Extending the science fiction and the Loehr--Warrington formula

TL;DR

This paper introduces a new polynomial generalization related to Macdonald polynomials, establishing connections with existing formulas and conjectures, and extends the science fiction conjecture and Macdonald positivity.

Contribution

It defines the Macdonald piece polynomial $ ext{I}_{mma,da,k}[X;q,t]$, linking it to key Macdonald-related formulas and conjectures, and extends the science fiction conjecture.

Findings

Established the connection between $ ext{I}_{mma,da,k}$ and $ abla s_{da}$ using plethystic formulas.

Linked $ ext{I}_{mma,da,k}$ to the Loehr--Warrington formula via combinatorics of $P$-tableaux.

Extended the science fiction conjecture and Macdonald positivity using the new polynomial.

Abstract

We introduce the Macdonald piece polynomial $\operatorname{I}_{\mu,\lambda,k}[X;q,t]$, 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 $\operatorname{I}_{\mu,\lambda,k}$, $\nabla s_{\lambda}$, and the Loehr--Warrington formula $\operatorname{LW}_{\lambda}$, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect $\operatorname{I}_{\mu,\lambda,k}$ and $\nabla s_{\lambda}$, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect $\operatorname{I}_{\mu,\lambda,k}$ and $\operatorname{LW}_{\lambda}$, 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 $\operatorname{I}_{\mu,\lambda,k}$.