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 ${\Phi }^{p^{m}k}={\Phi }^{p^m}\circ {\Phi }^k$, separating out the key case of $p$-power degree. Finally we study certain functions on power series over the Green ring, denoted by ${\mathbf{S}}^{\ast }$ and ${\mathbf{L}}^{\ast }$, which encode symmetric powers and Lie powers, respectively. In characteristic 0, ${\mathbf{L}}^{\ast }$ is the inverse of ${\mathbf{S}}^{\ast }$. In characteristic $p$, the composite ${\mathbf{L}}^{\ast }\circ {\mathbf{S}}^{\ast }$ maps any $p$-typical power series to a $p$-typical power series.

20 pages