Book contents
- Frontmatter
- Contents
- Preface
- List of talks
- List of participants
- Representations of groups on finite simplicial complexes
- Coxeter groups and matroids
- Finite groups and geometries: A view on the present state and on the future
- Groups acting simply transitively on the vertices of a building of type Ãn
- Finite simple subgroups of semisimple complex Lie groups – a survey
- Flag-transitive extensions of buildings of type G2 and C3*
- Disconnected linear groups and restrictions of representations
- Products of conjugacy classes in algebraic groups and generators of dense subgroups
- Monodromy groups of polynomials
- Subgroups of exceptional algebraic groups
- The geometry of traces in Ree octagons
- Small rank exceptional Hurwitz groups
- The direct sum problem for chamber systems
- Embeddings and hyperplanes of Lie incidence geometries
- Intermediate subgroups in Chevalley groups
- On a certain class of Frattini extensions of finite Chevalley groups
- Economical generating sets for finite simple groups
Finite simple subgroups of semisimple complex Lie groups – a survey
Published online by Cambridge University Press: 06 January 2010
- Frontmatter
- Contents
- Preface
- List of talks
- List of participants
- Representations of groups on finite simplicial complexes
- Coxeter groups and matroids
- Finite groups and geometries: A view on the present state and on the future
- Groups acting simply transitively on the vertices of a building of type Ãn
- Finite simple subgroups of semisimple complex Lie groups – a survey
- Flag-transitive extensions of buildings of type G2 and C3*
- Disconnected linear groups and restrictions of representations
- Products of conjugacy classes in algebraic groups and generators of dense subgroups
- Monodromy groups of polynomials
- Subgroups of exceptional algebraic groups
- The geometry of traces in Ree octagons
- Small rank exceptional Hurwitz groups
- The direct sum problem for chamber systems
- Embeddings and hyperplanes of Lie incidence geometries
- Intermediate subgroups in Chevalley groups
- On a certain class of Frattini extensions of finite Chevalley groups
- Economical generating sets for finite simple groups
Summary
Abstract
We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type.
Introduction
Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F) and positive integers n. The topic of this survey is motivated by the question as to what happens if GL(n,·) is replaced by another algebraic group G(·).
We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k. A further restriction of our discussion concerns the field k. It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G(ℂ). (See below for some exceptions in §3 and §5.)
For G(·) of classical type, the theory for representations L → G(ℂ) differs little from the usual one for GL(n, ℂ). Indeed, a representation L → GL(n, ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.
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- Information
- Groups of Lie Type and their Geometries , pp. 77 - 96Publisher: Cambridge University PressPrint publication year: 1995
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