Free Lie algebras and formal power series
Let \(G\) be a group and \(K\) a field. Suppose that \(V\) is a graded \(KG\)-module of the form \(V=V_1\oplus V_2\oplus\dots \), where each \(V_n\) is finite dimensional. Then the free Lie algebra \(L(V)\) acquires the structure of a graded \(KG\)-module \(L(V)=L^1(V)\oplus L^2(V)\oplus\dots \). The isomorphism types of \(V\) and \(L(V)\) may be described by the power series \(V(t)=\sum_{n\geq 1}[V_n]t^n\) and \(L(V)(t)=\sum_{n\geq 1}[L^n(V)]t^n\), where \([V_n]\) and \([L^n(V)]\) are the coefficients in the Green ring \(\Gamma_{KG}\). The object of study is the Lie module function \({\mathcal L}_{KG}: V(t)\to L(V)(t)\). It is a particular case of a logarithmic function introduced by the author. The Lie module function is difficult to describe in the general case. In a similar setting of the Grothendieck algebra \(\overline{\Gamma}_{KG}\), an explicit formula for the function \({\overline{\mathcal L}}_{KG}\) is given using Adams operators. If the field is of characteristic zero, formulae are given that are closely related to known character formulae of Brandt and others. The virtue of the paper is that it develops methods to attack the case of an arbitrary field.
scientific article; zbMATH DE number 1836724
scientific article