Irreducible subgroups of exceptional algebraic groups

• Main Author: Thomas, Adam Robert
• Other Authors: Liebeck, Martin
• Published: Imperial College London 2014
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.676768

Let $G$ be a semisimple algebraic group over an algebraically closed field $K$, of characteristic $p \geq 0$. A closed subgroup of $G$ is said to be irreducible if it does not lie in any proper parabolic subgroup of $G$. In this thesis we address the following problem: classify the connected irreducible subgroups of $G$, up to conjugacy, where $G$ is of exceptional type. Work of Liebeck and Seitz classifies the conjugacy classes of simple, connected irreducible subgroups of rank at least 2, with a restriction on the characteristic of the underlying field ($p > 7$ is sufficient). When $G$ is of type $F_4$, Stewart has classified the conjugacy classes of simple, connected irreducible subgroups of rank at least 2 in all characteristics. We classify the conjugacy classes of simple, connected irreducible subgroups, of rank at least 2 for $E_6$, $E_7$ and $E_8$. Our approach works in all characteristics, rather than starting from the characteristics excluded in the result of Liebeck and Seitz. We use these classifications to prove corollaries concerning the representation theory of such irreducible subgroups. For example, with one exception, two simple irreducible connected subgroups of rank at least 2 are $G$-conjugate if and only if they have the same composition factors on the adjoint module of $G$. We also consider connected subgroups of rank 1. Work of Lawther and Testerman classifies conjugacy classes of rank 1 connected irreducible subgroups, with a restriction on $p$ ($p>7$ is sufficient). The connected irreducible subgroups of rank 1 were found, in arbitrary characteristic, by Amende for all but $E_8$. We give a new proof of this, finding a set of conjugacy class representatives without repetition. We prove corollaries on the overgroups of irreducible $A_1$ subgroups. For example, we prove that if $p=2$ or $3$ then any irreducible $A_1$ subgroup of $E_7$ is contained in $A_1 D_6$. Finally, consider the semisimple, non-simple connected irreducible subgroups. We classify these, up to conjugacy, for $G_2$, $F_4$ and $E_6$. So in conclusion, we classify conjugacy classes of connected irreducible subgroups of $G_2$, $F_4$ and $E_6$, all simple, connected irreducible subgroups of $E_7$ and all simple, connected irreducible subgroups of $E_8$ of rank at least 2.