Torus fixed point sets of Hessenberg Schubert varieties in regular semisimple Hessenberg varieties

It is well-known that the T -fixed points of a Schubert variety in the flag variety GLn(C)/B can be characterized purely combinatorially in terms of Bruhat order on the symmetric group Sn. In a recent preprint, Cho, Hong, and Lee give a combinatorial description of the T -fixed points of Hessenberg analogues of Schubert varieties (which we call Hessenberg Schubert varieties) in a regular semisimple Hessenberg variety. This note gives an interpretation of their result in terms of Bruhat order by making use of a partition of the symmetric group defined using so-called subsets of Weyl type. The Appendix, written by Michael Zeng, proves a lemma concerning subsets of Weyl type which is required in our arguments.

众所周知，旗簇$GL_n(\mathbb{C})/B$中舒伯特簇的$T$-不动点可以纯粹用对称群$S_n$上的布吕阿序进行组合刻画。在最近的一篇预印本中，曹（Cho）、洪（Hong）和李（Lee）给出了正则半单赫森伯格簇中舒伯特簇的赫森伯格类似物（我们称之为赫森伯格舒伯特簇）的$T$-不动点的组合描述。本文通过利用由所谓的外尔型子集定义的对称群的一个划分，用布吕阿序对他们的结果进行了解释。由迈克尔·曾（Michael Zeng）撰写的附录证明了我们的论证中所需的一个关于外尔型子集的引理。