Modular Lie representations of finite groups

Abstract

Let K be a field of prime characteristic p and let G be a finite group with a Sylow p -subgroup of order p. For any finite-dimensional K G-module V and any positive integer n, let L"( V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then L n ( V) can be considered as a K G-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of L"( V) up to isomorphism.

Modular Lie powers and the Solomon descent algebra

“…Recently, Bryant [7][8][9] introduced the Lie resolvents φ n FG : R FG → R FG , n ≥ 1, to study the structure of L n (V ). These can be described by The question was raised in [7] whether such a factorisation rule might hold for arbitrary groups G.…”

Modular Lie representations of groups of prime order

“…It turns out that a reasonably simple closed formula can be obtained in this way. We obtain this formula for n KQ in [6]. However, the proof is rather long and complicated, so in the remainder of the present paper we restrict attention to the n KP .…”

“…We also have ψ n ∧ = ψ n S when char(K) n. Then ψ m ∧ • ψ n ∧ = ψ mn ∧ and ψ m S • ψ n S = ψ mn S . For the purposes of the present paper and its sequel [6] it is more natural to work with the ψ n S than the ψ n ∧ because of the importance of the power series S(V , t) in §4 below. Now let G = P , where P has prime order p and char(K) = p. We shall establish some facts about the ψ n S in this case.…”

“…(This, in turn, is based on [8].) Rather than restrict attention to P we shall initially work with a larger group Q in order to prepare the way for further developments in [6].…”

Factorisation of Lie resolvents

“…The Factorisation Theorem was conjectured in [3] and proved in [7] in the special case of m = 1. It is interesting that the Witt polynomials (as used to define the operations on Witt vectors) arise here in connection with the Factorisation Theorem.…”