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. © 2002 Elsevier Science (USA). All rights reserved.