Title: Extending the science fiction and the Loehr--Warrington formula Show Chinese title

Title: 扩展科幻小说和Loehr--Warrington公式

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 $\operatorname{I}_{\mu,\lambda,k}$. Show Chinese abstract

Abstract: 我们引入了麦克唐纳分段多项式$\operatorname{I}_{\mu,\lambda,k}[X;q,t]$，这是伯杰龙和加西亚在科幻猜想中提出的麦克唐纳交点多项式的广泛推广。我们展示了$\operatorname{I}_{\mu,\lambda,k}$、$\nabla s_{\lambda}$与洛尔-瓦灵顿公式$\operatorname{LW}_{\lambda}$之间的显著联系，从而得到了洛尔-瓦灵顿猜想作为推论。 为了连接$\operatorname{I}_{\mu,\lambda,k}$和$\nabla s_{\lambda}$，我们采用 Garsia--Haiman--Tesler 的麦当劳多项式的 plethystic 公式，为了连接$\operatorname{I}_{\mu,\lambda,k}$和$\operatorname{LW}_{\lambda}$，我们利用关于$P$-tableaux 的组合学的新发现以及列交换规则。我们还通过利用$\operatorname{I}_{\mu,\lambda,k}$扩展了科幻猜想和麦当劳正性。