We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group 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 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 -representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.

中文翻译：

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广义 Hessenberg 簇的双有理几何和广义 Shareshian-Wachs 猜想

我们介绍广义的 Hessenberg 簇并建立基本事实。我们证明对称群对 Hessenberg 簇上同调的 Tymoczko 作用延伸到广义 Hessenberg 簇，并且它们之间的自然态射保留了该作用。通过分析广义Hessenberg簇中的自然态射和双有理映射，我们给出了Shareshian-Wachs猜想的基本证明。此外，我们提出了 Shareshian-Wachs 猜想的自然推广，其中涉及广义 Hessenberg 簇并提供了基本证明。作为副产品，我们提出了图的广义斯坦利-斯坦布里奇猜想。我们对广义 Hessenberg 簇的双有理几何的研究使我们能够通过显式的放大序列或射影束映射将它们修改为更简单的簇，例如射影空间或置换面簇。利用这一点，我们提供了两种算法来计算广义 Hessenberg 簇上同调的 - 表示。作为一个应用，我们计算一些 Hessenberg 簇的低度上同调的表示。