Application of Koszul complex to Wronski relations for $U({\mathfrak{gl}}_n)$

Abstract

Explicit relations between two families of central elements in the universal enveloping algebra $U({\mathfrak{gl}}_n)$ of the general linear Lie algebra ${\mathfrak{gl}}_n$ are presented. The two families of central elements in question are the ones expressed respectively by the determinants and the permanents: the former are known as the Capelli elements, and the latter are the central elements obtained by Nazarov. The proofs given are based on the exactness of the Koszul complex and the Euler-Poincaré principle.