Signature characters of highest-weight representations of $U_{q}(\mathfrak{gl}_{n})$

TL;DR

This paper derives formulas for signature characters of highest-weight modules of the quantum group $U_q(gl_n)$ at $|q|=1$, using Gelfand-Tsetlin bases, and explores unitarity conditions.

Contribution

It introduces combinatorial formulas for signature characters of irreducible modules of $U_q(gl_n)$ at generic $q$, and classifies unitarity spectra.

Findings

Formulas for signature characters of finite-dimensional irreducible modules.

Classification of the unitarity locus for arbitrary $q$.

Explicit examples illustrating the signature characters.

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 $q \rightarrow 1$ that were obtained by different means. Finally, we provide several explicit examples of signature characters.