Free Lie algebras and formal power series

Let G be a group and K a field. If V is a graded KG-module of the form V=V1 V2 , where each Vn is finite dimensional, then the free Lie algebra L(V) acquires the structure of a graded KG-module, L(V)=L1(V) L2(V) . The isomorphism types of V and L(V) may be described by the power series ∑n 1[Vn]tn and ∑n 1[Ln(V)]tn with coefficients from the Green ring. The main object of study is the function on power series which maps ∑[Vn]tn to ∑[Ln(V)]tn for every graded KG-module V. Closed formulae are given in certain cases, and these are closely related to character formulae of Brandt and others.