Regular semisimple Hessenberg varieties with cohomology rings generated in degree two

TL;DR

This paper characterizes when the cohomology rings of regular semisimple Hessenberg varieties are generated in degree two, identifying a specific class of Hessenberg functions called double lollipops.

Contribution

It provides a complete characterization of Hessenberg functions for which the cohomology ring is generated in degree two, focusing on the class called double lollipops.

Findings

Cohomology rings are generated in degree two for double lollipop Hessenberg functions.

The paper identifies the class of double lollipop functions as precisely those with this property.

Provides a new understanding of the algebraic structure of Hessenberg varieties.

Abstract

A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the flag variety determined by a square matrix $S$ with distinct eigenvalues and a Hessenberg function $h$. The cohomology ring $H^*(\mathrm{Hess}(S,h))$ is independent of the choice of $S$ and is not explicitly described except for a few cases. In this paper, we characterize the Hessenberg function $h$ such that $H^*(\mathrm{Hess}(S,h))$ is generated in degree two as a ring. It turns out that such $h$ is what is called a (double) lollipop.