# 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.

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