Expanding the quasisymmetric Macdonald polynomials in the fundamental basis

TL;DR

This paper derives an expansion of quasisymmetric Macdonald polynomials in the fundamental basis, providing a new combinatorial understanding of these refined symmetric functions.

Contribution

It introduces an explicit expansion of quasisymmetric Macdonald polynomials in the fundamental basis, enhancing the combinatorial framework of these polynomials.

Findings

Explicit expansion formula derived

Connections to quasisymmetric Schur polynomials established

Provides tools for further combinatorial analysis

Abstract

The quasisymmetic Macdonald polynomials $G_{\gamma}(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_{\lambda}(X; q, t)$ with the property that $G_{\gamma}(X; 0, 0)$ equals $QS_{\gamma}(X)$, the quasisymmetric Schur polynomial of [9]. We derive an expansion for $G_{\gamma}(X; q, t)$ in the fundamental basis of quasisymmetric functions.