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Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture

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Title
Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture
Author(s)
Kiem, Young-Hoon; Donggun Lee
Publication Date
2024-08
Journal
Journal of Combinatorial Theory - Series A, v.206
Publisher
Academic Press
Abstract
We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group Sn on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the Sn-representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties. © 2024 Elsevier Inc.
URI
https://pr.ibs.re.kr/handle/8788114/15083
DOI
10.1016/j.jcta.2024.105884
ISSN
0097-3165
Appears in Collections:
Center for Complex Geometry (복소기하학 연구단) > 1. Journal Papers (저널논문)
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