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.

49 pages