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

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.

A survey of recent developments on Hessenberg varieties

“…In [5], the Poincaré dual of a regular Hessenberg variety Hess(R, h) in H * (F l(C n )) was computed in terms of positive roots associated the Hessenberg function h, and E. Insko, J. Tymoczko, and A. Woo gave a combinatorial formula for this class using Schubert polynomials ( [46]). Also, the cohomology class [Hess(R, h)] ∈ H * (F l(C n )) does not depend on a choice of a regular matrix R if h(i) ≥ i + 1 for all 1 ≤ i ≤ n (See for details [5,46]).…”

“…Woo gave a combinatorial formula for this class using Schubert polynomials ( [46]). Also, the cohomology class [Hess(R, h)] ∈ H * (F l(C n )) does not depend on a choice of a regular matrix R if h(i) ≥ i + 1 for all 1 ≤ i ≤ n (See for details [5,46]). 5.5.…”

The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

“…Let us also mention the work [34] which gives another polynomial representative for Σ h : consider the permutation w h ∈ S 2n given by w h (i + h(i)) = n + i for i ∈ [n] and put the values 1, . .…”

On the T-equivariant cohomology of Hessenberg varieties

“…The paper is organized as follows: Section 2 recalls two basic definitions of the Hessenberg variety and contains a description of the defining ideal of Hess(X, H) due to Insko [3,Theorem 10]. We give a simpler proof of this theorem using determinantal conditions.…”