Factorisation of Lie resolvents

Abstract

Let G be a group, F a field of prime characteristic p, and V a finite-dimensional F G-module. For each positive integer r , the r th homogeneous component of the free Lie algebra on V is an F G-module called the r th Lie power of V . Lie powers are determined, up to isomorphism, by certain functions Φ r on the Green ring of F G, called 'Lie resolvents'. Our main result is the factorisation Φ p m k = Φ p m • Φ k whenever k is not divisible by p. This may be interpreted as a reduction to the key case of p-power degree.

Bases, filtrations and module decompositions of free Lie algebras

“…In [8] this result was used to determine the indecomposable direct summands of L p and their Krull-Schmidt multiplicities as modules for G = G L(r, K ) in the case where K is an infinite field of characteristic p. For Lie powers L n with (n, p) = 1 this had been accomplished earlier by Donkin and Erdmann [13], while [14] deals with the case where n = mp with (m, p) = 1. Further progress has recently been made by Bryant and Schocker [9,10].…”

Free centre-by-nilpotent-by-abelian groups

“…Theorem 1.1 is the first addition to (1.4) since the result on c = 4 was published in 1993. It has been made possible by recent results on modular Lie representations due to Bryant, Erdmann, and Schocker [2][3][4][5] (see Theorem 3.2 in Section 3). We hope that our approach to the torsion problem via modular Lie powers of relation modules will lead to further additions to (1.4), which can now be restated as follows.…”

Lie powers and Witt vectors

“…It involves the Witt polynomials (as used to define operations on the ring of Witt vectors) and gives the B p m k only as the components of a Witt vector with known 'ghost' components. The results of [3] and [4] give no explicit information about the modules B p m k except that, as already mentioned, B p m k is a direct summand of V ⊗p m k . In this paper we shall give much more precise information.…”

“…Let us write n = p m k, where k is not divisible by p. In [4], a recursive formula was given for the modules B k , B pk , B p 2 k , . .…”

“…It is well known and easily verified that V ⊗2 ∼ = S 2 (V ) ⊕ ∧ 2 (V ), where S 2 (V ) is the symmetric square of V and ∧ 2 (V ) is the exterior square. The recursive formula from [4] gives B 2 ∼ = L 2 (V ) and…”