On the completely-reducible subgroups of 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.