Complete reducibility in bad characteristic

Let \(G\) be a simple algebraic group of exceptional type over an algebraically closed field of characteristic \(p > 0\). This paper continues a long-standing effort to classify the connected reductive subgroups of \(G\). Having previously completed the classification when \(p\) is sufficiently l...

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Bibliographic Details
Published in:arXiv.org 2023-04
Main Authors: Litterick, Alastair J, Thomas, Adam R
Format: Article
Language:English
Subjects:
Classification
Modules
Subgroups
Online Access:Get full text
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Summary:Let \(G\) be a simple algebraic group of exceptional type over an algebraically closed field of characteristic \(p > 0\). This paper continues a long-standing effort to classify the connected reductive subgroups of \(G\). Having previously completed the classification when \(p\) is sufficiently large, we focus here on the case that \(p\) is bad for \(G\). We classify the connected reductive subgroups of \(G\) which are not \(G\)-completely reducible, whose simple components have rank at least \(3\). For each such subgroup \(X\), we determine the action of \(X\) on the adjoint module \(L(G)\) and on a minimal non-trivial \(G\)-module, and the connected centraliser of \(X\) in \(G\). As corollaries we obtain information on: subgroups which are maximal among connected reductive subgroups; products of commuting \(G\)-completely reducible subgroups; subgroups with trivial connected centraliser; and subgroups which act indecomposably on an adjoint or minimal module for \(G\).
ISSN:2331-8422