Factorisation of Lie Resolvents

Abstract

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 $L^r(V)$ be the $r$th homogeneous component of $L(V)$, called the $r$th Lie power of $V$. In a previous paper we obtained a decomposition of $L^r(V)$ as a direct sum of modules of the form $L^s(W)$, 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' $\Phi^r $: 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 $\Phi^{p^mk} = \Phi^{p^m} \circ \Phi^k$, separating out the key case of $p$-power degree. Finally we study certain functions on power series over the Green ring, denoted by ${\bf S}^*$ and ${\bf L}^*$, which encode symmetric powers and Lie powers, respectively. In characteristic 0, ${\bf L}^*$ is the inverse of ${\bf S}^*$. In characteristic $p$, the composite ${\bf L}^* \circ {\bf S}^*$ maps any $p$-typical power series to a $p$-typical power series.