The reductive subgroups of \(F_4\). (English) Zbl 1295.20049
Mem. Am. Math. Soc. 1049, vi, 88 p. (2013).
Summary: 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, we consider the case that \(G=F_4(K)\).
We find all conjugacy classes of closed, connected, semisimple \(G\)-reducible subgroups \(X\) of \(G\). Thus we also find all non-\(G\)-completely reducible closed, connected, semisimple subgroups of \(G\). When \(X\) is closed, connected and simple of rank at least two, we find 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 \(G\).
Amongst the classification of subgroups of \(G=F_4(K)\) we find infinite varieties of subgroups \(X\) of \(G\) which are maximal amongst all reductive subgroups of \(G\) but not maximal subgroups of \(G\); thus they are not contained in any reductive maximal subgroup of \(G\). The connected, semisimple subgroups contained in no maximal reductive subgroup of \(G\) are of type \(A_1\) when \(p=3\) and of type \(A_1^2\) or \(A_1\) when \(p=2\). Some of those which occur when \(p=2\) act indecomposably on the 26-dimensional irreducible representation of \(G\).
We also use 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.
We find all conjugacy classes of closed, connected, semisimple \(G\)-reducible subgroups \(X\) of \(G\). Thus we also find all non-\(G\)-completely reducible closed, connected, semisimple subgroups of \(G\). When \(X\) is closed, connected and simple of rank at least two, we find 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 \(G\).
Amongst the classification of subgroups of \(G=F_4(K)\) we find infinite varieties of subgroups \(X\) of \(G\) which are maximal amongst all reductive subgroups of \(G\) but not maximal subgroups of \(G\); thus they are not contained in any reductive maximal subgroup of \(G\). The connected, semisimple subgroups contained in no maximal reductive subgroup of \(G\) are of type \(A_1\) when \(p=3\) and of type \(A_1^2\) or \(A_1\) when \(p=2\). Some of those which occur when \(p=2\) act indecomposably on the 26-dimensional irreducible representation of \(G\).
We also use 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.
Keywords:
exceptional groups; subgroup structure; prime characteristic; algebraic groups of exceptional type; maximal closed connected subgroups; embeddings; closed connected semisimple subgroups; reductive subgroups; parabolic subgroups; Levi subgroups; conjugacy classes of subgroups; completely reducible subgroups; irreducible subgroups; simple subgroupsReferences:
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