The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials

TL;DR

This paper explores the relationship between the cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials, providing explicit polynomial presentations and expressing these polynomials as sums of Schubert polynomials.

Contribution

It establishes that the polynomials defining the cohomology ring can be expressed as alternating sums of Schubert polynomials, linking geometric and combinatorial structures.

Findings

Polynomials $f_{i,j}$ are expressed as alternating sums of Schubert polynomials.

Provides an explicit presentation of the cohomology ring of Hessenberg varieties.

Connects the algebraic structure of Hessenberg varieties with Schubert calculus.

Abstract

In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.