Geodesic
The containment poset of type $A$ Hessenberg varieties
Algebra and discrete mathematics, Tome 29 (2020) no. 2, pp. 195-210

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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 $\mathbf 1$ 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.
Keywords: Hessenberg variety, root space, poset. 
@article{ADM_2020_29_2_a5,
     author = {E. Drellich},
     title = {The containment poset of type $A$ {Hessenberg} varieties},
     journal = {Algebra and discrete mathematics},
     pages = {195--210},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_29_2_a5/}
}
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E. Drellich. The containment poset of type $A$ Hessenberg varieties. Algebra and discrete mathematics, Tome 29 (2020) no. 2, pp. 195-210. http://geodesic.mathdoc.fr/item/ADM_2020_29_2_a5/