弱序下$132$ -回避区间的Sperner性质
The Sperner property for $132$-avoiding intervals in the weak order
摘要：
Stanley在1980年的一个著名结果暗示了对称群$S_n$的极大抛物商上的弱序具有Sperner性质; Gaetz和Gao最近为所有$S_n$上的弱序建立了相同的性质，解决了一个长期悬而未决的问题。在本文中，我们插入这些结果之间的弱顺序的任何抛物商的$S_n$（更一般地说，任何$132$ -避免间隔）的Sperner属性。这个结果是证明了展示的作用$\mathfrak{sl}_2$尊重弱序在这些区间。作为推论，我们得到了Schubert多项式的主特化的一个新公式。我们的公式可以看作是一个强大的Bruhat顺序模拟麦克唐纳的减少字公式。这种证明技巧和公式推广了Hamaker、Pechenik、Speyer、Weigandt、Gaetz和Gao的工作。
英文摘要：
A well-known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group $S_n$ has the Sperner property; this same property was recently established for the weak order on all of $S_n$ by Gaetz and Gao, resolving a long-open problem. In this paper we interpolate between these results by showing that the weak order on any parabolic quotient of $S_n$ (and more generally on any $132$-avoiding interval) has the Sperner property. This result is proven by exhibiting an action of $\mathfrak{sl}_2$ respecting the weak order on these intervals. As a corollary we obtain a new formula for principal specializations of Schubert polynomials. Our formula can be seen as a strong Bruhat order analogue of Macdonald's reduced word formula. This proof technique and formula generalize work of Hamaker, Pechenik, Speyer, and Weigandt and Gaetz and Gao.
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