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

Horiguchi, Tatsuya ; Shirato, Tomoaki

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e44

Résumé

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\operatorname {\mathrm {Pet}}_n$ and the opposite Schubert cell associated with the identity element $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result.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 $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ 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 $\operatorname {\mathrm {Hess}}(N,h)$ and $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. 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 $\operatorname {\mathrm {Hess}}(N,h_m)$ 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 $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ is related with the cyclic quotient singularity.

Détail

DOI : 10.1017/fms.2024.142

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