Minimal semisimple Hessenberg schemes

TL;DR

This paper investigates minimal semisimple Hessenberg varieties in type A, proving they are unions of Richardson varieties, explicitly describing these unions, and establishing their reduced scheme structure using algebraic and combinatorial methods.

Contribution

It characterizes minimal semisimple Hessenberg varieties as unions of Richardson varieties and proves their schemes are reduced, providing explicit descriptions and algebraic insights.

Findings

Minimal semisimple Hessenberg varieties are unions of Richardson varieties

All minimal semisimple matrix Hessenberg schemes are reduced

Explicit descriptions of the Richardson variety unions

Abstract

We study a collection of Hessenberg varieties in the type A flag variety associated to a nonzero semisimple matrix whose conjugacy class has minimal dimension. We prove each such minimal semisimple Hessenberg variety is a union Richardson varieties and compute this set of Richardson varieties explicitly. Our methods leverage the notion of matrix Hessenberg schemes to answer questions about the geometry of minimal semisimple Hessenberg varieties using commutative algebra and known results on Schubert determinantal ideals. In particular, we show that all type A minimal semisimple matrix Hessenberg schemes are reduced.