Lie elements and the matrix-tree theorem

Yurii Burman^ab, Valeriy Kulishov<sup>a

a</sup> National Research University Higher School of Economics, 119048, 6 Usacheva str., Moscow, Russia
^b Independent University of Moscow, 119002, 11 B.Vlassievsky per., Moscow, Russia

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 $\mathcal{L}(V) \subset k[G]$ 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 $G = S_n$.

Key words and phrases: matrix-tree theorem, multigraphs, generalized determinants.

MSC: 15A15

Language: English

DOI: 10.17323/1609-4514-2023-23-1-47-58