A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
复制标题
正则幂零Hessenberg簇上同调环的过滤
DOI：
10.1007/s00209-020-02646-x
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发表时间：
2021
影响因子：
0.8
通讯作者：
Tymoczko Julianna
中科院分区：
文献类型：
--
作者：
Harada Megumi;Horiguchi Tatsuya;Murai Satoshi;Precup Martha;Tymoczko Julianna
取一个正整数。本文的主要结果是在正则幂零Hessenberg变异体的上同环上构造了一个滤除，使得其相关的分级环上有与Hessenberg变异体的上同环同构的分级块（即齐次分量），显示了这些环的归纳性。在之前的工作中，前两位作者与Abe和Masuda一起，从生成和关系的角度给出了这些上同环的明确表示。我们引入了与上述关系密切相关的一组新的多项式，得到了它们所满足的一系列等价关系；这允许我们推导过滤。此外，我们得到了以下三个推论。首先，我们给出了这些变量的庞卡罗多项式的一个归纳公式。其次，我们给出了正则幂零Hessenberg变的上同环的显式单基。第三，我们导出了正则幂零Hessenberg变的上同环上的Schubert类象所满足的线性关系集的一个基。最后，我们的方法和结果为未来的工作提出了许多方向；特别地，我们在正则幂零Hessenberg变分的背景下提出了“Hessenberg - Schubert多项式”的定义，它推广了经典的Schubert多项式。我们还概述了与它们有关的几个悬而未决的问题。
Letnbe a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety insuch that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.