Gelfand-Tsetlin variety for $\mathfrak{gl}_n$

TL;DR

This paper generalizes Ovsienko's theorem on the equidimensionality of the Gelfand-Tsetlin variety for gl_n, extending its applicability and addressing a weaker version that includes the regular case, with implications in representation theory.

Contribution

It extends Ovsienko's result to a broader context and addresses a weaker version of the theorem including the regular case.

Findings

Gelfand-Tsetlin variety for gl_n is equidimensional with dimension n(n-1)/2

Generalization of Ovsienko's theorem to a broader setting

Includes a weak version of the theorem covering the regular case

Abstract

S. Ovsienko proved that the Gelfand-Tsetlin variety for $\mathfrak{gl}_n$ is equidimensional (i.e. all its irreducible components have the same dimension) with dimension equals $\frac{n(n-1)}{2}$. This result has important consequences in Representation Theory of Algebras, implying, in particular, the equidimensionality of the nilfiber of the Kostant-Wallach map. In this paper we will present the generalization of this result and will address a weak version of Ovsienko's Theorem which includes the regular case.