A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties

TL;DR

This paper extends the Garsia-Procesi basis from Springer varieties to Hessenberg varieties, proposing a conjectural presentation of their cohomology rings and providing explicit bases for certain cases.

Contribution

It generalizes the Garsia-Procesi construction to Hessenberg varieties and offers conjectures and explicit bases for their cohomology rings.

Findings

Conjectured a quotient presentation for Hessenberg varieties' cohomology rings.

Provided explicit bases for regular nilpotent Hessenberg varieties.

Presented new evidence supporting the conjecture for Peterson varieties.

Abstract

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.