A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

TL;DR

This paper constructs a filtration on the cohomology rings of regular nilpotent Hessenberg varieties, revealing their inductive structure and providing explicit bases, formulas, and relations, with implications for future polynomial definitions.

Contribution

It introduces a new filtration on these cohomology rings, linking them inductively to lower-dimensional cases and providing explicit bases and relations.

Findings

Established a filtration with graded pieces related to lower-dimensional Hessenberg varieties.

Derived an explicit monomial basis for the cohomology rings.

Provided an inductive formula for the Poincaré polynomial.

Abstract

Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincar\'e polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of "Hessenberg Schubert polynomials" in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.