Lie Powers of Modules for Groups of Prime Order

Abstract

Let L(V) be the free Lie algebra on a finite‐dimensional vector space V over a field K, with homogeneous components Ln(V) for n ⩾ 1. If G is a group and V is a KG‐module, the action of G extends naturally to L(V), and the Ln(V) become finite‐dimensional KG‐modules, called the Lie powers of V. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable KG‐modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where G has prime order p, and K has characteristic p. As is well known, there are p indecomposables, denoted here by J1,…,Jp, where Jr has dimension r. A theory is developed which provides information about the overall module structure of LV) and gives a recursive method for finding the multiplicities of J1,…,Jp in the Lie powers Ln(V). For example, the theory yields decompositions of L(V) as a direct sum of modules isomorphic either to J1 or to an infinite sum of the form Jr ⊕ J{p‐1} ⊕ J{p‐1} ⊕ … with r ⩾ 2. Closed formulae are obtained for the multiplicities of J1,…, Jp in Ln(Jp and Ln(J{p‐1). For r < p‐1, the indecomposables which occur with non‐zero multiplicity in Ln(Jr) are identified for all sufficiently large n. 2000 Mathematical Subject Classification: 17B01, 20C20.

The Decomposition of Lie Powers

“…The proof of the Decomposition Theorem uses the Filtration Theorem of §3 as well as Lazard's "Elimination Theorem" (see §2 below). This last result has proved to be fundamental in the theory and has been repeatedly used in earlier work such as [7].…”

Modular Lie representations of groups of prime order

“…In characteristic 0, L n (V ) can be completely described by Brandt's character formula [3]. The study of L n (V ) in non-zero characteristic is much more difficult, though there has been considerable progress in recent years: see [8] and the citations there.…”

“…Most of this paper is concerned with the important special case G = P , where P is a group of prime order p and K is a field of characteristic p. This case was studied in [8] where the author, together with Kovács and Stöhr, gave a method for obtaining, recursively, the Krull-Schmidt multiplicities of the indecomposable KP -modules in L n (V ). Closed formulae could only be given for certain modules V , and it was believed that if a closed formula for an arbitrary module V could be found it would be extremely complicated and useless as a basis for calculation.…”

Lie powers of free modules for certain groups of prime power order

“…In the case where K has prime characteristic p, and p divides the order of G, it is much more difficult to obtain information about the module structure of L"( V). Recent progress is described in [4] and [5]. The reader is also referred to [4] or [5] for further details of the background and underlying concepts.…”

“…Recent progress is described in [4] and [5]. The reader is also referred to [4] or [5] for further details of the background and underlying concepts.…”