Factorisation of Lie Resolvents

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Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V, regarded as an FG-module, and, for each positive integer r, let Lʳ(V) be the rth homogeneous component of L(V), called the rth Lie power of V. In a previous paper we obtained a decomposition of Lʳ(V) as a direct sum of modules of the form Lˢ(W), where s is a power of p. Here we derive some consequences. First we obtain a similar result for restricted Lie powers of V. Then we consider the `Lie resolvents' Φʳ: certain functions on the Green ring of FG which determine Lie powers up to isomorphism. For k not divisible by p, we obtain the factorisation Φpᵐk = Φᵖᵐ ∘ Φᵏ, separating out the key case of p-power degree. Finally we study certain functions on power series over the Green ring, denoted by S* and L*, which encode symmetric powers and Lie powers, respectively. In characteristic 0, L* is the inverse of S*. In characteristic p, the composite L* ∘ S* maps any p-typical power series to a p-typical power series.