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

Abstract

It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $G{L}_n(\mathbb{C})/B$ can be characterized purely combinatorially in terms of Bruhat order on the symmetric group ${\mathfrak{S}}_n$. 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.

13 pages. Major revision, due to an error discovered in the proof of the main theorem of the previous version. The current version states and proves a revised result, which concerns the T-fixed points of the Hessenberg Schubert variety instead of the variety itself