Complete reducibility and subgroups of exceptional algebraic groups

TL;DR

This survey explores Serre's concept of G-complete reducibility in connected reductive algebraic groups, focusing on its implications for understanding subgroup structures in exceptional simple groups, and provides updates and corrections to existing literature.

Contribution

It offers a comprehensive overview of G-complete reducibility and its application to subgroup classification in exceptional algebraic groups, including recent developments and corrections.

Findings

Clarifies the role of G-complete reducibility in subgroup structure analysis.

Summarizes known results on subgroups of exceptional groups.

Provides corrections to previous literature.

Abstract

This survey article has two components. The first part gives a gentle introduction to Serre's notion of $G$-complete reducibility, where $G$ is a connected reductive algebraic group defined over an algebraically closed field. The second part concerns consequences of this theory when $G$ is simple of exceptional type, specifically its role in elucidating the subgroup structure of $G$. The latter subject has a history going back about sixty years. We give an overview of what is known, up to the present day. We also take the opportunity to offer several corrections to the literature.