发表机构
National Institute of Technology, Akashi College（明石国立高等专门学校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究下幺幂群坐标环与带量子参数多项式环的联系，构造对称群作用并定义差分算子，进而研究 Schubert 多项式的量子化。
AI 中文摘要
Givental-Kim 和 Ciocan-Fontanine 给出了旗簇量子上同调环的一个显式表示。作者与 Shirato 在正则幂零 Hessenberg 簇的坐标环背景下引入了该表示的代数推广。特别地，他们建立了一般线性群中下幺幂群的坐标环与带量子参数的多项式环之间的联系。本文考虑该联系的对偶，并看到 Plücker 坐标对应于 Schur 多项式的量子化。作为该联系的一个应用，我们构造了对称群在带量子参数的多项式环上的作用。利用对称群作用，可以在带量子参数的多项式环上定义差分算子。我们结合差分算子研究 Schubert 多项式的量子化。
英文摘要
Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of their presentation in the context of the coordinate rings of regular nilpotent Hessenberg varieties. In particular, they connect the coordinate ring of the lower unipotent group in a general linear group and the polynomial ring with quantum parameters. In this paper we consider the dual of the connection and we see that the Plücker coordinates correspond to the quantizations of Schur polynomials. As an application of the connection, we construct an action of the symmetric group on the polynomial ring with quantum parameters. Using the symmetric group action, one can define the divided difference operators on the polynomial ring with quantum parameters. We study the quantizations of Schubert polynomials in relation to the divided difference operators.
Comments40 pages, 4 figures
