ(𝔽_{G}). □

Lemma 2.3. Let 𝔽 be the trivial 𝔽𝔖_{p}-module.

(i) For p = 2, Ω(𝔽) = 𝔽.

(ii) For p ≥ 3 and i ≡ j (mod 2p − 2) where j ∈ {0, 1, …, 2p − 3}, we have

Ω^{i}(𝔽) = \{ \begin{matrix}S^{(p−j,1^{j})} & if 0 ≤ j ≤ p − 1, \\ S_{(j−p+2,1^{2p−j−2})} & if p ≤ j ≤ 2p − 3. \end{matrix}

In particular, the trivial 𝔽𝔖_{p}-module 𝔽 has period \begin{cases}2p − 2, & if p is odd, \\ 1, & if p = 2. \end{cases}

Proof. For p = 2, 𝔽𝔖_{p} is projective indecomposable and so Ω(𝔽) = 𝔽.

For p ≥ 3 and j = 0, 1, …, p − 1, the Specht modules S^{(p−j,1^{j})} are pairwise nonisomorphic, and similarly, the dual Specht modules S_{(p−j,1^{j})} ≅ (S^{(p−j,1^{j})})^{∗} are also pairwise non-isomorphic. Furthermore, S^{(p−j,1^{j})} ≅ S_{(p−ℓ,1^{ℓ})} if and only if j = ℓ ∈ {0, p − 1}. It is well-known that Ω(S^{(p−j,1^{j})}) = S^{(p−j−1,1^{j+1})} for 0 ≤ j ≤ p − 2 while Ω(S_{(p−ℓ,1^{ℓ})}) = S_{(p−ℓ+1,1^{ℓ−1})} for 1 ≤ ℓ ≤ p − 1. Thus we obtain part (ii). The final assertion is now clear. □

Proposition 2.4. Suppose that p is odd, and let C_{p} be a subgroup of 𝔖_{p} of order p. Then

Ind_{C_{p}}^{𝔖_{p}} 𝔽 ≅ 𝔽 ⊕ Ω^{2}(𝔽) ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽) ⊕ P

for some projective 𝔽𝔖_{p}-module P.

Proof. In this proof, for a subgroup H of 𝔖_{p}, we denote the trivial 𝔽H-module as 𝔽_{H}.

Let L = N_{𝔖_{p}}(C_{p}). Since Res_{L}^{𝔖_{p}} 𝔽_{𝔖_{p}} = 𝔽_{L}, and 𝔽_{𝔖_{p}} has vertex C_{p}, we see that 𝔽_{𝔖_{p}} and 𝔽_{L} are Green correspondents of each other. Thus Ω^{i}(𝔽_{𝔖_{p}}) and Ω^{i}(𝔽_{L}) are Green correspondents of each other for all i ∈ ℤ^{+} (see [A, §20, Proposition 7]). In particular, since Green correspondence is bijective, this shows that 𝔽_{L} is periodic, having the same period as 𝔽_{𝔖_{p}}, i.e. 2p − 2 by Lemma 2.3(ii).

It is well-known that the trivial 𝔽C_{p}-module 𝔽_{C_{p}} is periodic of period 2. Thus Ind_{C_{p}}^{L} 𝔽_{C_{p}} is periodic of period 1 or 2 by Lemma 2.1(iv). Since 𝔽_{L} is a direct summand of Ind_{C_{p}}^{L}(𝔽_{C_{p}}) by Lemma 2.2, this implies that Ω^{2i}(𝔽_{L}) is a direct summand of Ind_{C_{p}}^{L}(𝔽_{C_{p}}) for all i ∈ ℤ. Since 𝔽_{L} has period 2p − 2, and each Ω^{2i}(𝔽_{L}) is indecomposable (see [A, §20, Theorem 5(1)]), we see that 𝔽_{L} ⊕ Ω^{2}(𝔽_{L}) ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽_{L}) is a direct summand of Ind_{C_{p}}^{L} 𝔽_{C_{p}}. Thus

p − 1 ≤ dim_{𝔽}(𝔽_{L} ⊕ Ω^{2}(𝔽_{L}) ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽_{L})) ≤ dim_{𝔽}(Ind_{C_{p}}^{L} 𝔽_{C_{p}}) = p − 1

(the last equality follows from the fact that L = N_{𝔖_{p}}(C_{p}) is a semidirect product of C_{p} with a cyclic group of order p − 1), forcing equality throughout. Hence

Ind_{C_{p}}^{L} 𝔽_{C_{p}} ≅ 𝔽_{L} ⊕ Ω^{2}(𝔽_{L}) ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽_{L}).

Since Ω^{i}(𝔽_{𝔖_{p}}) is the Green correspondent of Ω^{i}(𝔽_{L}), we get

Ind_{C_{p}}^{𝔖_{p}} 𝔽_{C_{p}} = Ind_{L}^{𝔖_{p}}(Ind_{C_{p}}^{L} 𝔽_{C_{p}}) = Ind_{L}^{𝔖_{p}}(𝔽_{L} ⊕ Ω^{2}(𝔽_{L}) ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽_{L}))
≅ 𝔽_{𝔖_{p}} ⊕ Ω^{2}(𝔽_{𝔖_{p}}) ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽_{𝔖_{p}}) ⊕ P,

where P is a projective 𝔽𝔖_{p}-module. □

3. Heller translates of Lie(pk)

In this section, we describe the Heller translates of the periodic Lie module Lie(pk) when p ∤ k (see Theorem 3.5). Our strategy is based on an analysis of the pth symmetrization S^{p}(Lie(k)) of Lie(k) that is defined in [ES]. We begin with a lemma.

Lemma 3.1. Let k ∈ ℤ^{+} with p ∤ k, and let C_{k} be a cyclic subgroup of 𝔖_{k} generated by a k-cycle. Then Lie(k) ≅ Ind_{C_{k}}^{𝔖_{k}} 𝔽_{δ} for any faithful one-dimensional 𝔽C_{k}-module 𝔽_{δ}.

Proof. This was proved in [R, Corollary 8.7] when 𝔽 has characteristic zero using the argument of ordinary characters. In prime characteristic, since both modules are projective (as is well-known), they are liftable and thus have equal ordinary characters. Hence these modules are equal in the Grothendieck group K_{0}(𝔽𝔖_{k}) and hence isomorphic since the set of indecomposable projective modules is a ℤ-basis for K_{0}(𝔽𝔖_{k}).

Alternatively, Klyachko [K, Theorem] proved that the modular Lie power L^{k}(V) is isomorphic to V^{⊗k} ⊗_{𝔽C_{k}} 𝔽_{δ} as left modules for the Schur algebra S(n, k), where V is a 𝔽-vector space of dimension n and V^{⊗k} is a (S(n, k), 𝔽𝔖_{k})-bimodule with 𝔖_{k} acting on the right by place permutations. Applying the Schur functor when n = k, which sends L^{k}(V) to Lie(k) and V^{⊗k} to 𝔽𝔖_{k}, we get the required isomorphism. □

The pth symmetrization of Lie(k), denoted as S^{p}(Lie(k)), is first introduced in [ES] to study the Lie module Lie(pk). It is an 𝔽𝔖_{pk}-module, and may be defined by

S^{p}(Lie(k)) = Ind_{𝔖_{k}≀𝔖_{p}}^{𝔖_{pk}}(Lie(k)^{⊗p})

where 𝔖_{p} acts on Lie(k)^{⊗p} by place permutation, and 𝔖_{k}≀𝔖_{p} is identified with the subgroup (∏_{i=1}^{p} 𝔖_{k}[i])𝔖_{p}^{[k]} of 𝔖_{pk}. This module is instrumental in the study of Lie(pk) because of the following result:

Theorem 3.2 ([ES, Theorem 10]). Let k ∈ ℤ^{+} with p ∤ k. There is a short exact sequence of left 𝔽𝔖_{pk}-modules

0 → Lie(pk) → 𝔽𝔖_{pk} e → S^{p}(Lie(k)) → 0,

where e is an idempotent of 𝔽𝔖_{pk}.

The module S^{p}(Lie(k)) is studied in detail in [ES] in a more general setting as S^{p}(U) where U is any projective 𝔽𝔖_{k}-module. In particular, we have the following:

Proposition 3.3. Let C_{p} be a fixed p-subgroup of 𝔖_{p}. Then

Ω^{0}(S^{p}(Lie(k))) ≅ Ω^{0}(Ind_{𝔖_{k}×N_{𝔖p}(C_{p})}^{𝔖_{pk}}(Lie(k) ⊠ 𝔽)) ≅ Ω^{0}(Ind_{𝔖_{k}×𝔖_{p}}^{𝔖_{pk}}(Lie(k) ⊠ 𝔽))

where we identify 𝔖_{k} ×N_{𝔖_{p}}(C_{p}) and 𝔖_{k} ×𝔖_{p} with the subgroups Δ_{p}(𝔖_{k})(N_{𝔖_{p}}(C_{p}))^{[k]} and Δ_{p}(𝔖_{k})𝔖_{p}^{[k]} of 𝔖_{pk} respectively.

Proof. Let M = Lie(k)^{⊗p}, an 𝔽(𝔖_{k} ≀ 𝔖_{p})-module, so that S^{p}(Lie(k)) = Ind_{𝔖_{k}≀𝔖_{p}}^{𝔖_{pk}} M, where

𝔖_{k} ≀ 𝔖_{p} is identified with the subgroup (∏_{i=1}^{p} 𝔖_{k}[i])𝔖_{p}^{[k]}. Then

M ≅ (Ind_{C_{k}}^{𝔖_{k}} 𝔽_{δ})^{⊗p} ≅ Ind_{C_{k}≀𝔖_{p}}^{𝔖_{k}≀𝔖_{p}} ((𝔽_{δ})^{⊗p})

by Lemma 3.1 and [ELT, Lemma 5]. In particular, M is relatively C_{p}-projective since C_{p} is the Sylow p-subgroup of C_{k} ≀ 𝔖_{p}. Note also that M is not projective since its dimension is not divisible by the order of a Sylow p-subgroup of 𝔖_{k} ≀ 𝔖_{p}.

Let

M = (⨁_{i=1}^{ℓ} X_{i}) ⊕ P_{H},

where each X_{i} is non-projective indecomposable and P_{H} is projective. Then each X_{i} is relatively C_{p}-projective (since M is) and hence it has a vertex C_{p}. In addition, by [ES, proof of Lemma 13], we also have

Ind_{𝔖_{k}≀𝔖_{p}}^{𝔖_{pk}} X_{i} = Y_{i} ⊕ Q_{i}

for some indecomposable non-projective 𝔽𝔖_{pk}-module Y_{i} and projective 𝔽𝔖_{pk}-module Q_{i}.

By [ES, proof of Proposition 12], M is a summand of a permutation module . In particular its direct summands X_{i} are all p-permutation modules. Thus each Brauer quotient X_{i}(C_{p}), as an 𝔽N_{𝔖_{k}≀𝔖_{p}}(C_{p})-module, is the Green correspondent of X_{i} [T, Ex. 27.4], so that

Ind_{N_{𝔖_{k}≀𝔖p}(C_{p})}^{𝔖_{k}≀𝔖_{p}} X_{i}(C_{p}) ≅ X_{i} ⊕ R_{i}

for some projective 𝔽(𝔖_{k}≀𝔖_{p})-module R_{i}. Consequently, identifying an element (σ_{1}, …, σ_{p})τ of 𝔖_{k} ≀ 𝔖_{p} with (∏_{i=1}^{p} σ_{i}[i])τ^{[k]} ∈ (∏_{i=1}^{p} 𝔖_{k}[i]) 𝔖_{p}^{[k]} ⊆ 𝔖_{pk}, we have

Ind_{N_{𝔖_{k}≀𝔖p}(C_{p})}^{𝔖_{pk}} M(C_{p}) = Ind_{N_{𝔖_{k}≀𝔖p}(C_{p})}^{𝔖_{pk}} (⨁_{i=1}^{ℓ} X_{i}(C_{p}))

≅ Ind_{𝔖_{k}≀𝔖_{p}}^{𝔖_{pk}} (⨁_{i=1}^{ℓ}(X_{i} ⊕ R_{i})) = (⨁_{i=1}^{ℓ} Y_{i}) ⊕ P

for some projective 𝔽𝔖_{pk}-module P, so that

Ω^{0} (Ind_{N_{𝔖_{k}≀𝔖p}(C_{p})}^{𝔖_{pk}} M(C_{p})) ≅ ⨁_{i=1}^{ℓ} Y_{i} ≅ Ω^{0} (Ind_{𝔖_{k}≀𝔖_{p}}^{𝔖_{pk}} M) = Ω^{0} (S^{p}(Lie(k))) .

The first isomorphism now follows, since

N_{𝔖_{k}≀𝔖_{p}}(C_{p}) = 𝔖_{k} × N_{𝔖_{p}}(C_{p})

and M(C_{p}) is isomorphic to Lie(k)⊠𝔽 as 𝔽(𝔖_{k} ×N_{𝔖_{p}}(C_{p}))-modules [ES, between Lemma 14 and 15, and Lemma 15].

For the second isomorphism, note first that the trivial 𝔽N_{𝔖_{p}}(C_{p})-module is the Green correspondent of the trivial 𝔽𝔖_{p}-module by Lemma 2.2. Thus, for each projective indecomposable 𝔽𝔖_{k}-module P, the 𝔽(𝔖_{k}×N_{𝔖_{p}}(C_{p}))-module P ⊠𝔽 is the Green correspondent of the 𝔽(𝔖_{k} × 𝔖_{p})-module P ⊠ 𝔽. Since Lie(k) is projective as an 𝔽𝔖_{k}-module, this shows that

Ω^{0} (Ind_{𝔖_{k}×N_{𝔖p}(C_{p})}^{𝔖_{k}×𝔖_{p}}(Lie(k) ⊠ 𝔽)) = Lie(k) ⊠ 𝔽,

and the second asserted isomorphism follows. □

Corollary 3.4. Let k ∈ ℤ^{+} with p ∤ k. Then

Ω^{0}(Lie(pk)) ≅ Ω (Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(𝔽 ⊠ Lie(k)))

where we identify 𝔖_{p} × 𝔖_{k} with the subgroup Δ_{k}(𝔖_{p})𝔖_{k}^{[p]} of 𝔖_{pk}.

Proof. By Theorem 3.2 and Proposition 3.3,

Ω^{0}(Lie(pk)) ≅ Ω (Ind_{𝔖_{k}×𝔖_{p}}^{𝔖_{pk}}(Lie(k) ⊠ 𝔽))

where 𝔖_{k} × 𝔖_{p} is identified with Δ_{p}(𝔖_{k})𝔖_{p}^{[k]}. But Ind_{𝔖_{k}×𝔖_{p}}^{𝔖_{pk}}(Lie(k) ⊠ 𝔽) is isomorphic

to Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(𝔽 ⊠ Lie(k)), where 𝔖_{k} × 𝔖_{p} is identified with Δ_{p}(𝔖_{k})𝔖_{p}^{[k]} in the former and

𝔖_{p} × 𝔖_{k} is identified with Δ_{k}(𝔖_{p})𝔖_{k}^{[p]} in the latter, via the map sending g ⊗ (x ⊗ 1) ↦

gz^{−1} ⊗(1⊗x) for g ∈ 𝔖_{pk} and x ∈ Lie(k), where z ∈ 𝔖_{pk} such that z (Δ_{p}(𝔖_{k})𝔖_{p}^{[k]}) z^{−1} =

Δ_{k}(𝔖_{p})𝔖_{k}^{[p]}. The corollary thus follows. □

Remark. In [ET], S^{p}(Lie(k)) was defined as Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(𝔽⊠Lie(k)) (where 𝔖_{p}×𝔖_{k} is identi-

fied with the subgroup Δ_{k}(𝔖_{p})𝔖_{k}^{[p]} of 𝔖_{pk}). This suited their purposes and they remarked that their definition is the ‘same’ as that used in [ES] without giving any justification. Corollary 3.4 now provides the omitted details.

We note that part (i) of Theorem 1.1 now follows from Corollary 3.4 and [Ca, Proposition 4.4(viii)]. In fact, we are able to give a more precise description of the Heller translates of Lie(pk) as follows:

Theorem 3.5. Let p ∤ k and let Lie(k) = ⨁_{s=1}^{m_{k}} P_{s} be a decomposition of Lie(k) into its indecomposable (projective) summands. We identify 𝔖_{p}×𝔖_{k} with the subgroup Δ_{k}(𝔖_{p})𝔖_{k}^{[p]} of 𝔖_{pk}.

(i) If p = 2 then, for all i ∈ ℤ,

Ω^{i}(Lie(2k)) ≅ ⨁_{s=1}^{m_{k}} Ω^{0}(Ind_{𝔖_{2}×𝔖_{k}}^{𝔖_{2k}}(𝔽 ⊠ P_{s})).

Furthermore, for each s ∈ {1, …, m_{k}}, the 𝔽𝔖_{2k}-module Ω^{0}(Ind_{𝔖_{2}×𝔖_{k}}^{𝔖_{2k}}(𝔽 ⊠ P_{s})) is indecomposable.

(ii) If p is odd and i ≡ j (mod 2p − 2) where j ∈ {0, 1, …, 2p − 3}, then

Ω^{i}(Lie(pk)) ≅ \begin{cases}⨁_{s=1}^{m_{k}} Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(S^{(p−j−1,1^{j+1})} ⊠ P_{s})), & if 0 ≤ j ≤ p − 2; \\ ⨁_{s=1}^{m_{k}} Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(S_{(j−p+3,1^{2p−j−3})} ⊠ P_{s})), & if p − 1 ≤ j ≤ 2p − 3. \end{cases}

Furthermore, for any admissible j, ℓ and s, the 𝔽𝔖_{pk}-modules

Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(S^{(p−j−1,1^{j+1})} ⊠ P_{s})) and Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(S_{(ℓ−p+3,1^{2p−ℓ−3})} ⊠ P_{s}))

are indecomposable.

Proof. Since Lie(k) = ⨁_{s=1}^{m_{k}} P_{s}, we have, by Corollary 3.4 and [Ca, Proposition 4.4(viii)],

Ω^{i}(Lie(pk)) ≅ Ω^{i+1} (Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(𝔽 ⊠ Lie(k)))

= Ω^{i+1} (⨁_{s=1}^{m_{k}} Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(𝔽 ⊠ P_{s}))

≅ ⨁_{s=1}^{m_{k}} Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{i+1}(𝔽) ⊠ P_{s})).

Now the first assertions of both parts (i) and (ii) follow from Lemma 2.3.

By [ES, Theorem 2], there is a one-to-one multiplicity-preserving correspondence between the non-projective indecomposable summands of Lie(pk) and the indecomposable summands of the projective 𝔽𝔖_{k}-module Lie(k). This implies that Lie(pk), and hence Ω^{i}(Lie(pk)) for any i ∈ ℤ, each has exactly m_{k} non-projective indecomposable summands in its decomposition into indecomposable summands.

Since Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{i+1}(𝔽) ⊠ P_{s}) is non-projective (as Ω^{i+1}(𝔽) is non-projective), we see

that Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{i+1}(𝔽) ⊠ P_{s})) ≠ 0 for all s ∈ {1, …, m_{k}}. Since Ω^{i}(Lie(pk)) has

exactly m_{k} non-projective summands, it follows that all Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{i+1}(𝔽) ⊠ P_{s})) are indecomposable. □

We remarked that the indecomposable summands of Lie(k) have been described in [DE, §3.3, Theorem].

Theorem 3.5(i) implies that Lie(2k) has period 1 when p = 2. When p ≥ 3, we may conclude directly (and independently of the results in [ELT]) that Lie(pk) is periodic, and its period divides 2p − 2. However, it is not immediately clear that its period is exactly 2p − 2, as we have yet to determine if it is possible for Ω^{i}(Lie(pk)) ≅ Ω^{j}(Lie(pk)) when 0 ≤ i, j ≤ 2p−3 with i ≠ j. The following corollary shows that Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{ℓ}(𝔽)⊠P_{s})) share a common vertex and fairly common sources, lending some evidence that such an undesirable isomorphism may occur.

Corollary 3.6. For all i ∈ ℤ, any indecomposable non-projective summand of the 𝔽𝔖_{pk}- module Ω^{i}(Lie(pk)) has a vertex Δ_{k}(C_{p}), and Δ_{k}(C_{p})-source 𝔽 when i is odd and Ω(𝔽) when i is even.

Proof. By Theorem 3.5, we may assume that i = 0 if p = 2 and i = 0, 1, …, 2p − 3 if p ≥ 3. In addition, any indecomposable non-projective summand of Ω^{i}(Lie(pk)) is of the form Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{i+1}(𝔽)⊠P_{s})) for some s ∈ {1, …, m_{k}}. Let ℓ = i+1. By Proposition 2.4, Ω^{ℓ}(𝔽_{𝔖_{p}}) has a vertex C_{p} and C_{p}-source 𝔽 if ℓ is even. Also, we have

Ω^{0} (Ind_{C_{p}}^{𝔖_{p}} Ω(𝔽)) ≅ Ω (Ind_{C_{p}}^{𝔖_{p}} 𝔽) ≅ Ω(𝔽) ⊕ Ω^{3}(𝔽) ⊕ ⋯ ⊕ Ω^{2p−3}(𝔽).

This shows that Ω^{ℓ}(𝔽_{𝔖_{p}}) has a vertex C_{p} and C_{p}-source Ω(𝔽) if ℓ is odd. Let S_{ℓ} be the C_{p}-source of Ω^{ℓ}(𝔽). Since P_{s} is projective, and hence injective, we see that P_{s} is a direct summand of 𝔽𝔖_{k}. Thus we obtain Ω^{0}(Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}}(Ω^{ℓ}(𝔽) ⊠ P_{s})) is a direct summand of

Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}} ((Ind_{C_{p}}^{𝔖_{p}} S_{ℓ}) ⊠ 𝔽𝔖_{k}) ≅ Ind_{Δ_{k}(C_{p})}^{𝔖_{pk}} S_{ℓ}. This completes our proof. □

Remark. In [ES, para. before Lemma 13], it is shown that C_{p}^{[k]} is an vertex of any nonprojective indecomposable summand of S^{p}(U), where U is a projective 𝔽𝔖_{k}-module. Since vertices are invariant under Heller translates, we may alternatively obtain C_{p}^{[k]} as an vertex of any indecomposable non-projective summand of Ω^{i}(Lie(pk)) at once by specialising this result to the case when U = Lie(k).

We end this section with a result which may be of independent interest.

Corollary 3.7. The Lie module Lie(pk) is an endo-p-permutation module.

Proof. By Corollary 3.6 , since Ω^{−1}(Lie(pk)) is a direct sum of trivial source modules, it is a p-permutation module and hence an endo-p-permutation module. By definition, Lie(pk) ≅ Ω(Ω^{−1}(Lie(pk))) ⊕ Q for some projective 𝔽𝔖_{pk}-module Q. Since the class of endo-p-permutation is closed under taking Heller translations and taking direct sums with projective modules, it follows that Lie(pk) is endo-p-permutation. □

4. The period of Lie(pk)

In this section, we prove Theorem 1.1(ii). Throughout this section, fix k ∈ ℤ^{+} with p ∤ k, C_{p} and C_{k} denote fixed cyclic subgroups of 𝔖_{p} and 𝔖_{k} generated by a p-cycle and a k-cycle respectively, and 𝔽_{δ} denotes a fixed faithful one-dimensional 𝔽C_{k}-module. Also, let Δ = Δ_{k}(𝔖_{p}) and D = Δ_{k}(𝔖_{p})C_{k}^{[p]} (= 𝔖_{p} × C_{k}).

By Theorem 1.1(i) and Lemma 2.3, we already knew that Lie(pk) has period 1 when p = 2, and that Lie(pk) has period dividing 2p − 2 when p ≥ 3, with

Ω^{i}(Lie(pk)) ≅ Ω^{i+1} (Ind_{𝔖_{p}×𝔖_{k}}^{𝔖_{pk}} (𝔽 ⊠ Lie(k))) ≅ Ω^{i+1} (Ind_{𝔖_{p}×C_{k}}^{𝔖_{pk}} (𝔽 ⊠ 𝔽_{δ})) ,

where the last isomorphism is given by Lemma 3.1. To complete the proof of Theorem 1.1(ii), we proceed with a careful analysis of the module

Λ_{k} := Ind_{𝔖_{p}×C_{k}}^{𝔖_{pk}} (𝔽 ⊠ 𝔽_{δ}) = Ind_{D}^{𝔖_{pk}} (𝔽 ⊠ 𝔽_{δ}) .

By Mackey decomposition,

Res_{Δ_{k}(𝔖_{p})}^{𝔖_{pk}} Λ_{k} ≅ Res_{Δ}^{𝔖_{pk}} Ind_{D}^{𝔖_{pk}}(𝔽 ⊠ 𝔽_{δ}) ≅ ⨁_{x∈Δ\𝔖_{pk}/D} Ind_{xDx^{−1}∩Δ}^{Δ} (x ⊗ (𝔽 ⊠ 𝔽_{δ})).

Clearly, if p ∤ |xDx^{−1} ∩ Δ|, then x ⊗ (𝔽 ⊠ 𝔽_{δ}) is projective as an 𝔽(xDx^{−1} ∩ Δ)-module, and hence Ind_{xDx^{−1}∩Δ}^{Δ} (x ⊗ (𝔽 ⊠ 𝔽_{δ})) is projective. It remains to describe Ind_{xDx^{−1}∩Δ}^{Δ} (x ⊗ (𝔽 ⊠ 𝔽_{δ})) when p | |xDx^{−1} ∩ Δ|. We begin with a preliminary result.

Proposition 4.1. Let x ∈ 𝔖_{pk}.

(i) If p | |xDx^{−1}∩Δ|, then there exists x_{0} ∈ (∏_{r=1}^{k} C_{p}[r])𝔖_{k}^{[p]} such that Δx_{0}D = ΔxD.

(ii) If x ∈ (∏_{r=1}^{k} C_{p}[r])𝔖_{k}^{[p]}, say x = yz where y ∈ ∏_{r=1}^{k} C_{p}[r] and z ∈ 𝔖_{k}^{[p]}, then

xDx^{−1} ∩ Δ = yΔy^{−1} ∩ Δ = \begin{cases}Δ, & if y ∈ Δ_{k}(C_{p}), \\ Δ_{k}(C_{p}), & otherwise. \end{cases}

Proof.

(i) Let a ∈ xDx^{−1} ∩Δ be an element of order p. Then a = Δ_{k}(σ) = xΔ_{k}(ρ)τ^{[p]}x^{−1} for some σ, ρ ∈ 𝔖_{p} and τ ∈ C_{k}. Since a has order p, while the order of xΔ_{k}(ρ)τ^{[p]}x^{−1} is the least common multiple of those of ρ and τ, and τ has order coprime to p, we see that τ = 1 and ρ has order p. Let π be a generator of C_{p}. Then π, σ and ρ are p-cycles in 𝔖_{p}, so there exist ξ, η ∈ 𝔖_{p} such that σ = ξπξ^{−1} and ρ = ηπη^{−1}. Thus

Δ_{k}(ξ)Δ_{k}(π)Δ_{k}(ξ)^{−1} = Δ_{k}(σ) = xΔ_{k}(ρ)x^{−1} = xΔ_{k}(η)Δ_{k}(π)Δ_{k}(η)^{−1}x^{−1}.

Let x_{0} = Δ_{k}(ξ^{−1})xΔ_{k}(η). Then Δx_{0}D = ΔxD, and Δ_{k}(π) = x_{0}Δ_{k}(π)x_{0}^{−1}, so that x_{0} ∈ C_{𝔖_{pk}}(Δ_{k}(C_{p})) = (∏_{r=1}^{k} C_{p}[r])𝔖_{k}^{[p]}.

(ii) Let a ∈ xDx^{−1} ∩ Δ. Then a = Δ_{k}(σ) = xΔ_{k}(ρ)τ^{[p]}x^{−1} for some σ, ρ ∈ 𝔖_{p} and τ ∈ C_{k}. Since z commutes with Δ_{k}(ρ), we have

∏_{r=1}^{k} 𝔖_{p}[r] ∋ Δ_{k}(ρ)^{−1}y^{−1}Δ_{k}(σ)y = zτ^{[p]}z^{−1} ∈ 𝔖_{k}^{[p]},

so that τ = 1 and a = Δ_{k}(σ) = yΔ_{k}(ρ)y^{−1}. Thus a ∈ Δ ∩ yΔy^{−1}. Conversely, xΔx^{−1} = yΔy^{−1} since z centralises Δ, so that

yΔy^{−1} ∩ Δ = xΔx^{−1} ∩ Δ ⊆ xDx^{−1} ∩ Δ.

This proves the first equality.

Now we prove the second equality. If y ∈ Δ_{k}(C_{p}), then

yΔy^{−1} ∩ Δ = Δ ∩ Δ = Δ.

Assume thus y ∉ Δ_{k}(C_{p}). Let y = ∏_{r=1}^{k} y_{r}[r] where y_{r} ∈ C_{p} for all r. By assumption, y_{i} ≠ y_{j} for some i and j. Let a ∈ yΔy^{−1} ∩ Δ. Then a = Δ_{k}(σ) = yΔ_{k}(ρ)y^{−1} for some σ, ρ ∈ 𝔖_{p}. Thus y_{i}ρy_{i}^{−1} = σ = y_{j}ρy_{j}^{−1}, so that ρ ∈ C_{𝔖_{p}}(y_{i}^{−1}y_{j}) = C_{p}. Hence, a = yΔ_{k}(ρ)y^{−1} = Δ_{k}(ρ) ∈ Δ_{k}(C_{p}). Consequently, yΔy^{−1} ∩ Δ ⊆ Δ_{k}(C_{p}). The reverse inclusion is clear.

□

Theorem 4.2. Let

Γ_{1} = {x ∈ Δ\𝔖_{pk}/D : xDx^{−1} ∩ Δ = Δ},
Γ_{2} = {x ∈ Δ\𝔖_{pk}/D : |xDx^{−1} ∩ Δ| = p}.

Then

Res_{Δ_{k}(𝔖_{p})}^{𝔖_{pk}} Λ_{k} ≅ 𝔽^{⊕|Γ_{1}|} ⊕ (Ind_{C_{p}}^{𝔖_{p}} 𝔽)^{⊕|Γ_{2}|} ⊕ P
≅ 𝔽^{⊕(|Γ_{1}|+|Γ_{2}|)} ⊕ Ω^{2}(𝔽)^{⊕|Γ_{2}|} ⊕ Ω^{4}(𝔽)^{⊕|Γ_{2}|} ⊕ ⋯ ⊕ Ω^{2p−4}(𝔽)^{⊕|Γ_{2}|} ⊕ P

for some projective 𝔽Δ_{k}(𝔖_{p})-module P. Here, Ind_{C_{p}}^{𝔖_{p}} 𝔽 is an 𝔽Δ_{k}(𝔖_{p})-module via the isomorphism 𝔖_{p} ≅ Δ_{k}(𝔖_{p}).

Proof. By Mackey decomposition,

Res_{Δ}^{𝔖_{pk}} Λ_{k} ≅ ⨁_{x∈Δ\𝔖_{pk}/D} Ind_{xDx^{−1}∩Δ}^{Δ} (x ⊗ (𝔽 ⊠ 𝔽_{δ})).

We have already seen that Ind_{xDx^{−1}∩Δ}^{Δ}(x ⊗ (𝔽 ⊠ 𝔽_{δ})) is projective if p ∤ |xDx^{−1} ∩ Δ|. Note that the statement of our theorem involves only the cardinality of Γ_{1} and Γ_{2} which are independent of the actual choice of double coset representatives Δ\𝔖_{pk}/D. As such we may assume that, by Proposition 4.1(i), whenever x ∈ Δ\𝔖_{pk}/D such that p | |xDx^{−1}∩Δ|, we have x ∈ (∏_{r=1}^{k} C_{p}[r])𝔖_{k}^{[p]}. Thus, by Proposition 4.1(ii), we have

Res_{Δ}^{𝔖_{pk}} Λ_{k} ≅ ⨁_{x∈Γ_{1}} Ind_{Δ}^{Δ} (x ⊗ (𝔽 ⊠ 𝔽_{δ})) ⊕ ⨁_{x∈Γ_{2}} Ind_{Δ_{k}(C_{p})}^{Δ} (x ⊗ (𝔽 ⊠ 𝔽_{δ})) ⊕ P
≅ 𝔽^{⊕|Γ_{1}|} ⊕ (Ind_{C_{p}}^{𝔖_{p}} 𝔽)^{⊕|Γ_{2}|} ⊕ P

for some projective 𝔽Δ-module P. The final isomorphism is obtained by using Proposition 2.4. □

Keeping the notations of Theorem 4.2, we may assume that 1 ∈ Γ_{1}. Thus Γ_{1} ≠ ∅. Using Theorem 4.2 and Lemma 2.3, we obtain the follow corollary.

Corollary 4.3. Let k ∈ ℤ^{+} with p ∤ k. Then Res_{Δ_{k}(𝔖_{p})}^{𝔖_{pk}} Λ_{k} has the same period as the trivial 𝔽𝔖_{p}-module 𝔽.

We are now ready to prove Theorem 1.1(ii).

Proof of Theorem 1.1(ii). Let the actual period of Lie(pk) be l_{p} and the asserted period be n_{p}. By Theorem 1.1(i), we have l_{p} ≤ n_{p}. On the other hand, Lemma 2.1(ii,iii) and Corollary 4.3 show that l_{p} ≥ n_{p}. Thus l_{p} = n_{p} as desired. □

We end the paper with the following remarks.

Remarks.

(1) One may use Theorem 4.2 to show that, when p is odd, Ω^{i}(Res_{Δ_{k}(𝔖_{p})}^{𝔖_{pk}} Λ_{k}) ≅

Ω^{j}(Res_{Δ_{k}(𝔖_{p})}^{𝔖_{pk}} Λ_{k}) if 0 < |i − j| < 2p − 2. This implies that

Ω^{i}(Lie(pk)) ≅ Ω^{i+1}(Λ_{k}) ≅ Ω̸^{j+1}(Λ_{k}) ≅ Ω^{j}(Lie(pk))

whenever 0 < |i − j| < 2p − 2. Together with Theorem 1.1(i) (or Theorem 3.5), this gives an alternative proof of Theorem 1.1(ii).

(2) If the conjecture made in [ELT] on the complexities of the Lie modules holds true; namely, the complexity of the Lie module Lie(p^{m}k) is m whenever p ∤ k, then Lie(pk) with p ∤ k are the only ones with complexity 1, and hence our results deal with all periodic Lie modules.

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(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

(K. M. Tan) Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076.

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