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It is well-known that the $T$-fixed points of a Schubert variety in the flagvariety $GL_n(\mathbb{C})/B$ can be characterized purely combinatorially interms of Bruhat order on the symmetric group $\mathfrak{S}_n$. In a recentpreprint, Cho, Hong, and Lee give a combinatorial description of the $T$-fixedpoints of Hessenberg analogues of Schubert varieties (which we call HessenbergSchubert varieties) in a regular semisimple Hessenberg variety. This note givesan interpretation of their result in terms of Bruhat order by making use of apartition of the symmetric group defined using so-called subsets of Weyl type.The Appendix, written by Michael Zeng, proves a lemma concerning subsets ofWeyl type which is required in our arguments.

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