A free Lie algebra as a module over the full linear group

Abstract

In this paper we consider a tree Lie algebra over a field of characteristic zero. This algebra is a module over the full linear group, and the spaces of homogeneous elements are invariant under this action. We study the decomposition of the homogeneous spaces into irreducible components and calculate their multiplicities. One method for calculating these multiplicities involves their connection with the values of the irreducible characters of the symmetric group on conjugacy classes of elements corresponding to a product of independent cycles of the same length. In the second section we give an explicit formula for calculating such character values. This formula is analogous to the hook formula for the dimension of the irreducible modules of the symmetric group. In the second method for calculating multiplicities we make use of Witt's formula for the dimensions of the polyhomogeneous components of a free Lie algebra. The rest of this paper deal with relations between the Hilbert series of a free two-generator Lie algebra and the generating series of the multiplicities of the irreducible modules in this algebra.