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

TL;DR

This paper presents a new proof of the branching rule for Macdonald polynomials using representation theory, specifically through intertwiners of quantum groups and double affine Hecke algebras.

Contribution

It introduces a novel representation-theoretic approach to the branching rule for Macdonald polynomials, leveraging the Etingof-Kirillov Jr. expression and Dunkl-Kasatani conjecture.

Findings

Diagonal matrix elements relate to Macdonald's operators

Map between spherical parts of double affine Hecke algebras established

New proof simplifies understanding of Macdonald polynomial branching

Abstract

We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of U_q(gl_n). In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.