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## 2006.269: Lie powers of modules for groups of prime order

2006.269: R. M. Bryant, L. G. Kovàcs and Ralph Stöhr (2002) Lie powers of modules for groups of prime order. Proceedings of the London Mathematical Society, 84 (2). pp. 343-374. ISSN 0024-6093

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## 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\$.

Item Type: Article free Lie algebras; cyclic groups; modular representations. MSC 2000 > 17 Nonassociative rings and algebrasMSC 2000 > 20 Group theory and generalizations 2006.269 Miss Louise Stait 09 August 2006

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