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

TL;DR

This paper generalizes Peterson's isomorphism between quantum cohomology rings and coordinate rings to regular nilpotent Hessenberg varieties, providing explicit presentations and analyzing their singular loci.

Contribution

It introduces quantizations of known presentations to connect Hessenberg varieties with coordinate rings, extending Peterson's result to a broader class of varieties.

Findings

Explicit presentation of quantized rings for Hessenberg varieties

Identification of singular loci with Schubert varieties

Relation of certain Hessenberg intersections to cyclic quotient singularities

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 This is an unpublished result, so papers of Kostant and Rietsch are referred for this result.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 $h_m=(m,n,\ldots,n)$ for $1<m<n$. We also see that $\mbox{Hess}(N,h_2) \cap \Omega_e^\circ$ is related with the cyclic quotient singularity.