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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| Datum: | 2020 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2020
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1216 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | 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. |
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