Signature characters of highest-weight representations of $U_q({\mathfrak{gl}}_n)$

Abstract

We consider $U_q({\mathfrak{gl}}_n)$, the quantum group of type $A$ for $|q|=1$, $q$ generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary $q$: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit $q\to 1$ that were obtained by different means. Finally, we provide several explicit examples of signature characters.

12 pages