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 $H\subseteq \mathfrak{g}$. 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.