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

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.

Keywords

• macdonald polynomials
• quantum groups
• gelfand-tsetlin basis
• double affine hecke algebras
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]