论文标题
以薄弱的范围$ 132 $避免销售的Sperner属性
The Sperner property for $132$-avoiding intervals in the weak order
论文作者
论文摘要
斯坦利（Stanley）的一个众所周知的结果表明，对称群体$ s_n $的最大抛物线商的弱点具有Sperner财产； Gaetz和Gao在所有$ S_N $上的弱点上建立了同样的属性，解决了一个漫长的问题。在本文中，我们通过表明$ s_n $的任何抛物线商的弱顺序（以及任何$ 132 $ - 避免$ 132 $的间隔）具有Sperner属性，从而在这些结果之间进行了插值。通过展示$ \ mathfrak {sl} _2 $在这些间隔上尊重弱顺序的动作来证明此结果。作为推论，我们获得了用于舒伯特多项式主要专业的新公式。我们的公式可以看作是麦克唐纳（MacDonald）简化的单词公式的强大序言类似物。这种证明技术和公式将Hamaker，Pechenik，Speyer和Weigandt和Gaetz和Gao的工作推广。
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.
