Macdonald's polynomials and representations of quantum groups

TL;DR

This paper derives a new formula for Macdonald's polynomials associated with the root system A(n-1), connecting them to quantum group representations and providing insights into their eigenfunctions and related hypergeometric functions.

Contribution

It introduces a trace formula for Macdonald's polynomials using quantum group representation theory, linking them to quantum Knizhnik-Zamolodchikov equations.

Findings

New trace formula for Macdonald's polynomials

Connection between Macdonald's difference operators and quantum group centers

Generalized q-hypergeometric functions as eigenfunctions

Abstract

In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functions which are related to solutions of the quantum Knizhnik-Zamolodchikov equations.