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.
Acknowledgments
This work was supported in part by the National Security Agency under Grant No.H98230-19-1-0119, The Lyda Hill Foundation, The McGovern Foundation, and Microsoft Research, as part of the Mathematical Sciences Research Institute Summer Research for Women program. The first author is supported by a Natural Science and Engineering Research Council Discovery Grant and a Canada Research Chair (Tier 2) from the Government of Canada. The second author is supported in part by NSF DMS-1954001. The Appendix is by Michael Zeng, and is part of an undergraduate research project conducted under the second author's supervision in 2020.
Citation
Megumi Harada. Martha Precup. "Torus fixed point sets of Hessenberg Schubert varieties in regular semisimple Hessenberg varieties." Osaka J. Math. 60 (3) 637 - 652, July 2023.
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