Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913696514572288 |
|---|---|
| author | Horiguchi, Tatsuya Shirato, Tomoaki |
| author_facet | Horiguchi, Tatsuya Shirato, Tomoaki |
| contents | 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 This is an unpublished result, so papers of Kostant and Rietsch are referred for this result.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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06041 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell Horiguchi, Tatsuya Shirato, Tomoaki Algebraic Geometry Combinatorics 14M15, 14N35, 14B05 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 This is an unpublished result, so papers of Kostant and Rietsch are referred for this result.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. |
| title | Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell |
| topic | Algebraic Geometry Combinatorics 14M15, 14N35, 14B05 |
| url | https://arxiv.org/abs/2302.06041 |