Free Lie algebras as modules for symmetric groups

Research Report MRR99-003

Free Lie algebras as modules for symmetric groups

R.M. Bryant, L.G. Kovács and Ralph Stöhr

Abstract: Let r be a positive integer, $\Bbb F$ a field of odd prime characteristic p, and L the free Lie algebra of rank r over $\Bbb F$ . Consider L a module for the symmetric group $\frak S_r$ of all permutations of a free generating set of L. The homogeneous components

of L are finite dimensional submodules, and L is their direct sum. For r=p and r=p+1, the main results of this paper identify the non-projective indecomposable direct summands of the

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

are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p=2 have been obtained elsewhere.)

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