A formula for the cohomology and $K$-class of a regular Hessenberg variety

TL;DR

This paper provides a new explicit formula for the cohomology and K-theory classes of regular Hessenberg varieties, connecting them to Schubert and Grothendieck polynomials through specific substitutions.

Contribution

It introduces a novel formula for the classes of regular Hessenberg varieties in cohomology and K-theory, expanding previous results with fewer restrictions on the defining function.

Findings

The classes are given by substitutions in Schubert and Grothendieck polynomials.

The formula applies to regular Hessenberg varieties with minimal restrictions.

Methods differ from recent related work by Abe, Fujita, and Zeng.

Abstract

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on $h$. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on $h$ than here.