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- W4320854080 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 This is an unpublished result, so papers of Kostant and Rietsch are referred for this result.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." @default.
- W4320854080 created "2023-02-16" @default.
- W4320854080 creator A5065559361 @default.
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- W4320854080 date "2023-02-12" @default.
- W4320854080 modified "2023-09-29" @default.
- W4320854080 title "Quantization of Peterson's statement in Lie type A" @default.
- W4320854080 doi "https://doi.org/10.48550/arxiv.2302.06041" @default.
- W4320854080 hasPublicationYear "2023" @default.
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