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# Mathematical Research Letters

## Volume 23 (2016)

### Number 3

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

Pages: 887 – 927

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n3.a15

#### Author

#### 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 $U_q (\mathfrak{gl}_n)$ given in [11]. 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 of [8, 9, 13, 20].