Algebraic groups and $G$-complete reducibility: a geometric approach

Benjamin Martin* (Corresponding Author)

*Corresponding author for this work

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The notion of a \emph{$G$-completely reducible} subgroup is important in the study of algebraic groups and their subgroup structure. It generalizes the usual idea of complete reducibility from representation theory: a subgroup $H$ of a general linear group $G= {\rm GL}_n(k)$ is $G$-completely reducible if and only if the inclusion map $i\colon H\rightarrow {\rm GL}_n(k)$ is a completely reducible representation of $H$. In these notes I give an introduction to the theory of complete reducibility and its applications, and explain an approach to the subject using geometric invariant theory.
Original languageEnglish
Number of pages28
Publication statusPublished - 25 Jul 2022

Bibliographical note

Notes based on lectures given at the International Workshop on "Algorithmic problems in group theory, and related areas", held at the Oasis Summer Camp near Novosibirsk from July 26 to August 4, 2016. 28 pages
Acknowledgements: I am grateful to the workshop organisers Evgeny Vdovin, Alexey Galt, Fedor Dudkin, Maria Zvezdina, Andrey Mamontov, and Alexey Staroletov for their hospitality, and for permission to make these notes publicly available. I also used these notes as a basis for some lectures at a Spring School on Complete Reducibility at Bochum University in April 2018, and I thank the organisers Gerhard Ro ̈hrle, Falk Bannuscher, Maike Gruchot and Alastair Litterick for their hospitality. I’m also grateful to the workshop and spring school participants for their comments and for pointing out various mistakes.


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