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 Macdonald polynomials associated with the root system A_{n-1}, extending previous work on their construction via quantum group representations.

Contribution

It introduces a new proof of Macdonald's identities using representation theory, offering a different perspective from combinatorial approaches.

Findings

Representation-theoretic proof of Macdonald's inner product identities

Verification of symmetry identities for Macdonald polynomials

Extension of previous constructions to prove fundamental properties

Abstract

This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system A_n-1 one can obtain Macdonald's polynomials - a new interesting class of symmetric functions recently defined by I. Macdonald {M1] - 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. Macdonald's inner product identities (see [M2]) have been proved by combinatorial methods my Macdonald ([Macdonald, private communication]).