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arXiv · 1808.01719

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

Erik Insko·Julianna Tymoczko·Alexander Woo
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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.

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Erik Insko, Julianna Tymoczko, Alexander Woo. 2018-08-06. A formula for the cohomology and $K$-class of a regular Hessenberg variety. https://arxiv.org/abs/1808.01719

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