Positive expansions of permuted basement and quasisymmetric Macdonald polynomials at $t=0$

TL;DR

This paper investigates the $t=0$ specializations of quasisymmetric and nonsymmetric Macdonald polynomials, demonstrating their positive expansions into quasisymmetric Schur functions and Demazure atoms, respectively, and describing their structure coefficients.

Contribution

It establishes positive expansion properties of $t=0$ specialized quasisymmetric and nonsymmetric Macdonald polynomials into fundamental bases and characterizes their structure coefficients.

Findings

Quasisymmetric Macdonald polynomials expand positively into quasisymmetric Schur functions.

ASEP polynomials expand positively into Demazure atoms.

Structure coefficients are described via the charge statistic on semistandard tableaux.

Abstract

It is well known that the $q$-Whittaker polynomials, which are $t=0$ specializations of the Macdonald polynomials $P_\lambda(X;q,t)$, expand positively as the sum of Schur polynomials. Macdonald polynomials have a quasisymmetric refinement: the quasisymmetric Macdonald polynomials $G_\gamma(X;q,t)$, and a nonsymmetric refinement: the ASEP polynomials $f_\alpha(X;q,t)$. We study the $t=0$ specializations of both these families of polynomials and show analogous properties: the quasisymmetric Macdonald polynomials expand positively as a sum of quasisymmetric Schur functions, $\text{QS}_\gamma(X)$, and the ASEP polynomials expand positively as a sum of Demazure atoms, $\mathcal{A}_\alpha(X)$. As a corollary of the latter, we prove more generally that any permuted basement Macdonald polynomial has a positive expansion in the Demazure atoms at $t=0$. We give a description of the structure coefficients of $G_\gamma(X;q,0)$ and $f_\alpha(X;q,0)$ in both cases in terms of the charge statistic on a restricted set of semistandard tableaux.