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

TL;DR

This paper provides a combinatorial interpretation of the fixed point sets of Hessenberg Schubert varieties within regular semisimple Hessenberg varieties, extending known results from classical Schubert varieties using Bruhat order and Weyl type subsets.

Contribution

It offers a new interpretation of fixed point sets of Hessenberg Schubert varieties in terms of Bruhat order and Weyl type subsets, building on recent combinatorial descriptions.

Findings

Characterization of fixed points using Bruhat order

Partition of the symmetric group via Weyl type subsets

Proof of a key lemma on Weyl type subsets

Abstract

It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $GL_n(\mathbb{C})/B$ 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.