Schubert polynomials and classes of Hessenberg varieties

TL;DR

This paper establishes a formula connecting Hessenberg variety classes to Schubert polynomials, providing geometric interpretations, decompositions, and combinatorial formulas, with implications for reduced schemes.

Contribution

It introduces a Giambelli formula for Hessenberg varieties, linking their classes to Schubert polynomials and providing new combinatorial and geometric insights.

Findings

Classes are specializations of double Schubert polynomials

Coefficients in decompositions are nonnegative

Closed formulas for coefficients in many cases

Abstract

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced.