The containment poset of type \(A\) Hessenberg varieties
Flag varieties are well-known algebraic varieties with many important geometric, combinatorial, and representation theoretic properties. A Hessenberg variety is a subvariety of a flag variety identified by two parameters: an element \(X\) of the Lie algebra \(\mathfrak{g}\) and a Hessenberg subspace...
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| Дата: | 2020 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2020
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1216 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-12162020-07-08T07:13:20Z The containment poset of type \(A\) Hessenberg varieties Drellich, E. Hessenberg variety, root space, poset 14A25, 17B45, 05E99 Flag varieties are well-known algebraic varieties with many important geometric, combinatorial, and representation theoretic properties. A Hessenberg variety is a subvariety of a flag variety identified by two parameters: an element \(X\) of the Lie algebra \(\mathfrak{g}\) and a Hessenberg subspace \(H\subseteq \mathfrak{g}\). This paper considers when two Hessenberg spaces define the same Hessenberg variety when paired with \(X\). To answer this question we present the containment poset \(\mathcal{P}_X\) of type \(A\) Hessenberg varieties with a fixed first parameter \(X\) and give a simple and elegant proof that if \(X\) is not a multiple of the element \(\bf 1\) then the Hessenberg spaces containing the Borel subalgebra determine distinct Hessenberg varieties. Lastly we give a natural involution on \(\mathcal{P}_X\) that induces a homeomorphism of varieties and prove additional properties of \(\mathcal{P}_X\) when \(X\) is a regular nilpotent element. Lugansk National Taras Shevchenko University 2020-07-08 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1216 10.12958/adm1216 Algebra and Discrete Mathematics; Vol 29, No 2 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1216/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1216/425 Copyright (c) 2020 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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|
| datestamp_date |
2020-07-08T07:13:20Z |
| collection |
OJS |
| language |
English |
| topic |
Hessenberg variety root space poset 14A25 17B45 05E99 |
| spellingShingle |
Hessenberg variety root space poset 14A25 17B45 05E99 Drellich, E. The containment poset of type \(A\) Hessenberg varieties |
| topic_facet |
Hessenberg variety root space poset 14A25 17B45 05E99 |
| format |
Article |
| author |
Drellich, E. |
| author_facet |
Drellich, E. |
| author_sort |
Drellich, E. |
| title |
The containment poset of type \(A\) Hessenberg varieties |
| title_short |
The containment poset of type \(A\) Hessenberg varieties |
| title_full |
The containment poset of type \(A\) Hessenberg varieties |
| title_fullStr |
The containment poset of type \(A\) Hessenberg varieties |
| title_full_unstemmed |
The containment poset of type \(A\) Hessenberg varieties |
| title_sort |
containment poset of type \(a\) hessenberg varieties |
| description |
Flag varieties are well-known algebraic varieties with many important geometric, combinatorial, and representation theoretic properties. A Hessenberg variety is a subvariety of a flag variety identified by two parameters: an element \(X\) of the Lie algebra \(\mathfrak{g}\) and a Hessenberg subspace \(H\subseteq \mathfrak{g}\). This paper considers when two Hessenberg spaces define the same Hessenberg variety when paired with \(X\). To answer this question we present the containment poset \(\mathcal{P}_X\) of type \(A\) Hessenberg varieties with a fixed first parameter \(X\) and give a simple and elegant proof that if \(X\) is not a multiple of the element \(\bf 1\) then the Hessenberg spaces containing the Borel subalgebra determine distinct Hessenberg varieties. Lastly we give a natural involution on \(\mathcal{P}_X\) that induces a homeomorphism of varieties and prove additional properties of \(\mathcal{P}_X\) when \(X\) is a regular nilpotent element. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2020 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1216 |
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AT drelliche thecontainmentposetoftypeahessenbergvarieties AT drelliche containmentposetoftypeahessenbergvarieties |
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2025-07-17T10:33:21Z |
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2025-07-17T10:33:21Z |
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