On the completely-reducible subgroups of type F4

关于 F4 型的完全可约子群

• 批准号: 2443755
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2443755
• 关键词: completely; reducible; subgroups; type; F4

On the completely-reducible subgroups of type F4 Summary: A long-standing objective in the theory of algebraic groups is the classification of connected reductive subgroups of simple algebraic groups G of exceptional type G2, F4, E6, E7 and E8 over algebraically closed fields. An important concept by Serre, called G-complete reducibility (G-cr) translates the definition of complete reducibility of a representation of G to a group theoretic definition in terms of the subgroup structure of G. This concept has been influential in tackling the classification problem. The G-cr subgroups have been classified for all simple algebraic groups of exceptional type. Furthermore, Liebeck-Seitz proved that for large characteristics, every connected reductive subgroup is G-cr. Therefore, what remains as the only obstacle to completing the classification in low characteristics are the elusive non-G-cr subgroups. When G is of type G2, all reductive non-G-cr subgroups are well-understood. The next natural target is G of type F4, where the reductive non-G-cr subgroups were extensively studied, though the classification was incomplete. The aim of my project is to complete the classification and additionally study properties which enhance our understanding of each of the non-G-cr subgroups. In particular, I will compute their connected centralisers in G and the restrictions of the minimal and adjoint G-modules to the non-G-cr subgroups. The crux of the classification problem lies in translating a difficult group theoretic question into a combinatorial one, which is done by using deep cohomological results.

关于 F4 型的完全可约子群 摘要：代数群理论中的一个长期目标是对代数闭域上的特殊类型 G2、F4、E6、E7 和 E8 的简单代数群 G 的连通还原子群进行分类。 Serre 的一个重要概念，称为 G 完全可归约性 (G-cr)，将 G 表示的完全可归约性定义转化为根据 G 子群结构的群论定义。这个概念在解决分类问题方面具有影响力。 G-cr 子群已被分类为所有特殊类型的简单代数群。此外，Liebeck-Seitz 证明，对于大特征，每个连通的还原子群都是 G-cr。因此，完成低特征分类的唯一障碍是难以捉摸的非 G-cr 亚组。当 G 为 G2 型时，所有还原性非 G-cr 子群都可以很好地理解。下一个自然目标是 F4 型 G，其中还原性非 G-cr 亚组已被广泛研究，但分类并不完整。我的项目的目的是完成分类并另外研究属性，以增强我们对每个非 G-cr 亚组的理解。特别是，我将计算 G 中它们的连接集中器以及最小和伴随 G 模对非 G-cr 子群的限制。分类问题的关键在于将困难的群论问题转化为组合问题，这是通过使用深度上同调结果来完成的。