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

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.