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Free Lie algebras as modules for symmetric groups

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

Published online by Cambridge University Press:  09 April 2009

R. M. Bryant ,
L. G. Kovács  and
Ralph Stöhr
R. M. Bryant
Affiliation:
UMIST PO Box 88 Manchester M60 1QD England e-mail: bryamt@umist.ac.uk e-mail: r.stohr@umist.ac.uk
L. G. Kovács
Affiliation:
Australian National University Canberra ACT 0200 Australia e-mail: kovacs@maths.anu.edu.au
Ralph Stöhr
Affiliation:
UMIST PO Box 88 Manchester M60 1QD England e-mail: bryamt@umist.ac.uk e-mail: r.stohr@umist.ac.uk
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Abstract

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Let r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

MSC classification

Secondary: 17B01: Identities, free Lie (super)algebras 20C30: Representations of finite symmetric groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 2 , October 1999 , pp. 143 - 156
Copyright
Copyright © Australian Mathematical Society 1999

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