Extending the science fiction and the Loehr--Warrington formula

Abstract

We introduce the Macdonald piece polynomial $I_{\mu ,\lambda ,k}[X;q,t]$, which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between $I_{\mu ,\lambda ,k}$, $\nabla s_{\lambda }$, and the Loehr--Warrington formula ${\mathrm{LW}}_{\lambda }$, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect $I_{\mu ,\lambda ,k}$ and $\nabla s_{\lambda }$, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect $I_{\mu ,\lambda ,k}$ and ${\mathrm{LW}}_{\lambda }$, we use our new findings on the combinatorics of $P$-tableaux together with the column exchange rule. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting $I_{\mu ,\lambda ,k}$.

arXiv comment34 pages, comments are welcome