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The Reductive Subgroups of $$F_4$$
David I. Stewart, New College, Oxford, United Kingdom
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Memoirs of the American Mathematical Society
2013; 88 pp; softcover
Volume: 223
ISBN-10: 0-8218-8332-1
ISBN-13: 978-0-8218-8332-7
List Price: US$69 Individual Members: US$41.40
Institutional Members: US\$55.20
Order Code: MEMO/223/1049

Let $$G=G(K)$$ be a simple algebraic group defined over an algebraically closed field $$K$$ of characteristic $$p\geq 0$$. A subgroup $$X$$ of $$G$$ is said to be $$G$$-completely reducible if, whenever it is contained in a parabolic subgroup of $$G$$, it is contained in a Levi subgroup of that parabolic. A subgroup $$X$$ of $$G$$ is said to be $$G$$-irreducible if $$X$$ is in no proper parabolic subgroup of $$G$$; and $$G$$-reducible if it is in some proper parabolic of $$G$$. In this paper, the author considers the case that $$G=F_4(K)$$.

The author finds all conjugacy classes of closed, connected, semisimple $$G$$-reducible subgroups $$X$$ of $$G$$. Thus he also finds all non-$$G$$-completely reducible closed, connected, semisimple subgroups of $$G$$. When $$X$$ is closed, connected and simple of rank at least two, he finds all conjugacy classes of $$G$$-irreducible subgroups $$X$$ of $$G$$. Together with the work of Amende classifying irreducible subgroups of type $$A_1$$ this gives a complete classification of the simple subgroups of $$G$$.

The author also uses this classification to find all subgroups of $$G=F_4$$ which are generated by short root elements of $$G$$, by utilising and extending the results of Liebeck and Seitz.

• Reductive subgroups of $$F_4$$