Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials

TL;DR

This paper introduces new compact formulas for Macdonald polynomials and a novel quasisymmetric analogue, enhancing understanding and computation of these important symmetric functions.

Contribution

It provides new compact formulas for Macdonald polynomials and introduces a quasisymmetric analogue that refines and specializes to known quasisymmetric Schur polynomials.

Findings

New compact formulas for Macdonald polynomials

Introduction of quasisymmetric Macdonald polynomials

Refinement to quasisymmetric Schur polynomials

Abstract

We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These quasisymmetric Macdonald polynomials refine the (symmetric) Macdonald polynomials and specialize at $q=t=0$ to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg. This is an extended abstract.