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

TL;DR

This work generalizes Dale Peterson’s link between the Peterson variety and quantum cohomology to regular nilpotent Hessenberg varieties in type A. It introduces a quantized presentation of the quantum cohomology framework via a matrix $M_n$ and polynomials $^hE_i^{(n)}$, producing an isomorphism between the coordinate ring of the Hessenberg open intersection $ ext{Hess}(N,h)\,∩\,Ω_e^∘$ and a quotient of a polynomial ring by the ideal generated by $^hE_i^{(n)}$. The main theorem unifies the Peterson correspondence with regular nilpotent Hessenberg varieties and yields explicit descriptions of singular loci, including a cyclic quotient singularity in the $h_2$ case and Schubert-geometry descriptions for general $h_m$. The paper also develops the Hilbert-series, Jacobian, and regular-sequence arguments to certify the isomorphism and provides concrete singular-locus descriptions in terms of Schubert varieties. Overall, it offers a concrete, elementary pathway from quantum cohomology data to coordinate rings of Hessenberg intersections, enriching the geometric and combinatorial understanding of Hessenberg varieties and their 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 $Ω_e^\circ$ 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 This is an unpublished result, so papers of Kostant and Rietsch are referred for this result.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 $Ω_e^\circ$ 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 $Ω_e^\circ$ is the intersection of certain Schubert variety and $Ω_e^\circ$ where $h_m=(m,n,\ldots,n)$ for $1<m<n$. We also see that $\mbox{Hess}(N,h_2) \cap Ω_e^\circ$ is related with the cyclic quotient singularity.

Figures (4)

• Figure 1: The configuration corresponding to $h=(3,3,4,5,5)$.
• Figure 2: The decomposition of $h=(2,3,3,5,5)$ into $h_1=(2,3,3)$ and $h_2=(2,2)$.
• Figure 3: The polynomial $^h E_i^{(j)} \in \mathbb C[x_1,\ldots,x_5,q_{12},q_{23},q_{34},q_{35},q_{45}]$ for $h=(3,3,4,5,5)$.
• Figure 4: The values $2(i-j)$ and $2(i-j+1)$ for $1 \leq j < i \leq n$.

Theorems & Definitions (93)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Example 2.1
• Definition 2.2
• Lemma 2.3
• proof