Vertices of Lie Modules

Roger M. Bryant, Susanne Danz, Karin Erdmann, and Jürgen Müller September 9, 2013

Abstract

Let Lie_{F} (n) be the Lie module of the symmetric group 𝔖_{n} over a field F of characteristic p > 0, that is, Lie_{F} (n) is the left ideal of F 𝔖_{n} generated by the Dynkin–Specht–Wever element ω_{n}. We study the problem of parametrizing non-projective indecomposable summands of Lie_{F} (n), via describing their vertices and sources. Our main result shows that this can be reduced to the case when n is a power of p. When n = 9 and p = 3, and when n = 8 and p = 2, we present a precise answer. This suggests a possible parametrization for arbitrary prime powers. Keywords: Lie module, vertex, source, endo-permutation module, symmetric group MR Subject Classification: 20C20, 20C30, 20G43

1 Introduction

The Lie module of the symmetric group 𝔖_{n} occurs in various contexts within algebra and topology, where the name-giving property is its close relation to the free Lie algebra; for more details, see for example the introduction in [14]. In the present paper, letting F be an algebraically closed field of characteristic p > 0, we realize the Lie module Lie_{F} (n) of 𝔖_{n}, for n ⩾ 2, as the submodule F 𝔖_{n}ω_{n} of the regular F 𝔖_{n}-module, where

ω_{n} := (1 − c_{2})(1 − c_{3}) ⋯ (1 − c_{n}) ∈ F 𝔖_{n}

is the Dynkin–Specht–Wever element of F 𝔖_{n}, where in turn c_{k} ∈ 𝔖_{n} is the backward cycle (k, k − 1, …, 2, 1).

1.1. It is well known that ω_{n}^{2} = nω_{n} ∈ F 𝔖_{n}. Hence if p does not divide n, then ω_{n}/n ∈ F 𝔖_{n} is an idempotent, so that Lie_{F} (n) is then a direct summand of the regular F 𝔖_{n}-module and is, thus, projective. In the present paper we are interested in the case when p divides n, which we assume from now on in this section. Then Lie_{F} (n) cannot be projective; for otherwise dim(Lie_{F} (n)) = (n − 1)! would have to be divisible by the p-part of n!, which is not the case. Therefore, in this case Lie_{F} (n) admits a decomposition

Lie_{F} (n) = Lie_{F}^{pr}(n) ⊕ Lie_{F}^{pf}(n),

where Lie_{F}^{pr}(n) is a projective F 𝔖_{n}-module and where Lie_{F}^{pf}(n) ≠ {0} is a projective-free F 𝔖_{n}-module.

The asymptotic behaviour of the quotient dim(Lie_{F}^{pr}(n))/ dim(Lie_{F} (n)) has recently been studied by Erdmann–Tan [14], and by Bryant–Lim–Tan [5]. By [5, Thm. 1.2], one has

\frac{dim(Lie_{F}^{pr}(n))}{dim(Lie_{F} (n))} ⟶ 1,

as n ⟶ ∞ in ℕ ∖ {p^{k} | k ⩾ 0}. Moreover, it is conjectured in [5] that this should remain true when allowing n to vary over all natural numbers. This suggests that Lie_{F}^{pf}(n) should be small, compared with the entire Lie module Lie_{F} (n).

Moreover, by work of Erdmann–Tan [15], we also know that the projective-free part Lie_{F}^{pf}(n) of Lie_{F} (n) always belongs to the principal block of F 𝔖_{n}, and Bryant–Erdmann [4] have studied indecomposable direct sum decompositions of the, necessarily projective, part of Lie_{F} (n) not contained in the principal block of F 𝔖_{n}. This leaves open, next to Lie_{F}^{pf}(n), only the direct sum decompositions of the component of Lie_{F}^{pr}(n) belonging to the principal block of F 𝔖_{n}.

1.2. One key ingredient of our approach is a decomposition theorem, expressing Lie_{F} (n) as a direct sum of pieces related to Lie modules Lie_{F} (p^{d}), for various d such that p^{d} divides n. This is obtained by translating the Bryant–Schocker decomposition theorem [6] for Lie powers to Lie modules, using work of Lim–Tan [24]. This paves the way to reduce questions on Lie modules to the case when n is a power of p, and puts the Lie modules Lie_{F} (p^{d}) into the focus of study. In particular, one is tempted to ask whether there is a neat description of the indecomposable direct summands of Lie_{F} (n) in terms of those of Lie_{F} (p^{d}), where d varies as indicated above. This has been fully accomplished for the case where p divides n but p^{2} does not, with a different line of reasoning, by Erdmann–Schocker [13], while the general case remains a mystery and is subject to further investigations.

Very little information concerning the decomposition of the principal block component of Lie_{F} (p^{d}) is available in the literature, and the projective-free part Lie_{F}^{pf}(p^{d}) is very poorly understood, even for very small exponents d: to our knowledge, the only cases dealt with systematically are the modules Lie_{F}^{pf}(p), that is, the case d = 1, by Erdmann–Schocker [13]; and, apart from the easy case Lie_{F} (4) = Lie_{F}^{pf}(4), there are just partial results for Lie_{F} (8), by Selick–Wu [33]. The aim of this paper now is to investigate indecomposable direct summands of Lie_{F} (p^{d}), for a few further small values of p and d.

The major obstacle here is that, due to the exponential growth of the dimension of Lie modules in terms of n, these modules quickly become very large. Hence, to proceed further in this direction, we apply computational techniques. More precisely, by this approach we are now able to give a complete description of the Lie modules Lie_{F} (8) of dimension 5040, and Lie_{F} (9) of dimension 40320.

Actually, in both cases it turns out that the projective-free part of the Lie module is already indecomposable, where Lie_{F}^{pf}(8) has dimension 816, and Lie_{F}^{pf}(9) has dimension 1683. In view of these results, and those on Lie_{F}^{pf}(4) and Lie_{F}^{pf}(p) mentioned above, the question arises whether Lie_{F}^{pf}(p^{d}) is always indecomposable.

1.3. To analyze the projective-free part of Lie_{F} (n), we are, in particular, interested in the Green vertices and sources of the indecomposable direct summands of Lie_{F}^{pf}(n). Using the reduction result mentioned above, to some extent we are able to reduce this problem for arbitrary n to the case where n is a p-power.

Moreover, we are able to compute vertices and sources of Lie_{F}^{pf}(8) and Lie_{F}^{pf}(9). It turns out that both modules are endo-p-permutation modules, in the sense of Urfer [36], their vertices are regular elementary abelian subgroups of 𝔖_{8} and 𝔖_{9}, respectively, and their sources are endo-permutation modules, in the sense of Dade [10], whose class in the Dade group we are able to determine. It is surprising to us to see the class of endo-permutation modules appear in this context.

Hence, in view of these results, and those concerning Lie_{F}^{pf}(4) and Lie_{F}^{pf}(p), one may wonder whether Lie_{F}^{pf}(p^{d}), assumed to be indecomposable, always is an endo-p-permutation module having regular elementary abelian vertices and endo-permutation sources, and, if so, what the class of a source in the Dade group looks like. If this holds true, then, by our reduction results, any indecomposable direct summand of any Lie module will have vertices and sources sharing the same properties.

1.4. This paper is organized as follows: in Section 2 we provide the necessary prerequisites; in particular, we recall the notions of Green vertices and sources, endo-permutation and endo-p-permutation modules, and the Dade group. In Section 3 we introduce Lie modules; in order to make this paper sufficiently self-contained, we also discuss Lie powers and their relation to Lie modules via the Schur functor, as well as variations on the construction and basic properties of Lie modules.

In Section 4 we collect the explicit computational results we have obtained for specific examples; in particular, we present more details of the computational ideas and tools we have been using, we revisit Lie_{F} (p) and Lie_{F} (4), and discuss the major examples Lie_{F} (8) and Lie_{F} (9), whose indecomposable direct sum decomposition we determine, together with vertices and sources of the non-projective indecomposable direct summands occurring.

In Section 5 we present a reduction, eventually showing that vertices and sources of indecomposable direct summands of Lie modules in general can be described in terms of the results in the p-power case; in order to do so, in Theorem 5.4 we provide a description of vertices and sources of indecomposable direct summands of modules for wreath products, in Theorem 5.5 we present the decomposition theorem for Lie modules mentioned above, and in Theorem 5.7 these are combined to prove the reduction result.

Acknowledgement: The second author’s research has been supported through a Marie Curie Career Integration Grant (PCIG10-GA-2011-303774). The second and fourth authors have also been supported by the DFG Priority Programme ‘Representation Theory’ (grant # DA1115/3-1). The first and third authors have been supported by EPSRC Standard Research Grant # EP/G025487/1.

2 Prerequisites

2.1. Generalities. (a) Throughout this paper, without further notice, we assume that F is an algebraically closed field of characteristic p > 0. Whenever G is a finite group, an F G-module is always understood to be a left F G-module of finite F -dimension, unless stated otherwise. If M and N are F G-modules such that N is isomorphic to a direct summand of M then we write N | M.

Whenever G is a finite group, H is a normal subgroup of G, and M is an F [G/H]-module, we denote by Inf_{G/H}^{G}(M) the F G-module obtained from M via inflation. More generally, by abuse of notation, given a fixed epimorphism of groups G ↠ K and an F K-module M, we denote the F G-module obtained via inflation with respect to this epimorphism by Inf_{K}^{G}(M) as well.

(b) By 𝔖_{n} we denote the symmetric group of degree n ⩾ 1, where permutations in 𝔖_{n} are also multiplied from right to left. So, for instance, we have (1, 2)(2, 3) = (1, 2, 3).

We assume the reader to be familiar with the basic notions of the representation theory of the symmetric group. For detailed background information, we refer to [20, 21]. The Specht modules S^{λ} of the group algebra F 𝔖_{n} will, as usual, be labelled by the partitions λ of n, and the simple F 𝔖_{n}-modules D^{λ} by the p-regular partitions of n. Furthermore, we denote by P^{λ} a projective cover of D^{λ}.

(c) Whenever G is a finite group with subgroups H and K, we write H ⩽_{G} K if H is G-conjugate to a subgroup of K, and we write H =_{G} K if H is G-conjugate to K.

Next we recall the notions of vertices and sources of indecomposable modules over group algebras, and we summarize some basic properties of endo-permutation modules over finite p-groups. The latter class of modules has been introduced by Dade [10], as generalizations of permutation modules. They have proved to play an important role in modular representation theory of finite groups, and, as we will see in subsequent sections, also appear naturally in the context of Lie modules. For a detailed account on the theory of vertices and sources we refer the reader to [30, Chap. 4.3]. Background information concerning endo-permutation modules can be found in [10, 11] and in [34, §28].

2.2. Vertices and sources. (a) Let G be a finite group, and let M be an indecomposable F G-module. By Green’s Theorem [17], we can assign to M a G-conjugacy class of p-subgroups of G, the vertices of M. A vertex Q of M is characterized by the property that Q is minimal such that M is relatively Q-projective, that is, M is isomorphic to a direct summand of Ind_{Q}^{G}(N), for some indecomposable F Q-module N. In particular, M is projective if and only if Q = {1}.

Given a vertex Q of M, an indecomposable F Q-module L such that M is isomorphic to a direct summand of Ind_{Q}^{G}(L) is called a Q-source of M, and is unique up to isomorphism and conjugation with elements in N_{G}(Q). Moreover, Q is also a vertex of L.

(b) Let B be the block of F G containing M. If Q is a vertex of M then there are a defect group R of B and a Sylow p-subgroup P of G such that Q ⩽ R ⩽ P. Moreover, as a consequence of Green’s Indecomposability Theorem [17], |P : Q| divides dim(M).

(c) Suppose that H ⩽ G is any subgroup of G and that N is an indecomposable direct summand of Res_{H}^{G}(M) with vertex R and R-source L′. Then there are a vertex Q of M and a Q-source L of M such that R ⩽ Q and L′ | Res_{R}^{Q}(L). This is seen as follows:

Let Q be any vertex of M, and let L be any Q-source of M. Then we have L′ | Res_{R}^{H}(N) | Res_{R}^{G}(M) and M | Ind_{Q}^{G}(L), thus L′ | Res_{R}^{G}(Ind_{Q}^{G}(L)). Now, by Mackey’s Theorem and the fact that L′ has vertex R, this implies L′ | Res_{R}^{^{g}Q}(^{g}L), for some g ∈ G such that R ⩽ ^{g}Q. But gQ is also a vertex of M, and ^{g}L is a ^{g}Q-source of M, whence the claim.

In particular, if H ⩽ G is such that M is relatively H-projective, then M | Ind_{H}^{G}(Res_{H}^{G}(M)) implies that there is an indecomposable direct summand of Res_{H}^{G}(M) sharing a vertex and a source with M.

(d) Suppose, conversely, that H ⩾ G is a finite overgroup of G and that N is an indecomposable direct summand of Ind_{G}^{H}(M). Then, given a vertex Q of M and a Q-source L of M, there is a vertex P of N such that P ⩽ Q, and there is some P -source of N that is isomorphic to a direct summand of Res_{P}^{Q}(L). This is seen as follows:

Let P be any vertex of N, and let L′ be any P -source of N. Then, by Mackey’s Theorem again, we get L′ | Res_{P}^{^{h}Q}(^{h}L), for some h ∈ H such that P ⩽ ^{h}Q. In other words, we have h^{−1}L′ | Res_{^{h^{−1}}P}^{Q} (L), where ^{h^{−1}}P is also a vertex of N, and ^{h^{−1}}L′ is a ^{h^{−1}}P -source of N, whence the claim. (Note that we cannot conclude that ^{h^{−1}}L′ is an arbitrary ^{h^{−1}}P -source of N, since the conjugating element h might depend on the choice of L′.)

In particular, since M | Res_{G}^{H}(Ind_{G}^{H}(M)), there is some indecomposable direct summand of Ind_{G}^{H}(M) sharing a vertex and a source with M.

2.3. Endo-permutation modules. An F G-module M, where G is a finite group, is called an endo-permutation module if its F -endomorphism ring End_{F} (M) ≅ M ⊗M^{∗} is a permutation F G-module. We list some properties of endo-permutation modules that we will need later in this paper:

(a) Permutation modules are endo-permutation modules. The class of endo-permutation modules is closed under taking F -linear duals, direct summands, tensor products, restriction to subgroups, inflation from factor groups, and taking Heller translates Ω and Ω^{−1}, but it is neither closed under taking direct sums, nor under induction to finite overgroups. In particular, any indecomposable endo-permutation module has endo-permutation sources.

(b) The problem of classifying the indecomposable endo-permutation modules for finite p-groups P has been worked on by various people. The final classification result was obtained by Bouc [1], but when P is abelian, the following classification result is already due to Dade [11]. We will describe this result below; this is the version we will need.

Note that it is indeed sufficient to classify the indecomposable endo-permutation F P - modules with vertex P, since if M is an indecomposable endo-permutation F P -module with vertex Q < P then any Q-source S of M is an endo-permutation F Q-module with vertex Q, and by Green’s Indecomposability Theorem [17] we have M ≅ Ind_{Q}^{P}(S).

2.4 Theorem ([11, Thm. 12.5]). Let P be an abelian p-group, and let M be an indecomposable endo-permutation F P -module with vertex P. Then M is, up to isomorphism, the unique indecomposable direct summand of

⨂_{|P:Q|⩾3} Inf_{P/Q}^{P}(Ω^{n_{Q}}(F_{P/Q}))

having vertex P. Here n_{Q} ∈ ℤ, for Q < P, is uniquely determined by M if P/Q is non-cyclic; otherwise n_{Q} is uniquely determined modulo 2.

In other words, the isomorphism types of indecomposable endo-permutation F P -modules with abelian vertex P are in bijection with the elements of the Dade group

𝔇(P) ≅ (  ∑_{|P:Q|⩾3, P/Q non-cyclic} ℤ) ⊕ (  ∑_{|P:Q|⩾3, P/Q cyclic} ℤ/2ℤ) .

For a precise definition of the Dade group of an arbitrary p-group P and further details, see [10] and [34, §29]. Whenever S is an indecomposable endo-permutation F P -module with vertex P, its image in 𝔇(P) will be denoted by [S]. If S and S′ are indecomposable endopermutation F P -modules with vertex P then their tensor product S ⊗ S′ has a unique (up to isomorphism) indecomposable direct summand T with vertex P, and the multiplication in 𝔇(P) is then given by [S] · [S′] := [T].

2.5. Endo-p-permutation modules. According to Urfer [36, 35], one can weaken the notion of endo-permutation modules as follows: an F G-module M, where G is a finite group, is called an endo-p-permutation module, if its F -endomorphism ring End_{F} (M) ≅ M ⊗ M^{∗} is a p-permutation F G-module, that is, all its indecomposable direct summands are trivial-source modules.

(a) Again, one has the following standard properties of endo-p-permutation modules: endopermutation modules are endo-p-permutation modules. The class of endo-p-permutation modules is closed under taking F -linear duals, Heller translates, direct summands, tensor products, restriction to subgroups, and inflation from factor groups, but it is neither closed under taking direct sums, nor under induction to finite overgroups. Note that for finite p-groups the classes of endo-p-permutation modules and of endo-permutation modules coincide, thus any indecomposable endo-p-permutation module has endo-permutation sources.

(b) A characterization of indecomposable endo-p-permutation module in terms of vertices and sources is given in [36, Thm. 1.5]. For the cases of interest in the present paper it can be rephrased as follows:

Let P ⩽ G be a p-group. As before, for any indecomposable endo-permutation F P -module S with vertex P let [S] ∈ 𝔇(P) be the associated element of the Dade group. Then [S] ∈ 𝔇(P) is called G-stable if

Res_{P ∩^{g}P}^{P} ([S]) = Res_{P ∩^{g}P}^{^{g}P} ([^{g}S]) ∈ 𝔇(P ∩^{g}P) , for all g ∈ G .

Then, by [36, Thm. 1.5], an indecomposable F G-module with vertex P and P -source S is an endo-p-permutation module if and only if S is an endo-permutation module such that [S] is G-stable.

(c) As in part (b), let P ⩽ G be a p-group. In good situations the G-stable elements of 𝔇(P) are described by a Burnside-type fusion argument as follows:

Let N_{G}(P) control fusion in P, that is, whenever Q ⩽ P and g ∈ G are such that ^{g}Q ⩽ P, there are some h ∈ N_{G}(P) and z ∈ C_{G}(Q) such that g = hz. Then, by [36, La. 1.8, Prop. 1.9], an element [S] ∈ 𝔇(P) is G-stable if and only if it is fixed by the conjugation action of N_{G}(P) on 𝔇(P). (Note that we do not require an additional saturation condition here, as is done in [36]: an inspection of the proofs of [36, La. 1.8, Prop. 1.9] shows that they are valid under the assumptions made here.)

Moreover, if P is abelian then an element [S] ∈ 𝔇(P) is an N_{G}(P)-fixed point if and only if the associated function Q ⟶ n_{Q} is constant on the N_{G}(P)-orbits on {Q < P | |P : Q| ⩾ 3}.

3 The Lie Module of the Symmetric Group

We begin this section by introducing the Lie module for F 𝔖_{n}, we list some of its properties, and briefly discuss variations. Many of these observations are certainly well known to the experts, but explicit references are not too easy to find. Thus we recall them here for the readers’ convenience, and to make this paper as self-contained as possible.

3.1. Lie powers. (a) Let GL_{n}(F) be the general linear group over F, where n ⩾ 1, which acts naturally on F^{n}, and let {e_{1}, …, e_{n}} be the standard basis of F^{n}. We may view 𝔖_{n} as a subgroup of GL_{n}(F), by identifying a permutation π ∈ 𝔖_{n} with the corresponding permutation matrix in GL_{n}(F).

The r-th tensor power (F^{n})^{⊗r}, where r ⩾ 1, is an F [GL_{n}(F)]-module by way of the diagonal action. Thus, via restriction (F^{n})^{⊗r} becomes an F 𝔖_{n}-module, where the symmetric group 𝔖_{n} acts by substitutions, that is,

π : v_{1} ⊗ ⋯ ⊗ v_{r} ⟼ πv_{1} ⊗ ⋯ ⊗ πv_{r}, for v_{1}, …, v_{r} ∈ F^{n}, π ∈ 𝔖_{n}.

On the other hand, (F^{n})^{⊗r} also carries a right F 𝔖_{r}-action ‘∗’ via place permutations, which hence centralizes the F [GL_{n}(F)]-action:

σ : v_{1} ⊗ ⋯ ⊗ v_{r} ⟼ (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ σ = v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(r)}, for v_{1}, …, v_{r} ∈ F^{n}, σ ∈ 𝔖_{r}.

(b) Now we consider the Lie bracket

κ_{2} : (F^{n})^{⊗2} ⟶ (F^{n})^{⊗2}, v_{1} ⊗ v_{2} ⟼ [v_{1}, v_{2}] := v_{1} ⊗ v_{2} − v_{2} ⊗ v_{1}, for v_{1}, v_{2} ∈ F^{n}.

By iteration, this yields the (left-normed) Lie bracket

κ_{r} : (F^{n})^{⊗r} ⟶ (F^{n})^{⊗r}, v_{1} ⊗ ⋯ ⊗ v_{r} ⟼ [[v_{1}, v_{2}, …, v_{r}] := [⋯ [[v_{1}, v_{2}], v_{3}], …, v_{r}] ,

for all r ⩾ 2, and for completeness we also let κ_{1} := id.

Hence we have κ_{r} ∈ End_{F} ((F^{n})^{⊗r}), for all r ⩾ 1, where we assume End_{F} ((F^{n})^{⊗r}) to act on (F^{n})^{⊗r} from the right, the action also being denoted by ‘∗’. The image (v_{1} ⊗⋯⊗v_{r})∗κ_{r} ∈ (F^{n})^{⊗r} of a pure tensor v_{1} ⊗ ⋯ ⊗ v_{r} ∈ (F^{n})^{⊗r} is called an (iterated) Lie bracket of length r.

The Lie bracket induces the structure of a Lie algebra on the tensor algebra T (F^{n}) := ⨁_{r⩾1}(F^{n})^{⊗r}. Hence, by definition, for all r ⩾ 2 we have (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ κ_{r} = [(v_{1} ⊗ ⋯ ⊗ v_{r−1}) ∗ κ_{r−1}, v_{r}]. Moreover, for r ⩾ 1, the right adjoint action of T (F^{n}) translates into

(v ⊗ (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ κ_{r}) ∗ κ_{r+1} = [v, (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ κ_{r}] ∈ (F^{n})^{⊗(r+1)}.  (1)

The map κ_{r} centralizes the F [GL_{n}(F)]-action, hence the image

L^{r}(F^{n}) := (F^{n})^{⊗r} ∗ κ_{r} ⊆ (F^{n})^{⊗r}

of κ_{r} is an F [GL_{n}(F)]-submodule of (F^{n})^{⊗r}, being called the r-th Lie power of F^{n}, where of course we have L^{1}(F^{n}) = F^{n} ∗ κ_{1} = F^{n} ∗ id = F^{n}. Thus we obtain the free Lie algebra on {e_{1}, …, e_{n}}

L(F^{n}) := ⨁_{r⩾1} L^{r}(F^{n}) ⊆ T (F^{n}) .

The fact that L(F^{n}) is free as a Lie algebra is well known, and is due to Witt.

(c) By Schur–Weyl duality, the action of κ_{r} is induced by the place permutation action of some element ω_{r} ∈ F 𝔖_{r}, which we are now going to determine:

For r ⩾ 1 let c_{r} := (r, r − 1, …, 1) ∈ 𝔖_{r}. Note that, of course, c_{1} = 1. Then the place permutation action yields

(v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ c_{r} = (v_{r} ⊗ v_{1} ⊗ ⋯ ⊗ v_{r−1}), for v_{1}, …, v_{r} ∈ F^{n} .

Now, for r = 2, we have (v_{1} ⊗ v_{2}) ∗ κ_{2} = [v_{1}, v_{2}] = v_{1} ⊗ v_{2} − v_{2} ⊗ v_{1} = (v_{1} ⊗ v_{2}) ∗ (1 − c_{2}), while for r ⩾ 3 and v_{1}, …, v_{r} ∈ F^{n} we get

(v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ κ_{r} = ((v_{1} ⊗ ⋯ ⊗ v_{r−1}) ∗ κ_{r−1}) ⊗ v_{r} − v_{r} ⊗ ((v_{1} ⊗ ⋯ ⊗ v_{r−1}) ∗ κ_{r−1})
= (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ (κ_{r−1} ⊗ id) − (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ (κ_{r−1} ⊗ id) ∗ c_{r}
= (v_{1} ⊗ ⋯ ⊗ v_{r}) ∗ (κ_{r−1} ⊗ id) ∗ (1 − c_{r}).

Thus, by induction on r ⩾ 2, this gives κ_{r} = ∗(1−c_{2})∗(1−c_{3})∗⋯∗(1−c_{r}) : (F^{n})^{⊗r} ⟶ (F^{n})^{⊗r}, so that, for r ⩾ 2, we have

ω_{r} := (1 − c_{2})(1 − c_{3}) ⋯ (1 − c_{r}) ∈ F 𝔖_{r} ,

and L^{r}(F^{n}) = (F^{n})^{⊗r} ∗ κ_{r} = (F^{n})^{⊗r} ∗ ω_{r}. The element ω_{r} is called the Dynkin–Specht–Wever element of F 𝔖_{r}; for completeness, since κ_{1} = id, we let ω_{1} := c_{1} ∈ F 𝔖_{1}. Note that we even have ω_{r} ∈ 𝔽_{p}𝔖_{r}, where 𝔽_{p} is the prime field of F.

3.2. Lie modules and the Schur functor. (a) Now let n ⩾ r. Then the classical Schur functor 𝔚^{r} takes homogeneous polynomial F [GL_{n}(F)]-modules of degree r to F 𝔖_{r}-modules, where, more precisely, an F [GL_{n}(F)]-module V is mapped to its (1^{r})-weight space 𝔚^{r}(V). In particular, for the F [GL_{n}(F)]-module (F^{n})^{⊗r} one more explicitly gets the following:

As mentioned above, the natural GL_{n}(F)-action on (F^{n})^{⊗r} induces a permutation action of 𝔖_{n}, and thus also a permutation action of 𝔖_{r}, on (F^{n})^{⊗r}. The vector e_{1} ⊗⋯⊗e_{r} ∈ (F^{n})^{⊗r} affords a regular 𝔖_{r}-orbit and, hence, induces an embedding of the regular F 𝔖_{r}-module into (F^{n})^{⊗r} via

F 𝔖_{r} ⟶ (F^{n})^{⊗r}, π ⟼ πe_{1} ⊗ ⋯ ⊗ πe_{r} = e_{π(1)} ⊗ ⋯ ⊗ e_{π(r)}, for π ∈ 𝔖_{r}.

The image of this embedding equals the (1^{r})-weight space

𝔚^{r}((F^{n})^{⊗r}) = Span_{F} ({e_{π(1)} ⊗ ⋯ ⊗ e_{π(r)} | π ∈ 𝔖_{r}}) ⊆ (F^{n})^{⊗r}

of the F [GL_{n}(F)]-module (F^{n})^{⊗r}. Moreover, the place permutation action of 𝔖_{r} on (F^{n})^{⊗r} restricts to 𝔚^{r}((F^{n})^{⊗r}), and via the above isomorphism 𝔚^{r}((F^{n})^{⊗r}) ≅ F 𝔖_{r} translates into right multiplication on F 𝔖_{r}.

From now on, suppose that n = r, which will be the case most relevant to us.

(b) Now one defines the Lie module Lie_{F} (n) of F 𝔖_{n} as the (1^{n})-weight space of the n-th Lie power L^{n}(F^{n}), that is,

Lie_{F} (n) := 𝔚^{n}(L^{n}(F^{n})) ⊆ (F^{n})^{⊗n} .

From L^{n}(F^{n}) = (F^{n})^{⊗n} ∗ κ_{n} ⊆ (F^{n})^{⊗n} one thus gets

Lie_{F} (n) = 𝔚^{n}((F^{n})^{⊗n} ∗ κ_{n}) = 𝔚^{n}((F^{n})^{⊗n}) ∗ κ_{n} = Span_{F} ({[[e_{π(1)}, …, e_{π(n)}] | π ∈ 𝔖_{n}}).

Via the isomorphism 𝔚^{n}((F^{n})^{⊗n}) ≅ F 𝔖_{n} of F 𝔖_{n}-modules, Lie_{F} (n) can be regarded as a submodule of the regular F 𝔖_{n}-module F 𝔖_{n}. Since the action of κ_{n} is induced by the place permutation action of ω_{n}, we get

Lie_{F} (n) ≅ F 𝔖_{n} · ω_{n} ⊆ F 𝔖_{n}.  (2)

Note that, in particular, Lie_{F} (1) ≅ F, the trivial F 𝔖_{1}-module.

Moreover, we observe that Lie_{F} (n) is already realized over the prime field 𝔽_{p} of F, that is, letting

Lie_{p}(n) := 𝔽_{p}𝔖_{n} · ω_{n} ⊆ 𝔽_{p}𝔖_{n} as 𝔽_{p}𝔖_{n}-modules,

we get Lie_{F} (n) ≅ F ⊗_{𝔽_{p}} Lie_{p}(n) as F 𝔖_{n}-modules. We will make use of this in order to facilitate explicit computations in Section 4.

3.3. Variations on Lie modules. Since there also exist slight modifications of the above modules in the literature, we briefly comment on variations of the construction:

(a) Firstly, starting with another vector e_{π(1)} ⊗ ⋯ ⊗ e_{π(n)} ∈ (F^{n})^{⊗n}, where π ∈ 𝔖_{n}, leads to a different identification of Lie_{F} (n) with a submodule of F 𝔖_{n}, namely to the F 𝔖_{n}-module F 𝔖_{n} · πω_{n}π^{−1}, that is, amounts to a renumbering.

(b) Secondly, taking right-normed Lie brackets instead, for r ⩾ 2 one gets

κ′_{r} : (F^{n})^{⊗r} ⟶ (F^{n})^{⊗r}, v_{1} ⊗ ⋯ ⊗ v_{r} ⟼ [v_{1}, v_{2}, …, v_{r}]] := [v_{1}, [v_{2}, …, [v_{r−1,}v_{r}] ⋯]];

we again let κ′_{1} := id. Since [v_{1}, v_{2}, …, v_{r}]] = (−1)^{r−1} · [[v_{r}, v_{r−1}, …, v_{1}], for r ⩾ 1, we get κ′_{r} = (−1)^{r−1} ∗ w_{r} ∗ κ_{r} : (F^{n})^{⊗r} ⟶ (F^{n})^{⊗r}, where w_{r} ∈ 𝔖_{r} is the longest element of 𝔖_{r} in the Coxeter sense, that is,

w_{r} = (1, r)(2, r − 1) ⋯

Thus this construction yields Lie′_{F} (n) = 𝔚^{n}((F^{n})^{⊗n}) ∗ κ′_{n} = 𝔚^{n}((F^{n})^{⊗n}) ∗ w_{n} ∗ κ_{n} = 𝔚^{n}((F^{n})^{⊗n}) ∗ κ_{n} = Lie_{F} (n).

(c) Lastly, we analyze the F 𝔖_{n}-module F 𝔖_{n} · ω_{n}^{ι} ⊆ F 𝔖_{n}, where

ω_{n}^{ι} := (1 − c_{n}^{−1})(1 − c_{n−1}^{−1}) ⋯ (1 − c_{2}^{−1}) ∈ F 𝔖_{n}

is the image of ω_{n} under the F -algebra anti-automorphism ι : F 𝔖_{n} ⟶ F 𝔖_{n} defined by ι : π ⟼ π^{−1}, for π ∈ 𝔖_{n}. Then we have an isomorphism of left F 𝔖_{n}-modules

F 𝔖_{n} · ω_{n}^{ι} ≅ (F 𝔖_{n} · ω_{n})^{∗},

where the latter F 𝔖_{n}-module denotes the contragredient dual of F 𝔖_{n} · ω_{n}. This fact is most elegantly established by recalling that group algebras are, in particular, symmetric algebras, and using the general isomorphism (3) in Remark 3.4 below. Thus we briefly deviate to establish this:

3.4 Remark. Let A be a finite-dimensional symmetric F -algebra with symmetrizing F - bilinear form ⟨·|·⟩. That is, ⟨·|·⟩ is associative, symmetric, and non-degenerate. Hence

A ⟶ Hom_{F} (A, F), a ⟼ (b ⟼ ⟨b|a⟩), for a, b ∈ A,

is an isomorphism of (A, A)-bimodules. Letting ω ∈ A, this induces an isomorphism

Hom_{F} (Aω, F) ≅ A/(Aω)^{⊥}

of right A-modules, where ^{⊥} denotes taking orthogonal spaces with respect to ⟨·|·⟩. Moreover,

(Aω)^{⊥} = {a ∈ A | ⟨Aω|a⟩ = 0} = {a ∈ A | ⟨A|ωa⟩ = 0} = {a ∈ A | ωa = 0} = ker(ω ·).

Since A/ker(ω ·) ≅ im(ω ·) = ωA, this yields an isomorphism of right A-modules

Hom_{F} (Aω, F) ≅ A/(Aω)^{⊥} = A/ker(ω · ) ≅ im(ω ·) = ωA.

Finally, suppose that there is an involutory F -algebra anti-automorphism ι : A ⟶ A, a ⟼ a^{ι}. Then, whenever M is a right A-module, one can define a left A-module structure on M by a · x := xa^{ι}, for x ∈ M, a ∈ A. Denoting the resulting module by M^{ι}, one, in particular, gets (ωA)^{ι} ≅ Aω^{ι} as left A-modules. Thus one has an isomorphism of left A-modules

Hom_{F} (Aω, F)^{ι} ≅ (ωA)^{ι} ≅ Aω^{ι} .  (3)

3.5. Properties of Lie modules. We collect a couple of properties of Lie modules.

(a) We exhibit an explicit F -basis of Lie_{F} (n). Firstly, any element of Lie_{F} (n) is an F - linear combination of Lie brackets of the form [[e_{n}, e_{π(1)}, …, e_{π(n−1)}], where π ∈ 𝔖_{n−1}. This is clear for n ⩽ 2 anyway, and for n ⩾ 3 is seen as follows: letting w be a Lie bracket involving a subset of {e_{1}, …, e_{i−1}, e_{i+1}, …, e_{n−1}}, where 1 ⩽ i ⩽ n − 1, we have [[w, e_{i}, e_{n}] = [e_{n}, [e_{i}, w]], where, by (1), the latter can be written as a sum of Lie brackets having e_{n} as their first component.

Now, since expanding [[e_{n}, e_{π(1)}, …, e_{π(n−1)}] into the standard F -basis of (F^{n})^{⊗n} yields a unique summand having e_{n} as its first component, namely e_{n} ⊗ e_{π(1)} ⊗ ⋯ ⊗ e_{π(n−1)}, we conclude that

{[[e_{n}, e_{π(1)}, …, e_{π(n−1)}] | π ∈ 𝔖_{n−1}} ⊆ Lie_{F} (n)

is F -linearly independent, thus is an F -basis. Moreover, since

[[e_{n}, e_{π(1)}, …, e_{π(n−1)}] = π · [[e_{n}, e_{1}, …, e_{n−1}] = π · c_{n} · [[e_{1}, e_{2}, …, e_{n}] for π ∈ 𝔖_{n−1},

the above F -basis can also be written as

{π · c_{n} · [[e_{1}, …, e_{n}] | π ∈ 𝔖_{n−1}} ⊆ Lie_{F} (n).

Thus Res_{𝔖_{n−1}}^{𝔖_{n}}(Lie_{F} (n)) is isomorphic to the regular module F 𝔖_{n−1}, in particular saying that dim_{F} (Lie_{F} (n)) = (n − 1)!. Moreover, since [[e_{1}, …, e_{n}] = (e_{1} ⊗ ⋯ ⊗ e_{n}) ∗ ω_{n} ∈ Lie_{F} (n) is sent to ω_{n} ∈ F 𝔖_{n} via the isomorphism (2), this means that an F -basis of F 𝔖_{n} ·ω_{n} is obtained as

{π · c_{n} · ω_{n} | π ∈ 𝔖_{n−1}}.

This F -basis will be particularly useful to facilitate the explicit computations in Section 4; note that, by the observations in 3.2, this is even an 𝔽_{p}-basis of Lie_{p}(n) = 𝔽_{p}𝔖_{n} · ω_{n}.

(b) We now show that ω_{n}^{2} = n · ω_{n} ∈ F 𝔖_{n}, in particular implying that \frac{1}{n} · ω_{n} ∈ F 𝔖_{n} is an idempotent whenever p ∤ n. We proceed in various steps:

Firstly, we show that for n ⩾ 1 we have ω_{n} · (e_{1} ⊗ ⋯ ⊗ e_{n}) = [[e_{1}, …, e_{n}] ∈ (F^{n})^{⊗n}: this is clear for n = 1 anyway, and for n = 2 we have ω_{2}·(e_{1}⊗e_{2}) = (1−c_{2})·(e_{1}⊗e_{2}) = e_{1}⊗e_{2}−e_{2}⊗e_{1} = (e_{1} ⊗ e_{2}) ∗ κ_{2}. For n ⩾ 3, arguing by induction and using ω_{n} = ω_{n−1} · (1 − c_{n}) ∈ F 𝔖_{n}, we get

ω_{n} · (e_{1} ⊗ ⋯ ⊗ e_{n}) = ω_{n−1} · (e_{1} ⊗ ⋯ ⊗ e_{n} − e_{n} ⊗ e_{1} ⊗ ⋯ ⊗ e_{n−1})

= ((e_{1} ⊗ ⋯ ⊗ e_{n−1}) ∗ κ_{n−1}) ⊗ e_{n} − e_{n} ⊗ ((e_{1} ⊗ ⋯ ⊗ e_{n−1}) ∗ κ_{n−1})

= [(e_{1} ⊗ ⋯ ⊗ e_{n−1}) ∗ κ_{n−1}, e_{n}]
= (e_{1} ⊗ ⋯ ⊗ e_{n}) ∗ κ_{n} = [[e_{1}, …, e_{n}] .

Secondly, we show that for n ⩾ 2 we have ω_{n−1}c_{n} · [[e_{1}, …, e_{n}] = −[[e_{1}, …, e_{n}] ∈ (F^{n})^{⊗n}: Recall that κ_{n} ∈ End_{F} ((F^{n})^{⊗n}) centralizes the F [GL_{n}(F)]-action. Then, using (1) we get

ω_{n−1}c_{n} · (e_{1} ⊗ ⋯ ⊗ e_{n}) ∗ κ_{n} = (e_{n} ⊗ ω_{n−1} · (e_{1} ⊗ ⋯ ⊗ e_{n−1})) ∗ κ_{n}

= (e_{n} ⊗ (e_{1} ⊗ ⋯ ⊗ e_{n−1}) ∗ κ_{n−1}) ∗ κ_{n} = [e_{n}, (e_{1} ⊗ ⋯ ⊗ e_{n−1}) ∗ κ_{n−1}] = −[(e_{1} ⊗ ⋯ ⊗ e_{n−1}) ∗ κ_{n−1}, e_{n}] = −(e_{1} ⊗ ⋯ ⊗ e_{n}) ∗ κ_{n}.

Combining these computations we get

ω_{n−1}c_{n} · ω_{n} · (e_{1} ⊗ ⋯ ⊗ e_{n}) = −ω_{n} · (e_{1} ⊗ ⋯ ⊗ e_{n}),

thus translating via the isomorphism 𝔚^{n}((F^{n})^{⊗n}) ≅ F 𝔖_{n} yields ω_{n−1}c_{n} · ω_{n} = −ω_{n} ∈ F 𝔖_{n}.

Thirdly, and finally, we show that ω_{n}^{2} = n · ω_{n} ∈ F 𝔖_{n}, for n ⩾ 1: this is clear for n = 1 anyway, and for n = 2 we have ω_{2}^{2} = (1 − c_{2})^{2} = ω_{2} − c_{2}ω_{2} = 2ω_{2}. Then, for n ⩾ 3 we have ω_{n}^{2} = ω_{n−1}(1 − c_{n})ω_{n} = ω_{n−1}ω_{n} − ω_{n−1}c_{n}ω_{n}, where, by induction, the first summand equals

ω_{n−1}ω_{n} = ω_{n−1}^{2}(1 − c_{n}) = (n − 1)ω_{n−1}(1 − c_{n}) = (n − 1)ω_{n}.

The second summand being −ω_{n−1}c_{n}ω_{n} = ω_{n}, this yields ω_{n}^{2} = (n − 1)ω_{n} + ω_{n} = nω_{n}.

4 Computational Data

In this section we summarize our computational results concerning the Lie modules for some symmetric groups. Before doing so, we need a few preparations.

4.1. Some subgroups of symmetric groups. (a) We will use the following convention for denoting the Sylow p-subgroups of the symmetric group 𝔖_{n}: suppose first that n = p^{d}, for some d ⩾ 0. Moreover, we set P_{1} := 1 and P_{p} := C_{p}, where C_{p} := ⟨(1, 2, …, p)⟩, as well as

P_{p^{i+1}} := P_{p^{i}} ≀ C_{p} = {(g_{1}, …, g_{p}; σ) | g_{1}, …, g_{p} ∈ P_{p^{i}}, σ ∈ C_{p}} for i ⩾ 1 .

Note that the multiplication in P_{p^{i+1}} is as explained in (6) below. For i ⩾ 0, we view P_{p^{i}} as a subgroup of 𝔖_{p^{i}} in the obvious way. Then, by [21, 4.1.22, 4.1.24], P_{n} is a Sylow p-subgroup of 𝔖_{n}, which can be generated by the following elements in 𝔖_{n}:

g_{j} := ∏_{k=1}^{p^{j−1}}(k, k + p^{j−1}, k + 2p^{j−1}, …, k + (p − 1)p^{j−1}) where j = 1, …, d .

Next suppose that p | n, but n is not necessarily a p-power. Consider the p-adic expansion n = ∑_{j=1}^{s} α_{j}p^{i_{j}}, for some s ⩾ 1, i_{1} > … > i_{s} ⩾ 1, and 1 ⩽ α_{j} ⩽ p − 1 for j = 1, …, s. By [21, 4.1.22, 4.1.24], P_{n} := ∏_{j=1}^{s} ∏_{l_{j}=1}^{α_{j}} P_{p^{i_{j}} ,l_{j}} is then a Sylow p-subgroup of 𝔖_{n}. Here, the direct factor P_{p^{i_{1}},1} is acting on {1, …, p^{i_{1}}}, P_{p^{i_{1}},2} is acting on {p^{i_{1}} + 1, …, 2p^{i_{1}}}, and so on.

If n is not divisible by p, let m < n be maximal such that p | m, and set P_{n} := P_{m}, so that P_{n} is a Sylow p-subgroup of 𝔖_{n} also in this case.

(b) For d ⩾ 1 we denote by E_{p^{d}} the unique maximal elementary abelian subgroup of P_{p^{d}} that acts regularly on {1, …, p^{d}}; in particular, |E_{p^{d}}| = p^{d}. Letting n := p^{d} and Q := E_{p^{d}}, we determine the structure of N_{𝔖_{n}}(Q):

Since Q ⩽ 𝔖_{n} is an abelian transitive subgroup, it is self-centralizing, and thus N_{𝔖_{n}}(Q)/Q is isomorphic to a subgroup of GL_{d}(p). Moreover, since the affine linear group AGL_{d}(p) ≅ C_{p}^{d} ⋊ GL_{d}(p) acts transitively and faithfully on its elementary abelian subgroup C_{p}^{d}, there is an embedding AGL_{d}(p) ⟶ 𝔖_{n}, mapping C_{p}^{d} to Q. Hence we conclude that

N_{𝔖_{n}}(Q) ≅ Q ⋊ GL_{d}(p).

Thus N_{𝔖_{n}}(Q) acts transitively on each of the sets {R ⩽ Q | |R| = p^{i}}, for 0 ⩽ i ⩽ d, and the stabilizer N_{N_{𝔖n}(Q)}(R) induces the full automorphism group on any subgroup R ⩽ Q. Hence N_{𝔖_{n}}(Q) controls fusion in Q, in the sense of 2.5.

4.2 Example. If p = 2 and n = 4 then

P_{4} = ⟨(1, 2), (1, 3)(2, 4)⟩ and
E_{4} = ⟨(1, 2)(3, 4), (1, 3)(2, 4)⟩.

If p = 2 and n = 8 then

P_{8} = ⟨(1, 2), (1, 3)(2, 4), (1, 5)(2, 6)(3, 7)(4, 8)⟩ and
E_{8} = ⟨(1, 2)(3, 4)(5, 6)(7, 8), (1, 3)(2, 4)(5, 7)(6, 8), (1, 5)(2, 6)(3, 7)(4, 8)⟩.

If p = 3 and n = 9 then

P_{9} = ⟨(1, 2, 3), (1, 4, 7)(2, 5, 8)(3, 6, 9)⟩ and
E_{9} = ⟨(1, 2, 3)(4, 5, 6)(7, 8, 9), (1, 4, 7)(2, 5, 8)(3, 6, 9)⟩.

4.3 Remark. (a) Before proceeding to computationally substantial examples, for later use we recall the following well-known case: by [13, Cor. 9] the projective-free part Lie_{F}^{pf}(p) of the Lie module Lie_{F} (p) is indecomposable, and actually isomorphic to the Specht module S^{(p−1,1)} of dimension p − 1, hence

Lie_{F}^{pf}(p) ≅ Ω(F) as F 𝔖_{p}-modules .

Hence Lie_{F}^{pf}(p) has vertex E_{p} = C_{p}, of course, and Ω(F) is an E_{p}-source, having dimension p − 1. Note that the F E_{p}-module Ω(F) is an endo-permutation module. Thus, from Theorem 2.4 and the remarks in 2.5 and 4.1(b) we conclude that Lie_{F}^{pf}(p) is an endo-p-permutation module.

(b) In view of the subsequent results, we ask ourselves whether Lie_{F}^{pf}(p) itself possibly is an endo-permutation F 𝔖_{p}-module. Indeed, for p = 2 we have Lie_{F} (2) ≅ Lie_{F}^{pf}(2) ≅ F, hence Lie_{F}^{pf}(2) is even a permutation F 𝔖_{2}-module.

For p = 3 we have Lie_{F} (3) ≅ Lie_{F}^{pf}(3) ≅ Ω(F), and from the theory of blocks of cyclic defect it is immediate that

Lie_{F}^{pf}(3) ⊗ Lie_{F}^{pf}(3)^{∗} ≅ Ω(F) ⊗ Ω(F)^{∗} ≅ F ⊕ P^{(2,1)} ,

where D^{(2,1)} is the sign representation. Note that, in accordance with part (a), all indecomposable direct summands of Lie_{F}^{pf}(3) ⊗ Lie_{F}^{pf}(3)^{∗} are trivial-source modules indeed.

To show that Lie_{F}^{pf}(3)⊗Lie_{F}^{pf}(3)^{∗} is not a permutation F 𝔖_{3}-module, assume to the contrary that it is. Thus, by dimension reasons we conclude that P^{(2,1)} is an indecomposable transitive permutation F 𝔖_{3}-module, but P^{(2,1)} does not have the trivial module as an epimorphic image, a contradiction.

4.4. Computational approach. We now give a description of the tools from computational group theory and computational representation theory we are employing, and indicate the computational ideas we are using to obtain the subsequent explicit results concerning some larger Lie modules. As a general background reference, see [25].

(a) To deal with finite groups, in particular permutation groups and matrix groups, we use the general purpose computer algebra systems GAP [16] and MAGMA [7]. In particular, we make use of the character table library CTblLib [2] of GAP, which provides electronic access to the data collected in the Atlas [8] and in the ModularAtlas [22, 37]; these databases, in particular, contain the explicit 2- and 3-modular decomposition matrices for various symmetric groups given in [20, App.] or [21, App. I].

Moreover, we have used the more specialized computer algebra system MeatAxe [31, 32], and its extensions [26, 27, 28, 29], to deal with various aspects concerning matrix representations over (small) finite fields. Apart from general linear algebra, these tools, in particular, allow us to find composition series and direct sum decompositions, including isomorphism checks of simple and indecomposable modules, respectively, and to find splitting fields and to check absolute indecomposability; moreover, they enable us to compute homomorphism spaces and endomorphism rings, to determine radical and socle series, and to compute submodule lattices; apart from these analytic capabilities, they also provide the constructions needed below, such as Kronecker products and the computation of Heller translates.

(b) To facilitate explicit computations, we make use of the observation in 3.2, saying that Lie_{F} (n) ≅ F ⊗_{𝔽_{p}} Lie_{p}(n) as F 𝔖_{n}-modules, where

Lie_{p}(n) := 𝔽_{p}𝔖_{n} · ω_{n} ⊆ 𝔽_{p}𝔖_{n} as 𝔽_{p}𝔖_{n}-modules.

Thus we are indeed reduced to considerations of permutation representations, and matrix representations over finite (prime) fields.

Having got hands on the 𝔽_{p}𝔖_{n}-module Lie_{p}(n), the task then is to find the decomposition Lie_{p}(n) ≅ Lie_{p}^{pf}(n) ⊕ Lie_{p}^{pr}(n) into its projective-free and projective part, respectively, to determine how Lie_{p}^{pr}(n) decomposes into projective indecomposable modules, and what the indecomposable direct summands of Lie_{p}^{pf}(n) look like. However, the examples of Lie modules Lie_{p}(n) to be dealt with here are too large to simply apply to them the general techniques available to compute direct sum decompositions. Hence we have to proceed otherwise to find Lie_{p}^{pf}(n) in the first place; after all, by the asymptotic results mentioned in the introduction, we expect Lie_{p}^{pf}(n) to be small compared with Lie_{p}(n), small enough to allow for a detailed analysis.

By [15], we know that, in order to detect Lie_{p}^{pf}(n), we only need to consider the component Lie_{p}^{pbl}(n) of Lie_{p}(n) belonging to the principal p-block of 𝔽_{p}𝔖_{n}. Using the p-modular decomposition matrix of 𝔖_{n}, which is available for all cases considered here, and [4, Cor. 3.4], we may determine dim(Lie_{p}^{pbl}(n)) in advance, and [4, Thm. 3.1] also tells us the projective indecomposable direct summands of Lie_{p}(n)/Lie_{p}^{pbl}(n), so that next to Lie_{p}^{pf}(n) only the projective indecomposable direct summands of Lie_{p}^{pbl}(n) have to be determined.

(c) To find an 𝔽_{p}-basis of Lie_{p}(n) or Lie_{p}^{pbl}(n) in the first place, let ϵ_{n} ∈ 𝔽_{p}𝔖_{n} be the centrally primitive idempotent belonging to the principal p-block of 𝔽_{p}𝔖_{n}; recall that ϵ_{n} can be computed from the ordinary character table of 𝔖_{n}. We now use the observation in 3.5, saying that an 𝔽_{p}-basis of Lie_{p}(n) is given as

{π · c_{n} · ω_{n} | π ∈ 𝔖_{n−1}} ⊆ Lie_{p}(n) ⊆ 𝔽_{p}𝔖_{n} ,

implying that an 𝔽_{p}-spanning set of Lie_{p}^{pbl}(n) is given as

{π · c_{n} · ω_{n} · ϵ_{n} | π ∈ 𝔖_{n−1}} ⊆ Lie_{p}(n) · ϵ_{n} = Lie_{p}^{pbl}(n) ⊆ 𝔽_{p}𝔖_{n} .

Hence our starting point is the regular representation 𝔽_{p}𝔖_{n} of 𝔖_{n}, being equipped with its natural 𝔽_{p}-basis. Determining the permutation action of elements of 𝔖_{n} on coordinate vectors with respect to this basis essentially amounts to computing with permutations in 𝔖_{n}. This allows us to apply successively all elements of 𝔖_{n−1} to c_{n} · ω_{n} ∈ 𝔽_{p}𝔖_{n} and c_{n} · ω_{n} · ϵ_{n} ∈ 𝔽_{p}𝔖_{n}, respectively; to do this efficiently, we first find a Schreier tree of 𝔖_{n−1} in terms of some generating set, our favourite one being {(1, …, n − 1), (1, 2)}.

Thus having found an 𝔽_{p}-basis of Lie_{p}(n), we directly determine the action of a generating set of 𝔖_{n}, our favourite one again being {(1, …, n), (1, 2)}. For Lie_{p}^{pbl}(n), before doing so, we pick an 𝔽_{p}-basis out of the 𝔽_{p}-spanning set obtained. These tasks are efficiently solved using the linear algebra routines available in the MeatAxe.

(d) Next we proceed to find the projective indecomposable direct summands of Lie_{p}^{pbl}(n). To do so, we apply a technique based on the considerations in [26]. In order to describe this we first recall the relevant notions:

Let A be a finite-dimensional K-algebra, where K is any field, and let S be a simple A-module. Then, the endomorphism algebra End_{A}(S) is a skew field, and for a ∈ A letting ker(a_{S}) denote the kernel of the K-endomorphism of S induced by the action of a, we have dim(End_{A}(S)) | dim(ker(a_{S})). Now a ∈ A is called an S-peakword, if dim(ker(a_{S}^{2})) = dim(End_{A}(S)), and ker(a_{T}) = {0} for all simple A-modules T not isomorphic to S. In particular, if K is a splitting field of A, then the first condition just becomes dim(ker(a_{S}^{2})) = dim(ker(a_{S})) = 1. In practice, peakwords are found by a random search, yielding a Monte Carlo method, which for the case of K being a (small) finite field is available in the MeatAxe.

Let now M be an A-module, and let a ∈ A be an S-peakword. Then, by [26, Thm.2.5], the set of all submodules L of M such that L/ Rad(L) ≅ S concides with the set of all cyclic submodules of M generated by some v ∈ ⋃_{i⩾1} ker(a_{M}^{i}) ∖ {0}. Thus, in particular, all submodules of M isomorphic to the projective cover P_{S} of S are found this way, and for a cyclic submodule L as above we have L ≅ P_{S} if and only if dim(L) = dim(P_{S}). Thus, if A is a self-injective algebra, a random search through ⋃_{i⩾1} ker(a_{M}^{i}) yields a Monte Carlo method to find a largest direct summand of M being the direct sum of copies of P_{S}. Note that dim(P_{S}) is indeed known in advance in all explict cases considered here, and that, if K is a (small) finite field, then techniques to compute cyclic submodules are available in the MeatAxe.

(e) Thus, quotienting out the projective direct summands found, we now have Lie_{p}^{pf}(n) in our hands, at least with high probability. In all cases considered here this module turns out to be small enough to apply to it the general techniques available in the MeatAxe to find direct sum decompositions. The latter techniques would also find a projective direct summand left over, thus providing a verification of the above Monte Carlo results. Actually, for the examples to be discussed below, this even shows that Lie_{p}^{pf}(n) is indecomposable and non-projective.

Hence we may now assume that we have got a non-projective indecomposable 𝔽_{p}𝔖_{n}- module M, for which we have to find a vertex and a source. In order to do so, we consider the restriction Res_{P_{n}}^{𝔖_{n}}(M) of M to the Sylow p-subgroup P_{n} of 𝔖_{n}; recall from 2.2 that, since M is relatively P_{n}-projective, Res_{P_{n}}^{𝔖_{n}}(M) has an indecomposable direct summand sharing a vertex and a source with M.

Hence we may assume that M is an 𝔽_{p}P -module, where P is a p-group. Again, we have to find direct sum decompositions, which can be speeded up by detecting particular indecomposable direct summands beforehand. Namely, in case of an 𝔽_{p}P -module the strategy described in part (d) specializes to the following: the set of all submodules L of M such that L/ Rad(L) is simple is precisely the set of all non-zero cyclic submodules of M. (Note that, in terms of the language used above, since the trivial module is the only simple 𝔽_{p}P -module, the zero element in 𝔽_{p}P is a peakword.) This leads to a straightforward Monte Carlo method to find a largest direct summand of M that is the direct sum of copies of the regular module 𝔽_{p}P; see [12, Sect. 3.2].

Quotienting out projective direct summands we again, in all cases considered here, end up with a module whose direct sum decomposition can be computed with the general techniques available in the MeatAxe.

Hence we may finally assume that M is an indecomposable non-projective 𝔽_{p}P -module such that we are in a position to use the techniques described in [12, Sect. 3.1], whose basic ingredient is Higman’s Criterion for relative projectivity. An implementation is available in MAGMA, which, in particular, employs its facilities to compute with finite p-groups, for example to determine subgroup lattices, and sets of subgroup coset representatives.

(f) To make sure that computational results are still valid when going over to the algebraically closed field F again, we always check that the indecomposable modules found are actually absolutely indecomposable; techniques to achieve that are available in the MeatAxe. Recall that it is well known that 𝔽_{p} is a splitting field of 𝔽_{p}𝔖_{n}, hence absolute indecomposability is automatic anyway for the simple modules and the projective indecomposable modules found.

4.5. Examining Lie_{2}(4). Let p = 2. We examine the 𝔽_{2}𝔖_{4}-module Lie_{2}(4).

(a) Recall from Remark 3.5(a) that dim(Lie_{2}(4)) = 3! = 6. A dimension consideration shows that Lie_{2}(4) cannot possibly contain a projective direct summand, hence Lie_{2}(4) coincides with its projective-free part Lie_{2}^{pf}(4). Moreover, it is easily checked computationally, that Lie_{2}^{pf}(4) is absolutely indecomposable, namely

Lie_{2}^{pf}(4) ≅ Ω^{−1}(D^{(3,1)}) ,

where D^{(3,1)} ≅ Inf_{𝔖_{3}}^{𝔖_{4}}(D^{(2,1)}) is the simple 𝔽_{2}𝔖_{4}-module of dimension 2, and the inflation is along the natural map 𝔖_{4}/E_{4} ≅ 𝔖_{3}. Since D^{(2,1)} is a projective simple 𝔽_{2}𝔖_{3}-module, D^{(3,1)} is a trivial-source module with vertex E_{4}. Thus we conclude that Lie_{2}^{pf}(4) has vertex E_{4}, and Ω^{−1}(𝔽_{2}) = 𝔽_{2}E_{4}/ Soc(𝔽_{2}E_{4}) is an E_{4}-source, having dimension 3.

Note that Ω^{−1}(𝔽_{2}) is an endo-permutation module. Thus from Theorem 2.4, and the remarks in 2.5 and 4.1(b), we conclude that Lie_{2}^{pf}(4) is an endo-p-permutation module.

(b) In view of the above and the subsequent results, it seems worthwhile to show that Lie_{2}^{pf}(4) is not an endo-permutation 𝔽_{2}𝔖_{4}-module. To this end, we compute an explicit indecomposable direct sum decomposition of Lie_{2}^{pf}(4) ⊗ Lie_{2}^{pf}(4)^{∗}:

Lie_{2}^{pf}(4) ⊗ Lie_{2}^{pf}(4)^{∗} ≅ D^{(3,1)} ⊕ Ind_{𝔄_{4}}^{𝔖_{4}}(𝔽_{2}) ⊕ P^{(4)} ⊕ 3 · P^{(3,1)} ,  (4)

where both projective indecomposable 𝔽_{2}𝔖_{4}-modules P^{(4)} and P^{(3,1)} have dimension 8. Note that, in accordance with part (a), we indeed observe that all indecomposable direct summands of Lie_{2}^{pf}(4) ⊗ Lie_{2}^{pf}(4)^{∗} are trivial-source modules.

To show that Lie_{2}^{pf}(4)⊗Lie_{2}^{pf}(4)^{∗} is not a permutation 𝔽_{2}𝔖_{4}-module, assume to the contrary that it is. Then there is some H ⩽ 𝔖_{4} such that D^{(3,1)} is isomorphic to a direct summand of Ind_{H}^{𝔖_{4}}(𝔽_{2}) and such that Ind_{H}^{𝔖_{4}}(𝔽_{2}) is isomorphic to a direct summand of Lie_{2}^{pf}(4) ⊗ Lie_{2}^{pf}(4)^{∗}. In particular, D^{(3,1)} is then relatively H-projective, and since, by (a), E_{4} ⊴ 𝔖_{4} is a vertex of D^{(3,1)}, we infer E_{4} ⩽ H. On the other hand, H cannot possibly contain a Sylow 2-subgroup of 𝔖_{4}, since otherwise Ind_{H}^{𝔖_{4}}(𝔽_{2}) has the trivial 𝔽_{2}𝔖_{4}-module as a direct summand. This leaves the cases H ∈ {E_{4}, 𝔄_{4}}. But if H = 𝔄_{4} then D^{(3,1)} ∤ Ind_{H}^{𝔖_{4}}(𝔽_{2}), and if H = E_{4} then 2 · D^{(3,1)} | Ind_{H}^{𝔖_{4}}(𝔽_{2}). In any case, we obtain a contradiction.

4.6. Examining Lie_{2}(8). Let p = 2. We examine the 𝔽_{2}𝔖_{8}-module Lie_{2}(8).

(a) Recall from Remark 3.5(a) that dim(Lie_{2}(8)) = 7! = 5040. Moreover, using the 2modular decomposition matrix of 𝔖_{8} and [4, Cor. 3.4], we find dim(Lie_{2}^{pbl}(8)) = 4016. By work of Selick–Wu [33], it is known that

Lie_{2}^{pbl}(8) ≅ Lie_{2}^{pf}(8) ⊕ 2 · P^{(6,2)} ⊕ P^{(5,3)} ⊕ 4 · P^{(4,3,1)} ,  (5)

so that we infer that dim(Lie_{2}^{pf}(8)) = 816 = 2^{4} · 3 · 17. We have verified the decomposition (5) independently, with the computational techniques described in 4.4. In addition to the calculations in [33], we have checked explicitly that Lie_{2}^{pf}(8) is actually absolutely indecomposable.

(b) We will subsequently describe the vertices and sources of the projective-free part Lie_{2}^{pf}(8). In order to do so, we consider the restriction of Lie_{2}^{pf}(8) to the Sylow 2-subgroup P_{8} of 𝔖_{8}; note that, since |P_{8}| = 2^{7}, from 2.2 we conclude that every vertex of Lie_{2}^{pf}(8) has order at least 8. Our computations yield the following decomposition:

Res_{P_{8}}^{𝔖_{8}}(Lie_{2}^{pf}(8)) = M_{1} ⊕ M_{2} ⊕ (cyc),

where ‘(cyc)’ denotes a direct sum of absolutely indecomposable 𝔽_{2}P_{8}-modules with vertex Z(P_{8}) of order 2, and with trivial sources.

The direct summand M_{2} is absolutely indecomposable of dimension 96, and has vertex

V := ⟨(1, 3)(2, 4)(5, 6)(7, 8), (1, 4)(2, 3)(5, 8)(6, 7)⟩ ≅ C_{2} × C_{2}

of order 4, and a V -source isomorphic to 𝔽_{2}V/ Soc(𝔽_{2}V) ≅ Ω^{−1}(𝔽_{2}). In particular, the sources of M_{2} are endo-permutation modules.

The remaining direct summand, M_{1}, is absolutely indecomposable of dimension 336, and has vertex E_{8} and an E_{8}-source S of dimension 21 satisfying

End_{𝔽_{2}}(S) ≅ S ⊗ S^{∗} ≅ 𝔽_{2} ⊕ ⨁_{Q<E_{8}, |Q|=2} 2 · Ind_{Q}^{E_{8}}(𝔽_{2}) ⊕ (proj),

where ‘(proj)’ denotes a projective 𝔽_{2}E_{8}-module. Consequently, End_{𝔽_{2}}(S) is a permutation 𝔽_{2}E_{8}-module, that is, S is an endo-permutation 𝔽_{2}E_{8}-module. In fact, by Theorem 2.4, the isomorphism type of S is determined by the following isomorphism, which is easily verified computationally, using the techniques in 4.4:

Ω^{3}(𝔽_{2}) ⊗ ⨂_{Q<E_{8}, |Q|=2} Inf_{E_{8}/Q}^{E_{8}}(Ω^{−1}((𝔽_{2})_{E_{8}/Q})) ≅ S ⊕ (proj).

Note that S is the only non-projective direct summand occurring.

In conclusion, this shows that Lie_{2}^{pf}(8) has vertex E_{8} and endo-permutation source S. In particular, by 2.5 and 4.1(b), we conclude that Lie_{2}^{pf}(8) is an endo-p-permutation module.

4.7. Examining Lie_{3}(9). Next let p = 3. We examine the 𝔽_{3}𝔖_{9}-module Lie_{3}(9).

(a) Recall from Remark 3.5(a) that dim(Lie_{3}(9)) = 8! = 40320. Moreover, using the 3-modular decomposition matrix of 𝔖_{9} and [4, Cor. 3.4], we find dim(Lie_{3}^{pbl}(9)) = 16020. Employing the techniques described in 4.4, we obtain the following decomposition

Lie_{3}^{pbl}(9) ≅ 2 · P^{(7,1^{2})} ⊕ 5 · P^{(6,3)} ⊕ 3 · P^{(6,2,1)} ⊕ 4 · P^{(5,2^{2})} ⊕ 2 · P^{(4,3,2)} ⊕ P^{(4^{2},1)} ⊕ 4 · P^{(3^{2},2,1)}
⊕ Lie_{3}^{pf}(9),

where hence Lie_{3}^{pf}(9) has dimension 1683 = 3^{2} · 11 · 17, and turns out to be absolutely indecomposable.

(b) To describe the vertices and sources of the projective-free part Lie_{3}^{pf}(9), we first note that from |P_{9}| = 3^{4} and 2.2 we conclude that every vertex of Lie_{3}^{pf}(9) has order at least 9. We determine an indecomposable direct sum decomposition of the restriction of Lie_{3}^{pf}(9) to P_{9}, and get

Res_{P_{9}}^{𝔖_{9}}(Lie_{3}^{pf}(9)) ≅ N_{1} ⊕ 2 · N_{2} ⊕ 4 · N_{3} ⊕ (proj),

where N_{2} ≅ N̸_{3} are absolutely indecomposable of dimension 54 each, having non-conjugate cyclic vertices of order 3, and endo-permutation sources of dimension 2.

The direct summand N_{1} is absolutely indecomposable of dimension 144, and has vertex E_{9} and an E_{9}-source S′ of dimension 16 satisfying

End_{𝔽_{3}}(S′) ≅ S′ ⊗ (S′)^{∗} ≅ 𝔽_{3} ⊕ ⨁_{Q<E_{9}, |Q|=3} Ind_{Q}^{E_{9}}(𝔽_{3}) ⊕ (proj).

Consequently, End_{𝔽_{3}}(S′) is a permutation 𝔽_{3}E_{9}-module, that is, S′ is an endo-permutation 𝔽_{3}E_{9}-module. Its isomorphism type, in the sense of Theorem 2.4, is determined by the following isomorphism, where again S′ is the only non-projective direct summand occurring:

Ω^{−2}(𝔽_{3}) ⊗ ⨂_{Q<E_{9}, |Q|=3} Inf_{E_{9}/Q}^{E_{9}}(Ω((𝔽_{3})_{E_{9}/Q})) ≅ S′ ⊕ (proj).

Also this decomposition is verified computationally via the techniques described in 4.4.

In conclusion, this shows that Lie_{3}^{pf}(9) has vertex E_{9} and endo-permutation source S′. In particular, by 2.5 and 4.1(b), we conclude that Lie_{3}^{pf}(9) is an endo-p-permutation module.

5 A Reduction Theorem

The aim of this section is to establish Theorem 5.7, which will allow for a partial reduction of the question concerning vertices and sources of indecomposable direct summands of Lie_{p}(n) to the case where n is a p-power. The key ingredients will be Theorem 5.5, and the results in [23] on vertices of indecomposable modules of wreath products. Therefore, we start out by collecting a number of facts on wreath products and their representations, which we will then apply in the context of Lie modules.

5.1. Wreath products and their modules. (a) Let G be a finite group, and consider the wreath product

G ≀ 𝔖_{n} := {(g_{1}, …, g_{n}; σ) | g_{1}, …, g_{n} ∈ G, σ ∈ 𝔖_{n}}.

Recall that the multiplication in G ≀ 𝔖_{n} is given by

(g_{1}, …, g_{n}; σ)(h_{1}, …, h_{n}; π) = (g_{1}h_{σ^{−1}(1)}, …, g_{n}h_{σ^{−1}(n)}; σπ) ,  (6)

for g_{1}, …, g_{n}, h_{1}, …, h_{n} ∈ G and σ, π ∈ 𝔖_{n}. Hence we have the natural epimorphism

: G ≀ 𝔖_{n} ⟶ 𝔖_{n}, (g_{1}, …, g_{n}; σ) ⟼ σ .

We denote by G^{n} the base group of G ≀ 𝔖_{n}, that is,

G^{n} = {(g_{1}, …, g_{n}; 1) | g_{1}, …, g_{n} ∈ G} ⊴ G ≀ 𝔖_{n}.

Moreover, letting σ^{♯} := (1, …, 1; σ) ∈ G ≀ 𝔖_{n}, for σ ∈ 𝔖_{n}, we get an isomorphism

𝔖_{n}^{♯} := {σ^{♯} | σ ∈ 𝔖_{n}} ≅ 𝔖_{n};

note that the map (−)^{♯} : 𝔖_{n} ⟶ G≀𝔖_{n} is a section for the natural epimorphism : G≀𝔖_{n} ⟶ 𝔖_{n}. More generally, if H ⩽ G and U ⩽ 𝔖_{n} then we further set U^{♯} := {σ^{♯} | σ ∈ U} ⩽ 𝔖_{n}^{♯}, as well as H^{n} := {(g_{1}, …, g_{n}; 1) | g_{1}, …, g_{n} ∈ H} ⩽ G^{n}, and

H ≀ U := {(g_{1}, …, g_{n}; σ) | g_{1}, …, g_{n} ∈ H, σ ∈ U} ⩽ G ≀ 𝔖_{n}.

(b) Let M be an F G-module. Then the (outer) tensor product M^{⊗n} = M ⊗ ⋯ ⊗ M becomes an F [G ≀ 𝔖_{n}]-module via

(g_{1}, …, g_{n}; σ)(x_{1} ⊗ ⋯ ⊗ x_{n}) := g_{1}x_{σ^{−1}(1)} ⊗ ⋯ ⊗ g_{n}x_{σ^{−1}(n)},

for g_{1}, …, g_{n} ∈ G, σ ∈ 𝔖_{n}, and x_{1}, …, x_{n} ∈ M. This module is called a tensor-induced module.

From now on, we denote by Λ(m, n) the set of compositions of n with at most m non-zero parts. If λ = (λ_{1}, …, λ_{m}) ∈ Λ(m, n) then we denote by 𝔖_{λ} the corresponding (standard) Young subgroup 𝔖_{λ_{1}} × ⋯ × 𝔖_{λ_{m}} of 𝔖_{n}. With this notation,

(G ≀ 𝔖_{λ_{1}}) × ⋯ × (G ≀ 𝔖_{λ_{m}}) ≅ G ≀ 𝔖_{λ} ⩽ G ≀ 𝔖_{n}.

Thus, if M_{1}, …, M_{m} are F G-modules, the (outer) tensor product

M^{⊗λ} := M_{1}^{⊗λ_{1}} ⊗ ⋯ ⊗ M_{m}^{⊗λ_{m}}

carries an F [G ≀ 𝔖_{λ}]-module structure.

Moreover, suppose again that H ⩽ G and U ⩽ 𝔖_{n}, and let L be an F U-module. Then, via the map (−)^{♯}, the F U-module L can be viewed as an F U^{♯}-module, which we denote by L^{♯}. Via inflation along the natural epimorphism , the F U-module L becomes also an F [H ≀ U]-module, which we denote by L̂ := Inf_{U}^{H≀U}(L). Thus we have Res_{U^{♯}}^{H≀U}(L̂) = L^{♯}.

(c) Let N be an F 𝔖_{n}-module, and again let M be an F G-module. In this section, we will describe vertices and sources of indecomposable direct summands of the F [G≀𝔖_{n}]-module M^{⊗n}⊗N̂ in terms of those of the indecomposable direct summands of M and N. We, therefore, recall the structure of the indecomposable direct summands of the F [G ≀ 𝔖_{n}]-modules M^{⊗n} and N̂, respectively:

Let N_{1}, …, N_{l} be pairwise non-isomorphic indecomposable F 𝔖_{n}-modules, and b_{1}, …, b_{l} ∈ ℕ be such that N ≅ b_{1}N_{1} ⊕ ⋯ ⊕ b_{l}N_{l}. Then we get

N̂ = Inf_{𝔖_{n}}^{G≀𝔖_{n}}(N) ≅ ⨁_{i=1}^{l} b_{i} Inf_{𝔖_{n}}^{G≀𝔖_{n}}(N_{i}) = ⨁_{i=1}^{l} b_{i}N̂_{i},

where the F [G≀𝔖_{n}]-modules N̂_{i} := Inf_{𝔖_{n}}^{G≀𝔖_{n}}(N_{i}) are pairwise non-isomorphic and indecomposable. Thus, the indecomposable direct summmands of the F 𝔖_{n}-module N and those of the F [G ≀ 𝔖_{n}]-module N̂ are in natural bijection, and hence in the sequel we may assume that N is indecomposable.

As for M^{⊗n}, let M_{1}, …, M_{m} be pairwise non-isomorphic indecomposable F G-modules, and let a_{1}, …, a_{m} ∈ ℕ be such that

M ≅ a_{1}M_{1} ⊕ ⋯ ⊕ a_{m}M_{m} .

Then we have the following well-known result; we include a proof for the readers’ convenience.

5.2 Lemma. With the notation as in 5.1(c),

M^{⊗n} ≅ ⨁_{λ∈Λ(m,n)} c_{λ} · Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M_{1}^{⊗λ_{1}} ⊗ ⋯ ⊗ M_{m}^{⊗λ_{m}})

is an indecomposable direct sum decomposition of the F [G ≀ 𝔖_{n}]-module M^{⊗n}, for suitable c_{λ} ∈ ℕ.

Proof. We have an isomorphism of F [G ≀ 𝔖_{n}]-modules

M^{⊗n} ≅ ⨁_{λ=(λ_{1},...,λ_{m})∈Λ(m,n)} c_{λ} · (⨁ M̃_{1} ⊗ ⋯ ⊗ M̃_{n}),

the inner sum being taken over all n-tuples (M̃_{1}, …, M̃_{n}) of F G-modules satisfying

|{1 ⩽ j ⩽ n | M̃_{j} = M_{i}}| = λ_{i}, for i = 1, …, m.

The respective coefficient c_{λ} equals a_{1}^{λ_{1}} ⋯ a_{m}^{λ_{m}}.

Given λ = (λ_{1}, …, λ_{m}) ∈ Λ(m, n), the sum ⨁(M̃_{1} ⊗ ⋯ ⊗ M̃_{n}) is a transitive imprimitive F [G ≀ 𝔖_{n}]-module, and the direct summands M̃_{1} ⊗ ⋯ ⊗ M̃_{n} form a system of imprimitivity. One of these direct summands equals M^{⊗λ} = M_{1}^{⊗λ_{1}} ⊗ ⋯ ⊗ M_{m}^{⊗λ_{m}}. Its restriction to the base group G^{n} of G ≀ 𝔖_{n} is indecomposable, and its inertial group in G ≀ 𝔖_{n} equals G ≀ 𝔖_{λ}. Thus, by [9, 50.2], we deduce that

⨁ M̃_{1} ⊗ ⋯ ⊗ M̃_{n} ≅ Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M^{⊗λ})

and, by [23, Prop. 4.1], Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M^{⊗λ}) is an indecomposable F [G ≀ 𝔖_{n}]-module.

5.3. Wreath products and vertices. We retain the notation from 5.1(c). In particular, we suppose that N is an indecomposable F 𝔖_{n}-module. We now want to examine the vertices and sources of the indecomposable direct summands of the F [G ≀ 𝔖_{n}]-module M^{⊗n} ⊗ N̂; the answer is given in Theorem 5.4 below. To this end, let P be a Sylow p-subgroup of G, and for j = 1, …, m let R_{j} be a vertex of the F G-module M_{j}.

(a) Now let L be an indecomposable direct summand of M^{⊗n} ⊗ N̂. Then, by 5.1(c) and Lemma 5.2, there is some λ = (λ_{1}, …, λ_{m}) ∈ Λ(m, n) such that L is isomorphic to a direct summand of

Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M^{⊗λ}) ⊗ N̂ ≅ Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M^{⊗λ} ⊗ Res_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(N̂))
≅ Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M^{⊗λ} ⊗ Inf_{𝔖_{λ}}^{G≀𝔖_{λ}}(Res_{𝔖_{λ}}^{𝔖_{n}}(N))) .

Then, by [23, Prop. 5.1] and the discussion preceding it, there is an indecomposable direct summand N′ of Res_{𝔖_{λ}}^{𝔖_{n}}(N) such that

L ≅ Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(L′), where L′ := M^{⊗λ} ⊗ N̂′ .

In particular, L′ is an indecomposable F [G ≀ 𝔖_{λ}]-module. Now, if Q′ ⩽ 𝔖_{λ} is a vertex of N′, then

Q := (R_{1}^{λ_{1}} × ⋯ × R_{m}^{λ_{m}}) ⋊ (Q′)^{♯} ⩽ G ≀ 𝔖_{λ} ⩽ G ≀ 𝔖_{n}  (7)

is a common vertex of L and L′.

(b) We consider a common Q-source S of the F [G ≀ 𝔖_{n}]-module L and the F [G ≀ 𝔖_{λ}]- module L′. To this end, we from now on additionally suppose that each of the F G-modules M_{1}, …, M_{m} has trivial sources. Note that this, in particular, includes the case that all these modules are projective.

Let P_{λ} = P_{λ_{1}} × ⋯ × P_{λ_{m}} be a Sylow p-subgroup of the Young subgroup 𝔖_{λ} = 𝔖_{λ_{1}} × ⋯ × 𝔖_{λ_{m}}. Then, in consequence of [23, Prop. 1.2, Prop. 3.1],

R_{λ} := (R_{1}^{λ_{1}} × ⋯ × R_{m}^{λ_{m}}) ⋊ P_{λ}^{♯} = (R_{1} ≀ P_{λ_{1}}) × ⋯ × (R_{m} ≀ P_{λ_{m}}) ⩽ G ≀ 𝔖_{λ}

is a vertex of the indecomposable F [G≀𝔖_{λ}]-module M^{⊗λ}, and M^{⊗λ} is a trivial-source module, that is, M^{⊗λ} | Ind_{R_{λ}}^{G≀𝔖_{λ}}(F). (Note that the assertion on vertices is just a special case of (7).)

Suppose that S′ is a Q′-source of the F 𝔖_{λ}-module N′. From [19, Prop. 2] we deduce that the F [G ≀ 𝔖_{λ}]-module N̂′ := Inf_{𝔖_{λ}}^{G≀𝔖_{λ}}(N′) has vertex P ≀ Q′, and Ŝ′ := Inf_{Q′}^{P ≀Q′}(S′) is a (P ≀ Q′)- source of N̂′. Thus we have N̂′ | Ind_{P ≀Q′}^{G≀𝔖_{λ}}(Ŝ′). Hence Mackey’s Tensor Product Theorem shows

that there is some g ∈ G ≀ 𝔖_{λ} such that L′ is a direct summand of IndG≀𝔖_{λ} ; Q̃(Res^{g}(P ≀Q′) ; Q̃(^{g}Ŝ′)), where

Q̃ := R_{λ} ∩^{g}(P ≀ Q′) ⩽ G ≀ 𝔖_{λ} .

Hence S is a direct summand of Res_{Q}^{G≀𝔖_{λ}}(IndG≀𝔖_{λ} ; Q̃(Res^{g}(P ≀Q′) ; Q̃(^{g}Ŝ′))), thus, by Mackey’s Theorem, there is some h ∈ G ≀ 𝔖_{λ} such that S is a direct summand of

Ind_{Q∩^{h}Q̃}^{Q}(Res_{Q∩^{h}Q̃}^{^{h}Q̃} ^{h}(Res^{g}(P ≀Q′) ; Q̃(^{g}Ŝ′))) = Ind_{Q∩^{h}Q̃}^{Q}(Res_{Q∩^{h}Q̃}^{^{hg}(P ≀Q′)}(^{hg}Ŝ′)).

Since S has vertex Q, we infer Q ∩^{h}Q̃ = Q, so S is a direct summand of Res_{Q}^{^{hg}(P ≀Q′)}(^{hg}Ŝ′). Now we consider the natural epimorphism ^{−} : G ≀ 𝔖_{λ} ⟶ 𝔖_{λ}, and let σ := \overline{hg} ∈ 𝔖_{λ}. Then

Q′ = \overline{(R_{1}^{λ_{1}} × ⋯ × R_{m}^{λ_{m}}) ⋊ (Q′)^{♯}} = \overline{Q} ⩽ \overline{^{hg}(P ≀ Q′)} = ^{σ}Q′ .

Hence we have σ ∈ N_{𝔖_{λ}}(Q′). Moreover, since the base group G^{n} acts trivially on Ŝ′, we infer that Q ∩ G^{n} = R_{1}^{λ_{1}} × ⋯ × R_{m}^{λ_{m}} acts trivially on S, and since S′ is an indecomposable F Q′-module, we finally conclude that

S ≅ Res_{Q}^{^{hg}(P ≀Q′)}(^{hg}Ŝ′) ≅ Inf_{Q′}^{Q}(^{σ}S′) .

Recall from 2.2 that, since S′ is a Q′-source of N′, so is ^{σ}S′.

(c) Keep the notation as in part (b), and suppose additionally that the F G-module M is projective, that is, R_{1} = ⋯ = R_{m} = {1}, thus Q = (Q′)^{♯}. Furthermore, we now get σ^{♯} = (1, …, 1; σ) ∈ N_{G≀𝔖_{λ}}((Q′)^{♯}), since σ ∈ N_{𝔖_{λ}}(Q′). Hence we have S ≅ (^{σ}S′)^{♯} = ^{σ^{♯}}((S′)^{♯}). Since S′ was an arbitrary Q′-source of N′, this shows that indeed every Q′-source of N′, in the way just described, yields a common Q-source of L and L′.

Altogether we have, in particular, now proved the following:

5.4 Theorem. Let M be an F G-module, let N be an indecomposable F 𝔖_{n}-module, and let L be an indecomposable direct summand of the F [G ≀ 𝔖_{n}]-module M^{⊗n} ⊗ N̂. Suppose that M ≅ ⨁_{j=1}^{m} a_{j}M_{j} is an indecomposable direct sum decomposition of the F G-module M. For j = 1, …, m, let R_{j} be a vertex of M_{j}.

(a) There are some λ = (λ_{1}, …, λ_{m}) ∈ Λ(m, n) and an indecomposable direct summand N′ of Res_{𝔖_{λ}}^{𝔖_{n}}(N) such that L ≅ Ind_{G≀𝔖_{λ}}^{G≀𝔖_{n}}(M^{⊗λ} ⊗ N̂′). For every vertex Q′ ⩽ 𝔖_{λ} of N′, the group

Q := (R_{1}^{λ_{1}} × ⋯ × R_{m}^{λ_{m}}) ⋊ (Q′)^{♯} ⩽ G ≀ 𝔖_{λ} ⩽ G ≀ 𝔖_{n}

is a vertex of L.

(b) Suppose in addition that M_{1}, …, M_{m} are trivial-source modules. Then there is, moreover, a Q′-source S′ of N′ such that Inf_{Q′}^{Q}(S′) is a Q-source of L. Here the inflation is taken with respect to the natural epimorphism Q ⟶ Q′.

(c) If M is a projective F G-module, and if λ and N′ are as is part (a), then (Q′)^{♯} is a vertex of L. If S′ is any Q′-source of N′ then (S′)^{♯} is also a (Q′)^{♯}-source of L.

The next theorem concerning decompositions of Lie modules is a consequence of the results in [3], [6] and [24]. This decomposition and Theorem 5.4 will then enable us to reduce the problem of determining vertices of indecomposable direct summands of Lie_{F}^{pf}(n) to the case where n is a p-power.

5.5 Theorem. Let k > 1 with p ∤ k. Then, for every s ⩾ 0, there is a projective F 𝔖_{kp^{s}}-module X_{kp^{s}} such that, for all d ⩾ 0, one has

Lie_{F} (kp^{d}) ≅ ⨁_{t=0}^{d} Ind_{𝔖_{kp^{t}}≀𝔖_{p^{d−t}}}^{𝔖_{kp^{d}}} (X_{kp^{t}}^{⊗p^{d−t}} ⊗ { \atop Lie_{F} (p^{d−t})}) .  (8)

Proof. We fix n := kp^{d}, for some d ⩾ 0, and consider the natural F [GL_{n}(F)]-module F^{n}. Recall from 3.2(b) that the Lie module Lie_{F} (n) is the image of the F [GL_{n}(F)]-module L^{n}(F^{n}) under the Schur functor 𝔚^{n}, taking n-homogeneous polynomial F [GL_{n}(F)]-modules to F 𝔖_{n}- modules.

By [3, Thm. 3.4], for all t ⩾ 0 there are idempotents f_{kp^{t}} in the group algebra F 𝔖_{kp^{t}}, only depending on p, k and t, but independent of n, such that there is an isomorphism of F [GL_{n}(F)]-modules

L^{n}(F^{n}) ≅ ⨁_{t=0}^{d} L^{p^{d−t}}((F^{n})^{⊗kp^{t}} ∗ f_{kp^{t}}) ;  (9)

recall from 3.1 that 𝔖_{kp^{t}} acts from the right on (F^{n})^{⊗kp^{t}} by place permutations.

Now fix some 0 ⩽ t ⩽ d, and set m := kp^{t} and q := p^{d−t}. Suppose that V is any mhomogeneous polynomial F [GL_{n}(F)]-module, so that the Schur functor can be applied to V, yielding the left F 𝔖_{m}-module 𝔚^{m}(V). Then, by [24, Cor. 3.2], there is an isomorphism of F 𝔖_{n}-modules

𝔚^{n}(L^{q}(V)) ≅ Ind_{𝔖_{m}≀𝔖_{q}}^{𝔖_{n}}(𝔚^{m}(V)^{⊗q} ⊗ { \atop Lie_{F} (q)}) .

We apply this to our fixed direct summand on the right-hand side of (9), with V := (F^{n})^{⊗m} ∗ f_{m}. Thus letting

X_{m} := 𝔚^{m}((F^{n})^{⊗m} ∗ f_{m})

yields the decomposition (8), and it remains to show that X_{m} is a projective F 𝔖_{m}-module, and does not depend on the F [GL_{n}(F)]-module F^{n} used to define it.

But, since n ⩾ m, the Schur functor 𝔚^{m} takes (F^{n})^{⊗m} to the regular module F 𝔖_{m}, and the isomorphism 𝔚^{m}((F^{n})^{⊗m}) ≅ F 𝔖_{m} translates the place permutation action into right multiplication. Thus we deduce

X_{m} = 𝔚^{m}((F^{n})^{⊗m} ∗ f_{m}) = 𝔚^{m}((F^{n})^{⊗m}) ∗ f_{m} ≅ F 𝔖_{m} · f_{m} ,

which is of course a projective F 𝔖_{m}-module, and independent of n.

Note that also for the case k = 1, which is excluded from the present discussion, we have a decomposition similar to (8), but in this case becoming trivial inasmuch as for t = 0 we get the trivial F 𝔖_{1}-module X_{1} ≅ F, and X_{p^{t}} = {0} for t ⩾ 1. Thus the crucial question arising now is whether the projective modules X_{kp^{t}} in the decomposition (8) could possibly be {0}. The next lemma shows that this is not the case, which will be essential for Theorem 5.7 below. We remark that, by the above proof, we have X_{kp^{t}} ≠ {0} if and only if the idempotent f_{kp^{t}} ∈ F 𝔖_{kp^{t}} is different from zero. This should follow from a close inspection of the results in [6] and [3], but unfortunately is not explicitly stated there. Hence we provide a straightforward proof, based on calculations in [5].

5.6 Lemma. Keep the notation as in Theorem 5.5. Then X_{kp^{t}} ≠ {0}, for all t ⩾ 0.

Proof. For t ⩾ 0, write

x_{kp^{t}} := \frac{dim(X_{kp^{t}})}{dim(Lie_{F} (kp^{t}))} = \frac{dim(X_{kp^{t}})}{(kp^{t} − 1)!} .

Note that, taking d = t in Theorem 5.5, we see that X_{kp^{t}} is isomorphic to a submodule of the Lie module Lie_{F} (kp^{t}). Thus 0 ⩽ x_{kp^{t}} ⩽ 1 and it suffices to show that x_{kp^{t}} > 0. Now observe that, in the notation of [5, page 851], we have X_{kp^{t}} = C(kp^{t}). Hence, by [5, (10)],

x_{kp^{t}} = 1 − ∑_{i=1}^{t} a′_{i}(x_{kp^{t−i}})^{p^{i}} ,

where a′_{i} = (kp^{t−i})^{−(p^{i}−1)}. Since k > 1, we have

∑_{i=1}^{t} a′_{i}(x_{kp^{t−i}})^{p^{i}} ⩽ ∑_{i=1}^{t} a′_{i} ⩽ ∑_{i=1}^{t} k^{−(p^{i}−1)} < ∑_{j=1}^{∞} k^{−j} = (k − 1)^{−1} ⩽ 1 .

Therefore, x_{kp^{t}} > 0.

We can now formulate the following reduction result:

5.7 Theorem. Let n = k · p^{d}, for some d ⩾ 0 and some k > 1 with p ∤ k.

(a) Let L be an indecomposable direct summand of Lie_{F} (n) with vertex Q. There are some integer t ∈ {0, …, d}, a composition λ of p^{d−t}, and an indecomposable direct summand L′ of Res_{𝔖_{λ}}^{𝔖_{p^{d−t}}} (Lie_{F} (p^{d−t})) such that

Q ⩽_{𝔖_{n}} (Q′)^{♯} ⩽ 𝔖_{λ}^{♯} ⩽ 𝔖_{kp^{t}} ≀ 𝔖_{p^{d−t}} ⩽ 𝔖_{n} , for every vertex Q′ of L′.

(b) In the situation of part (a), if S′ is a Q′-source of L′, then there is a Q-source S of L such that S | Res_{Q}^{(Q′)^{♯}}((S′)^{♯}). Moreover, there is an indecomposable direct summand K of Lie_{F} (p^{d−t}) having a vertex R with Q′ ⩽ R. Furthermore, there is an R-source T of K such that S′ | Res_{Q′}^{R}(T).

(c) Conversely, let 0 ⩽ t ⩽ d, and let K be any indecomposable direct summand of Lie_{F} (p^{d−t}) with vertex R. Then there is an indecomposable direct summand L of Lie_{F} (n) with vertex R^{♯}, and every R-source of K is then also an R^{♯}-source of L.

Proof. Parts (a) and (b) are immediate consequences of 2.2, Theorem 5.4 and Theorem 5.5. Note that here we need the fact that the F 𝔖_{kp^{t}}-modules X_{kp^{t}} in Theorem 5.5 are projective.

It remains to prove (c). So let t ∈ {0, …, d}. Let further X be any indecomposable direct summand of the F 𝔖_{kp^{t}}-module X_{kp^{t}}; note that here we need Lemma 5.6 to ensure that all the projective modules in Theorem 5.5 are indeed non-zero. Now consider the one-part partition λ = (p^{d−t}) of p^{d−t}. Then, by 5.3, we get the indecomposable F [𝔖_{kp^{t}} ≀ 𝔖_{p^{d−t}}]-module L′ := X^{⊗p^{d−t}} ⊗ K̂. By Theorem 5.4(c), L′ has vertex R^{♯}, and every R-source T of K yields the R^{♯}-source T^{♯} of L′. As we have remarked in 2.2(d), there is an indecomposable direct summand L of Ind_{𝔖_{kp^{t}}≀𝔖_{p^{d−t}}}^{𝔖_{n}} (L′) with vertex R^{♯} and R^{♯}-source T^{♯}. By Theorem 5.5, we have

Ind_{𝔖_{kp^{t}}≀𝔖_{p^{d−t}}}^{𝔖_{n}} (L′) | Lie_{F} (n), and hence assertion (c) follows.

Exploiting our computational data from Section 4 and Theorem 5.7 we obtain the following

5.8 Corollary. Let p = 2 and n = k · 2^{d}, where k ⩾ 1 is odd and 0 ⩽ d ⩽ 3, and let L be an indecomposable direct summand of Lie_{F} (n).

(a) Let Q ⩽ 𝔖_{n} be a vertex of L, and let S be a Q-source. Then Q is elementary abelian of order |Q| ⩽ 2^{d}, and S is an endo-permutation module.

(b) Suppose that |Q| is maximal amongst the orders of the vertices of all the indecomposable direct summands of Lie_{F} (n). Then one has Q =_{𝔖_{n}} E_{2^{d}}^{♯} ⩽ 𝔖_{2^{d}}^{♯} ⩽ 𝔖_{k} ≀ 𝔖_{2^{d}} ⩽ 𝔖_{n}; in particular, Q is uniquely determined up to 𝔖_{n}-conjugation. Moreover, every E_{2^{d}}-source of Lie_{F}^{pf}(2^{d}) is an E_{2^{d}}^{♯}-source of L.

Proof. To show (a), by Theorem 5.7 and 2.2(c), there is some integer t ∈ {0, …, d}, and there is an indecomposable direct summand L′ of Lie_{F} (2^{d−t}) with vertex R such that Q ⩽_{𝔖_{n}} R^{♯} ⩽ 𝔖_{k·2^{t}}≀𝔖_{2^{d−t}} ⩽ 𝔖_{n}. (Note that the image of the map (−)^{♯} depends on the particular choice of t.) Moreover, we observe from the results of 4.5, 4.6 and Remark 4.3 that all the indecomposable direct summands of Lie_{F} (2^{d−t}) are either projective, or are isomorphic to Lie_{F}^{pf}(2^{d−t}) and have elementary abelian vertex of order 2^{d−t} and endo-permutation sources. Finally, recall that the property of being an endo-permutation module is retained under restriction to subgroups and under taking direct summands.

To show (b), recall again that if L is an indecomposable direct summand of Lie_{F} (n) then there is some t ⩽ d such that the vertices of L are conjugate to subgroups of E_{2^{d−t}}^{♯}. Now note that, by Theorem 5.7(c), there indeed is an indecomposable direct summand of Lie_{F} (n) having a vertex that is 𝔖_{n}-conjugate to E_{2^{d}}^{♯}. Thus, if |Q| is maximal then we have |Q| = 2^{d}, and Q is 𝔖_{n}-conjugate to E_{2^{d}}^{♯}; hence we may assume that Q = E_{2^{d}}^{♯}. But this forces t = 0 and K = Lie_{F}^{pf}(2^{d}). By Theorem 5.7(c) again, every E_{2^{d}}-source of Lie_{F}^{pf}(2^{d}) is an E_{2^{d}}^{♯}-source of L.

The following result deals with the case where p = 3. The proof is completely analogous to that of Corollary 5.8, and is thus left to the reader.

5.9 Corollary. Let p = 3 and n = k · 3^{d}, where k ⩾ 1 is such that 3 ∤ k, and 0 ⩽ d ⩽ 2, and let L be an indecomposable direct summand of Lie_{F} (n).

(a) Let Q ⩽ 𝔖_{n} be a vertex of L, and let S be a Q-source. Then Q is elementary abelian of order |Q| ⩽ 3^{d}, and S is an endo-permutation module.

(b) Let |Q| be maximal amongst the orders of the vertices of all the indecomposable direct summands of Lie_{F} (n). Then one has Q =_{𝔖_{n}} E_{3^{d}}^{♯} ⩽ 𝔖_{3^{d}}^{♯} ⩽ 𝔖_{k} ≀ 𝔖_{3^{d}} ⩽ 𝔖_{n}; in particular, Q is uniquely determined up to 𝔖_{n}-conjugation. Moreover, every E_{3^{d}}-source of Lie_{F}^{pf}(3^{d}) is an E_{3^{d}}^{♯}-source of L.

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R.B.: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom roger.bryant@manchester.ac.uk

S.D.: Department of Mathematics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany danz@mathematik.uni-kl.de

K.E.: Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, United Kingdom, erdmann@maths.ox.ac.uk

J.M.: Lehrstuhl D für Mathematik, RWTH Aachen Templergraben 64, D-52062 Aachen, Germany Juergen.Mueller@math.rwth-aachen.de