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.