The containment poset of type \(A\) Hessenberg varieties

Abstract

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.