A summation formula for Macdonald polynomials

TL;DR

This paper presents an explicit summation formula for symmetric Macdonald polynomials, unifying and extending known formulas for various special cases such as Jack and $q$-Whittaker polynomials.

Contribution

It introduces a new explicit sum formula involving symmetric group sums and Hecke algebra actions, applicable to Macdonald and related polynomials.

Findings

Recover known formulas for monomial symmetric and Hall-Littlewood polynomials at special cases

Derive new expressions for Jack and $q$-Whittaker polynomials

Unify various polynomial formulas under a single summation framework

Abstract

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and $q$-Whittaker polynomials.