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 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.