Schubert-Galois群在\（\textit｛Gr｝\，（4,9）\）中的分类

Q3 Mathematics

Arnold Mathematical Journal Pub Date : 2023-01-17 DOI:10.1007/s40598-022-00221-2

Abraham Martín del Campo, Frank Sottile, Robert Lee Williams

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引用次数: 1

摘要

我们将9维空间中4平面的Grassmann中的Schubert问题按它们的Galois群进行分类。在这个格拉斯曼的31806个基本舒伯特问题中，只有149个问题的伽罗瓦群不包含交替群。我们确定了这149个群中的伽罗瓦群——每个群都是一个无印置换群。根据它们的几何形状，这149个分为两个家族。这项研究提出了一种可能的舒伯特问题的分类，其伽罗瓦群不是完全对称群，这是舒伯特微积分反伽罗瓦问题的第一步。

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Classification of Schubert Galois Groups in \(\textit{Gr}\,(4,9)\)

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.

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来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.