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

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.

A representation-theoretic proof of the branching rule for Macdonald polynomials

“…• In the specialization q = 0, the Macdonald polynomials become the Hall-Littlewood polynomials. In [Ven14], a summation expression was given for matrix elements of the U q (gl n )-intertwiners Φ n λ in the Gelfand-Tsetlin basis; this expression factors and becomes particularly simple in the Hall-Littlewood limit. In the notation of [Ven14], Ω β/µ (q 2 , q 2k ) can be non-zero only if µ i ≤ β i ≤ µ i +(k−1), meaning that the prelimit expression of [Ven14, Theorem 1.3] is a sum over an index set similar to that which appears in Proposition 4.3.…”

“…In [Ven14], a summation expression was given for matrix elements of the U q (gl n )-intertwiners Φ n λ in the Gelfand-Tsetlin basis; this expression factors and becomes particularly simple in the Hall-Littlewood limit. In the notation of [Ven14], Ω β/µ (q 2 , q 2k ) can be non-zero only if µ i ≤ β i ≤ µ i +(k−1), meaning that the prelimit expression of [Ven14, Theorem 1.3] is a sum over an index set similar to that which appears in Proposition 4.3. It would be interesting to understand if the factorization which results from degenerating [Ven14, Theorem 1.3] may be obtained by degenerating our Theorem 4.4.…”