Free Lie algebras as modules for symmetric groups

Abstract

Let r be a positive integer, double-struck F sign a field of odd prime characteristic p, and L the free Lie algebra of rank r over double-struck F sign. Consider L a module for the symmetric group G-fraktur signr 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-projective 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.).