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Finite simple subgroups of semisimple complex Lie groups – a survey

Published online by Cambridge University Press:  06 January 2010

Arjeh M. Cohen
Affiliation:
Fac. Wisk. en Inf. TUE P.O. Box 513 5600 MB Eindhoven The Netherlands
David B. Wales
Affiliation:
Sloan Lab Caltech Pasadena CA 91125 USA
William M. Kantor
Affiliation:
University of Oregon
Lino Di Martino
Affiliation:
Università degli Studi di Milano
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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 LGL(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 LG(ℂ) differs little from the usual one for GL(n, ℂ). Indeed, a representation LGL(n, ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

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Publisher: Cambridge University Press
Print publication year: 1995

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