Macdonald polynomials and chromatic quasisymmetric functions

TL;DR

This paper connects Macdonald polynomials to chromatic quasisymmetric functions, providing new formulas for Macdonald and Jack polynomials through graph-based expansions and known symmetric function decompositions.

Contribution

It introduces a novel expression of Macdonald polynomials as weighted sums of chromatic quasisymmetric functions, enabling new Schur and power sum formulas.

Findings

Derived Schur and power sum formulas for Macdonald polynomials

Extended formulas to Jack polynomials as special cases

Linked Macdonald polynomials to graph-based chromatic functions

Abstract

We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.