We introduce the Macdonald piece polynomial $\operatorname{I}_{\mu,\lambda,k}[X;q,t]$, a broad generalization of the Macdonald intersection polynomial from the science fiction conjecture of Bergeron and Garsia.
We first present an extension of the science fiction conjecture using $\operatorname{I}_{\mu,\lambda,k}[X;q,t]$. We then derive $\nabla s_{\lambda}$ algebraically from $\operatorname{I}_{\mu,\lambda,k}[X;q,t]$, and discuss how we can naturally obtain the reformulation of the Loehr–Warrington formula from $\operatorname{I}_{\mu,\lambda,k}[X;q,t]$ combinatorially. In particular, this provides an elementary proof of the Loehr–Warrington conjecture, previously proven by Blasiak–Haiman–Morse–Pun–Seelinger.
Additionally, we demonstrate the utility of this reformulation of the Loehr–Warrington formula, by proving a conjecture of Wilson regarding a combinatorial formula for $\nabla e_{(1^n)}$.
This talk is based on joint projects with Jaeseong Oh and Seung Jin Lee, as well as a separate collaboration with Jaeseong Oh.
