On group actions on free Lie algebras

Abstract

We first study the module structure of the free Lie algebra $L(V)$ in characteristic zero under the action of the general linear group. Here we give a new, purely combinatorial, proof of Klyachko's celebrated theorem on Lie representations using the Kra\\'{s}kiewicz-Weyman theorem.\n\nWe then give a new factorisation of the Dynkin-Specht-Wever\nidempotent and use this to prove that $L_2(L_k(V))$ is a $KG$-module direct summand of $L_{2k}(V)$, for $G$ an arbitrary group, $K$ a field of characteristic $p \\nmid k$ and $V$ a $KG$-module. For finite-dimensional modules $V$, this follows immediately from the Decomposition Theorem of Bryant and Schocker. We consider a small example of this theorem, namely the sixth Lie power over a field of\ncharacteristic $3$. Here we show explicitly that $L_6(V)$ decomposes into a direct sum of the modules $L_3(L_2(V))$ and $L_2(V) \\otimes S_2(V) \\otimes S_2(V)$, where $S_2(V)$ denotes the symmetric square of $V$. We give a description, up to isomorphism, of the modules $B_{p^mk}$ occurring in the Decomposition Theorem.\n\nFinally, we apply our knowledge of Lie powers to a group theoretic problem. We show that the torsion subgroup $t_c$ of the quotient $\\gamma_c R/ [\\gamma_c R, F]$ is bounded as follows, for $c=2p^m$ or $c=3p^m$, where $p$ is an arbitrary prime, $m\\geq 0$:\n\n\\begin{eqnarray*}\n2t_{2p^m} = 0 \\;\\;\\;\\mbox{provided } G=F/R \\mbox{ has no 2-torsion and no p-torsion,}\\\\\n3t_{3p^m} = 0 \\;\\;\\;\\mbox{provided } G=F/R \\mbox{ has no 3-torsion\nand no p-torsion.}\n\\end{eqnarray*}\n\nThus, we have that $\\gamma_6 R/ [\\gamma_6 R, F]$ is torsion-free, provided that $G=F/R$ has no elements of order $2$ or $3$.