Gelfand--Tsetlin bases for representations of ${\mathrm{U}}_q({\mathfrak{gl}}_n)$ at roots of unity

Abstract

In this paper, we construct Gelfand--Tsetlin bases for representations of ${\mathrm{U}}_q({\mathfrak{gl}}_n)$ at roots of unity. As an application, we obtain explicit bases for certain baby Verma modules of ${\mathrm{U}}_q({\mathfrak{gl}}_3)$ and determine their composition series.