A parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$

Abstract

Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, $\{C_{a}\}$ for any $a\in \mathbb{Z}$, in their compositional refinement of the shuffle (ex-)conjecture. For any $\alpha\vDash n$, the combinatorial formula for $\nabla C_{\alpha}$ is a weighted sum of parking functions. These summations can be converted to a weighted sum of certain LLT polynomials. Thus $\nabla C_{\alpha}$ is Schur positive since Grojnowski and Haiman proved that all LLT polynomials are Schur positive. In this paper, we obtain a recursion that implies the $C$-positivity of $(-1)^{k} m_{2^{k}1^{l}}$, and hence prove the Schur positivity of $(-1)^{k}\nabla m_{2^{k}1^{l}}$. As a corollary, a parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$ is obtained by using the compositional shuffle theorem of Carlsson and Mellit.