Application of Koszul complex to Wronski relations for \(U(\mathfrak{gl}_n)\)
It is well-known that the center \(Z({\mathfrak g}{\mathfrak l}_n)\) of the universal enveloping algebra \(U({\mathfrak g}{\mathfrak l}_n)\) of the general linear Lie algebra \({\mathfrak g}{\mathfrak l}_n\) is isomorphic to the polynomial ring in \(n\) variables. Furthermore, there are two classical sets of generators in \(Z({\mathfrak g}{\mathfrak l}_n)\), the first one consists of the so-called Capelli elements \(C_k\), which can be expressed via determinants, and the second one, discovered by \textit{M. L. Nazarov} in [Lett. Math. Phys. 21, No. 2, 123--131 (1991; Zbl 0722.17004)], consists of the elements \(D_k\), which can be expressed via permanents. There are classical formulae which express \(\{C_k\}\) via \(\{D_l\}\) and vice versa, usually called Wronski formulae, and there are also classical relations involving both \(\{C_k\}\) and \(\{D_l\}\), usually called Wronski relations. In the very interesting paper under review the author proves both Wronski formulae and Wronski relations using the exactness of the Koszul complex and the Euler-Poincaré principle.
center of the universal enveloping algebra
scientific article; zbMATH DE number 2038402