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Lie powers of free modules for certain groups of prime power order

Part of: Representation theory of groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

R. M. Bryant  and
I. C. Michos
R. M. Bryant
Affiliation:
UMIST, PO Box 88, Manchester M60 1QDEngland e-mail: bryant@umist.ac.uk
I. C. Michos
Affiliation:
LIAFA, Université Paris 775251 Paris, France e-mail: michos@liafa.jussieu.fr
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Abstract

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Let G be a finite group of order pk, where p is a prime and k ≥ 1, such that G is either cyclic, quaternion or generalised quaternion. Let V be a finite-dimensional free KG-module where K is a field of characteristic p. The Lie powers Ln(V) are naturally KG-modules and the main result identifies these modules up to isomorphism. There are only two isomorphism types of indecomposables occurring as direct summands of these modules, namely the regular KG-module and the indecomposable of dimension pk – pk−1 induced from the indecomposable K H-module of dimension p − 1, where H is the unique subgroup of G of order p. Formulae are given for the multiplicities of these indecomposables in Ln(V). This extends and utilises work of the first author and R. Stöhr concerned with the case where G has order p.

MSC classification

Secondary: 17B01: Identities, free Lie (super)algebras 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 2 , October 2001 , pp. 149 - 158
Copyright
Copyright © Australian Mathematical Society 2001

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