发表机构
Washington University in St. Louis（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明外尔群点作用表示限制到A型抛物子群时分解为A型点作用表示的直和，并为B/C和D型经典外尔群提供组合公式，进而证明Lesnevich猜想及加泰罗尼亚数计数。
AI 中文摘要
正则半单黑森堡簇是旗簇的子簇，对任何约化群都有定义。它们的上同调承载着相应的外尔群的一个表示，称为点作用表示。在$A$型中，这些表示可以通过自然标记的单位区间图的色拟对称函数来计算。我们证明，将任何外尔群的点作用表示限制到$A$型抛物子群，所得到的表示同构于$A$型点作用表示的直和。这项工作受到一个猜想的启发，即类似的性质对所有抛物子群都成立。特别地，对于$B/C$型和$D$型的经典外尔群，我们使用色对称函数，为限制到特定的极大$A$型抛物子群提供了一个简化的组合公式。作为应用，我们证明了Lesnevich关于在标准偏序集同构下识别的$B$型和$C$型理想所关联表示的猜想。具体来说，我们证明了相应的点作用表示同构当且仅当这些理想在各自的根系中确定相同的反射集。此外，我们证明了一个加泰罗尼亚数计数了此类理想的总数。
英文摘要
Regular semisimple Hessenberg varieties are subvarieties of flag varieties defined for any reductive group. Their cohomology carries a representation of the associated Weyl group, known as the dot action representation. In type $A$, these representations can be computed using the chromatic quasisymmetric functions of naturally labeled unit interval graphs. We prove that restricting the dot action representation of any Weyl group to a type $A$ parabolic subgroup yields a representation isomorphic to a direct sum of type $A$ dot action representations. This work is motivated by the conjecture that an analogous property holds for all parabolic subgroups. In particular, for classical Weyl groups of types $B/C$ and $D$, we use chromatic symmetric functions to provide a streamlined, combinatorial formula for the restriction to a specific maximal type $A$ parabolic subgroup. As an application, we prove a conjecture of Lesnevich regarding the representations associated with ideals of types $B$ and $C$ identified under the standard poset isomorphism. Specifically, we show that the corresponding dot action representations are isomorphic if and only if the ideals determine the same reflection sets in their respective root systems. Furthermore, we show that a Catalan number counts the total number of such ideals.
Comments23 pages
