LIE ELEMENTS AND THE MATRIX-TREE THEOREM

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) & SUB; 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 partic-ular, 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 = Sn.