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 $\hat{I}\pm \models n$, the combinatorial formula for $\nabla C_{\hat{I}}\pm$ is a weighted sum of parking functions. These summations can be converted to a weighted sum of certain LLT polynomials. Thus $\nabla C_{\hat{I}}\pm$ 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.

19 pages