The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$

Abstract

We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),n\dots ,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group ${\mathfrak{S}}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the ${\mathfrak{S}}_n$-invariant subring of the cohomology ring.

24 pages, 5 figures

References in corpus (5)

• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies
• Hessenberg varieties and hyperplane arrangements
• The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A

Cited by in corpus (6)

• The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture
• The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
• Geometry of regular Hessenberg varieties
• A survey of recent developments on Hessenberg varieties
• Hessenberg varieties, Slodowy slices, and integrable systems
• GKM spaces, and the signed positivity of the nabla operator