On the completely-reducible subgroups of type F4

关于F4型完全可约子群综述：代数群论的一个长期目标是在代数闭域上对例外类型为G2、F4、E6、E7和E8型的单代数群G的连通可约子群进行分类。Serre提出的一个重要概念--G-完全可约性(G-cr)将G的表示的完全可约性的定义转化为关于G的子群结构的群论定义。这一概念在解决分类问题中发挥了重要作用。对所有例外类型的单代数群，对G-cr子群进行了分类。此外，Liebeck-Seitz证明了对于较大的特征，每个连通约化子群都是G-cr。因此，在低特征下完成分类的唯一障碍是难以捉摸的非G-cr子群。当G是G2型时，所有可约的非G-cr子群都是很好理解的。下一个自然目标是F4型G，在那里还原的非G-cr子群得到了广泛的研究，尽管分类是不完整的。我的项目的目的是完成分类，并额外研究性质，以加强我们对每个非G-cr子群的理解。特别地，我将计算它们在G中的连通中心子，以及极小和伴随G-模对非G-cr子群的限制。分类问题的关键在于将一个困难的群论问题转化为一个组合问题，这是利用深度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.