# Irreducible Almost Simple Subgroups of Classical Algebraic Groups

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*Timothy C. Burness; Soumaïa Ghandour; Donna M. Testerman; Claude Marion*

Let \(G\) be a simple classical algebraic group over
an algebraically closed field \(K\) of characteristic
\(p\geq 0\) with natural module \(W\). Let \(H\)
be a closed subgroup of \(G\) and let \(V\) be a
nontrivial \(p\)-restricted irreducible tensor indecomposable
rational \(KG\)-module such that the restriction of
\(V\) to \(H\) is irreducible.

In this paper the authors classify the triples \((G,H,V)\)
of this form, where \(V \neq W,W^{*}\) and \(H\) is a
disconnected almost simple positive-dimensional closed subgroup of
\(G\) acting irreducibly on \(W\). Moreover, by
combining this result with earlier work, they complete the
classification of the irreducible triples \((G,H,V)\) where
\(G\) is a simple algebraic group over \(K\), and
\(H\) is a maximal closed subgroup of positive
dimension.

#### Table of Contents

# Table of Contents

## Irreducible Almost Simple Subgroups of Classical Algebraic Groups

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Acknowledgments 815 free

- Chapter 2. Preliminaries 916
- 2.1. Notation and terminology 916
- 2.2. Weights and multiplicities 1017
- 2.3. Tensor products and reducibility 1118
- 2.4. Invariant forms 1118
- 2.5. Connected subgroups 1219
- 2.6. Clifford theory 1320
- 2.7. Parabolic embeddings 1421
- 2.8. Some remarks on the case 𝐻=𝐻⁰.2 1724
- 2.9. Reducible subgroups 1825
- 2.10. Some 𝐴₁-restrictions 1926

- Chapter 3. The case 𝐻⁰=𝐴_{𝑚} 2330
- Chapter 4. The case 𝐻⁰=𝐷_{𝑚}, 𝑚≥5 6370
- Chapter 5. The case 𝐻⁰=𝐸₆ 7784
- Chapter 6. The case 𝐻⁰=𝐷₄ 8794
- Chapter 7. Proof of Theorem 5 105112
- Notation 107114
- Bibliography 109116
- Back Cover Back Cover1122