Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties
Abstract
A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type in this paper. We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety.
Keywords
Cite
@article{arxiv.2307.13334,
title = {Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties},
author = {Soojin Cho and JiSun Huh and Seonjeong Park},
journal= {arXiv preprint arXiv:2307.13334},
year = {2026}
}
Comments
Final version. 37 pages, 8 figures