Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell

Abstract

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\mbox{GL}_n(\mathbb{C})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\mbox{Pet}_n$ and the opposite Schubert cell associated with the identity element ${\Omega }_e^{\circ }$ in $\mbox{GL}_n(\mathbb{C})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of $\mbox{GL}_n(\mathbb{C})/B$ is given by Ciocan-Fontanine and Givental-Kim. In this paper we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties $\mbox{Hess}(N,h)$ and ${\Omega }_e^{\circ }$ in $\mbox{GL}_n(\mathbb{C})/B$. In other words, we generalize the Peterson's statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan-Fontanine and Givental-Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety $\mbox{Hess}(N,h_m)$ and ${\Omega }_e^{\circ }$ is the intersection of certain Schubert variety and ${\Omega }_e^{\circ }$ where $h_m=(m,n,\dots ,n)$ for $1<m<n$. We also see that $\mbox{Hess}(N,h_2) \cap Ω_e^\circ$ is related with the cyclic quotient singularity.

arXiv comment43 pages, 4 figures

References in corpus (7)

• Quantum cohomology of flag manifolds and Toda lattices
• Quantum cohomology of partial flag manifolds
• 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
• The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
• The equivariant cohomology rings of Peterson varieties
• Hessenberg varieties of codimension one in the flag variety