论文标题
普遍的黑森伯格品种和广义的夏雷什·雷克斯（Shareshian-Wachs）猜想
Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture
论文作者
论文摘要
我们介绍了广义的赫森伯格品种并建立基本事实。我们表明，对称群体$ s_n $对Hessenberg品种共同体的Tymoczko动作扩展到广义的Hessenberg品种，并且其中的自然形态保留了动作。通过分析广义赫森伯格品种之间的自然形态和生育图，我们给出了Shareshian-Wachs猜想的基本证明。此外，我们提出了Shareshian-Wachs猜想的自然概括，该猜想涉及普遍的Hessenberg品种并提供基本证明。作为副产品，我们为加权图提出了一个广义的Stanley-Stembridge猜想。我们对普通赫森伯格品种的生育几何形状的调查使我们能够通过明确的爆炸序列或投影束图将它们修改为更简单的品种，例如投射空间或置换式欧洲式品种。使用此功能，我们提供了两种算法来计算有关广义Hessenberg品种的共同体的$ S_N $代表。作为应用程序，我们计算某些黑森伯格品种的低度同谋的表示。
We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group $S_n$ on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the $S_n$-representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.
