On the completely-reducible subgroups of type F4

代数群理论中一个长期存在的目标是代数闭域上例外型G2，F4，E6，E7和E8的单代数群G的连通约化子群的分类。塞尔的一个重要概念，称为G-完全约化（G-cr），将G的表示的完全约化的定义转化为根据G的子群结构的群论定义。这一概念在处理分类问题方面具有影响力。对例外型单代数群的G-cr子群进行了分类。此外，Liebeck-Seitz证明了对于大特征，每个连通约化子群都是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是G2型时，所有的可约非G-cr子群都是很好理解的。下一个自然目标是F4型G，其中还原的非G-cr子群被广泛研究，尽管分类是不完整的。我的项目的目的是完成分类并进一步研究属性，以增强我们对每个非G-cr亚组的理解。特别地，我将计算它们在G中的连通中心化子以及极小G-模和伴随G-模对非G-cr子群的限制。分类问题的关键在于将一个困难的群论问题转化为一个组合问题，这是通过使用深上同调结果来完成的。

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.