On the expansion of certain vector-valued characters of $U_q(\mathfrak{gl}_n)$ with respect to the Gelfand-Tsetlin basis

TL;DR

This paper derives a branching rule for Macdonald polynomial characters of quantum group representations, expressing their expansion in the Gelfand-Tsetlin basis, with special focus on the $q=0$ case and its combinatorial interpretation.

Contribution

It introduces a new branching rule for Macdonald polynomial characters of quantum groups, connecting skew Macdonald polynomials and Gelfand-Tsetlin basis expansions.

Findings

Derived a branching rule using skew Macdonald polynomials

Provided explicit expansion in the Gelfand-Tsetlin basis

Analyzed the $q=0$ case with combinatorial interpretation

Abstract

Macdonald polynomials are an important class of symmetric functions, with connections to many different fields. Etingof and Kirillov showed an intimate connection between these functions and representation theory: they proved that Macdonald polynomials arise as (suitably normalized) vector-valued characters of irreducible representations of quantum groups. In this paper, we provide a branching rule for these characters. The coefficients are expressed in terms of skew Macdonald polynomials with plethystic substitutions. We use our branching rule to give an expansion of the characters with respect to the Gelfand-Tsetlin basis. Finally, we study in detail the $q=0$ case, where the coefficients factor nicely, and have an interpretation in terms of certain $p$-adic counts.