Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials

TL;DR

This paper provides a representation-theoretic proof of Macdonald's inner product and symmetry identities for the root system A_{n-1}, utilizing techniques from quantum group representations and ribbon graphs.

Contribution

It introduces a novel proof of Macdonald's identities using quantum group representations and ribbon graph techniques, extending previous work on Macdonald polynomials.

Findings

Representation-theoretic proof of Macdonald's identities

Recursive relations derived from symmetry identities

Application of ribbon graph techniques to Macdonald polynomials

Abstract

This paper is a continuation of our papers \cite{EK1, EK2}. In \cite{EK2} we showed that for the root system $A_{n-1}$ one can obtain Macdonald's polynomials as weighted traces of intertwining operators between certain finite-dimensional representations of $U_q(sl_n)$. The main goal of the present paper is to use this construction to give a representation-theoretic proof of Macdonald's inner product and symmetry identities for the root system $A_{n-1}$. The proofs are based on the techniques of ribbon graphs developed by Reshetikhin and Turaev. We also use the symmetry identities to derive recursive relations for Macdonald's polynomials.