The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture

We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the $S_n$-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian-Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and Brosnan-Chow, as well as results of the second author on the geometry and combinatorics of Hessenberg varieties. As part of our arguments, we obtain inductive formulas for the Poincare polynomials of regular abelian Hessenberg varieties.

受科斯坦特（Kostant）和彼得森（Peterson）所发展的李代数中阿贝尔理想理论的启发，我们定义了一类赫森伯格簇（Hessenberg varieties）的子类，称为阿贝尔赫森伯格簇。对于一个阿贝尔正则半单赫森伯格簇关于由泰莫茨科（Tymoczko）所定义的作用在其上同调上的$S_n$ -表示，我们给出了一个归纳公式。我们的结果意味着在阿贝尔情形下，斯坦利 - 斯坦布里奇（Stanley - Stembridge）猜想的一个分次版本成立，并且推广了沙雷希安 - 瓦克斯（Shareshian - Wachs）和特夫（Teff）所得到的结果。我们的证明使用了斯坦利、加沙罗夫（Gasharov）、沙雷希安 - 瓦克斯以及布罗斯南 - 周（Brosnan - Chow）之前的工作，以及第二作者关于赫森伯格簇的几何与组合方面的结果。作为我们论证的一部分，我们得到了正则阿贝尔赫森伯格簇的庞加莱多项式的归纳公式。