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

Erik Insko,Julianna Tymoczko,Alexander Woo

doi:10.1016/j.jpaa.2019.106230

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.

Highlights

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h
Regular semisimple Hessenberg varieties govern an important geometric representation connected to the Stanley-Stembridge conjecture through a conjecture of Shareshian and Wachs [SW16] that was recently proven by Brosnan and Chow [BC18] and almost simultaneously by Guay-Paquet [GP15]
Anderson and the second author gave a different formula for the cohomology class using degeneracy loci and degeneration arguments [AT10]

Summary

Comparing the group to the flag variety

Knutson and Miller proved that the multidegree (respectively K-polynomial) of the coordinate ring of Y represents the class of Y in the cohomology (respectively K-theory) of the flag variety of Y. Let K(Y ) be the K-polynomial and C(Y ) the multidegree of Y both with the grading coming from setting deg(zij) = xj. The graded module zij corresponds to the sheaf of T -equivariant sections of the T -equivariant line bundle Mn × K on which T acts on the right by (z, y) · t = (z · t, ytj), where tj is the j-th diagonal entry in T. On the other hand Lj is defined as the quotient of G × K by the equivalence relation (g, y) ∼ (gb, ej(b)y) where ej : B → K∗ picks out the j-th diagonal entry This description coincides with the definition of xj and proves the claim

Matrix Schubert varieties and Grothendieck and Schubert polynomials

Regular intersections with Cohen-Macaulay schemes

Hessenberg varieties and their equations

Matrix Hessenberg varieties and the main theorem