Bases de Gröbner de Variedades de Gelfand-Tsetlin

Abstract

In this work, we present the Gelfand-Tsetlin variety associated with the Lie algebra gl_n through the Gelfand-Tsetlin subalgebra, denoted by Γ. To achieve this, we conduct a detailed study of this subalgebra Γ, which definition is based on the centers of the universal enveloping algebra of gl_i, with i = 1, ..., n, which D. P. Zelobenko in [ZE73] proved to be described as polynomials in i variables. Additionally, we provide an alternative polynomial characterization for these centers that arises from the theory of quasideterminants, developed in the paper [GKL95] and the book [MO07]. The concept of Gröbner Basis is also introduced, along with the algorithm for calculating this basis, aiming to understand the process of classification of algebraic varieties, in this case, the Gelfand-Tsetlin variety for gl_n.