Lie elements and the matrix-tree theorem

TL;DR

This paper introduces Lie elements in the group algebra for finite groups, showing they form a Lie algebra with combinatorial properties, and proves a matrix-tree theorem analogue for permutation representations of symmetric groups.

Contribution

It defines Lie elements in group algebras, proves they form a Lie algebra and G-module, and establishes a matrix-tree theorem analogue for symmetric groups.

Findings

Lie elements form a Lie algebra and G-module

Characteristic polynomial formula for Lie elements in S_n

Analogue of the matrix-tree theorem for permutation representations

Abstract

For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra k[G]. The set L(V) \subset k[G] of Lie elements is a Lie algebra and a G-module acting on the original representation V. Lie elements often exhibit nice combinatorial properties. Thus, for G = S_n and V, a permutation representation, we prove a formula for the characteristic polynomial of a Lie element similar to the classical matrix-tree theorem.