$K$-Orbit closures and Hessenberg varieties

Author:

Mahir Bilen Can, Martha Precup, John Shareshian, Özlem Uğurlu

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)

journal:

date:

2023-09-10 16:00:00

Abstract

This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under which these semisimple Hessenberg varieties are irreducible. We determine the dimension of each irreducible Hessenberg variety under consideration and show that the number of such varieties is a Catalan number. We then apply a theorem of Brion to compute a polynomial representative for the cohomology class of each such variety. Additionally, we calculate the intersections of a standard (Schubert) hyperplane section of the flag variety with each of our Hessenberg varieties and prove this intersection possess a cohomological multiplicity-free property.

PDF: $K$-Orbit closures and Hessenberg varieties.pdf