# Moscow Mathematical Journal

Volume 23, Issue 1, January–March 2023  pp. 47–58.

Lie Elements and the Matrix-Tree Theorem

Authors:  Yurii Burman (1) and Valeriy Kulishov (2)
Author institution:(1) National Research University Higher School of Economics, 119048, 6 Usacheva str., Moscow, Russia, and Independent University of Moscow, 119002, 11 B.Vlassievsky per., Moscow, Russia
(2) National Research University Higher School of Economics, Moscow, Russia

Summary:

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 $\mathcal{L}(V) \subset k[G]$ of Lie elements is a Lie algebra and a $G$-module acting on the original representation $V$.

Lie elements often exhibit nice combinatorial properties. In particular, we prove a formula, similar to the classical matrix-tree theorem, for the characteristic polynomial of a Lie element in the permutation representation $V$ of the group $G = S_n$.

2020 Math. Subj. Class. 15A15.

Keywords: Matrix-tree theorem, multigraphs, generalized determinants.