Lie Powers of Modules for Groups of Prime Order

Abstract

Let $L(V)$ be the free Lie algebra on a finite-dimensional vector space $V$ over a field $K$, with homogeneous components $L^n(V)$ for $n \geq 1$. If $G$ is a group and $V$ is a $KG$-module, the action of $G$ extends naturally to $L(V)$, and the $L^n(V)$ become finite-dimensional $KG$-modules, called the Lie powers of $V$. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable $KG$-modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where $G$ has prime order $p$, and $K$ has characteristic $p$. As is well known, there are $p$ indecomposables, denoted here by $J_1,\dots,J_p$, where $J_r$ has dimension $r$. A theory is developed which provides information about the overall module structure of $L(V)$ and gives a recursive method for finding the multiplicities of $J_1,\dots,J_p$ in the Lie powers $L^n(V)$. For example, the theory yields decompositions of $L(V)$ as a direct sum of modules isomorphic either to $J_1$ or to an infinite sum of the form $J_r \oplus J_{p-1} \oplus J_{p-1} \oplus \ldots $ with $r \geq 2$. Closed formulae are obtained for the multiplicities of $J_1,\dots,J_p$ in $L^n(J_p)$ and $L^n(J_{p-1})$. For $r < p-1$, the indecomposables which occur with non-zero multiplicity in $L^n(J_r)$ are identified for all sufficiently large $n$.