Modular Lie powers and the Solomon descent algebra

Let $V$ be an $r$-dimensional vector space over an infinite field $F$ of prime characteristic $p$, and let $L_n(V)$ denote the $n$-th homogeneous component of the free Lie algebra on $V$. We study the structure of $L_n(V)$ as a module for the general linear group $GL_r(F)$ when $n=pk$ and $k$ is not divisible by $p$ and where $n \geq r$. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of $L_k(V)$ and the indecomposable direct summands of $L_n(V)$ which are not isomorphic to direct summands of $V^{\otimes n}$. The direct summands of $L_k(V)$ have been parametrised earlier, by Donkin and Erdmann. Bryant and St\"{o}hr have considered the case $n=p$ but from a different perspective. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups.