Combinatorial construction of Gelfand-Tsetlin modules for ${\mathfrak{gl}}_n$

Abstract

We propose a new effective method of constructing explicitly Gelfand -Tsetlin modules for ${\mathfrak{gl}}_n$. We obtain a large family of simple modules that have a basis consisting of Gelfand-Tsetlin tableaux, the action of the Lie algebra is given by the Gelfand-Tsetlin formulas and with all Gelfand-Tsetlin multiplicities equal $1$. As an application of our construction we prove necessary and sufficient condition for the Gelfand and Graev's continuation construction to define a module which was conjectured by Lemire and Patera.

References in corpus (2)

• Irreducible Generic Gelfand-Tsetlin Modules of $\mathfrak{gl}(n)$
• A new way to construct 1-singular Gelfand-Tsetlin modules

Cited by in corpus (2)

• Gelfand-Tsetlin modules of quantum gl_n$defined by admissible sets of relations
• Explicit construction of irreducible modules for U_q(gl_n)