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 g and a Hessenberg subspace H ⊆ g. This paper...
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| Veröffentlicht in: | Algebra and Discrete Mathematics |
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/188515 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | The containment poset of type A Hessenberg varieties / E. Drellich // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 195–210. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| 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 g and a Hessenberg subspace H ⊆ 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 Px 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 1 then the Hessenberg spaces containing the Borel subalgebra determine distinct Hessenberg varieties. Lastly we give a natural involution on Px that induces a homeomorphism of varieties and prove additional properties of Px when X is a regular nilpotent element.
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| ISSN: | 1726-3255 |