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Finite Reflection Groups By L.C. Grove

Finite Reflection Groups by L.C. Grove

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Summary

Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Throughout most of Chapter 2 we do not distinguish between groups that are geo metrically indistinguishable, that is, conjugate in the orthogonal group.

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Finite Reflection Groups Summary

Finite Reflection Groups by L.C. Grove

Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. Throughout most of Chapter 2 we do not distinguish between groups that are geo metrically indistinguishable, that is, conjugate in the orthogonal group. Very little of the material in Chapter 2 is actually required for the sub sequent chapters, but it serves two important purposes: It aids in the development of geometrical insight, and it serves as a source of illustrative examples. There is a discussion offundamental regions in Chapter 3. Chapter 4 provides a correspondence between fundamental reflections and funda mental regions via a discussion of root systems. The actual classification and construction of finite reflection groups takes place in Chapter 5. where we have in part followed the methods of E. Witt and B. L. van der Waerden. Generators and relations for finite reflection groups are discussed in Chapter 6. There are historical remarks and suggestions for further reading in a Post lude.

Table of Contents

1 Preliminaries.- 2 Finite Groups in Two and Three Dimensions.- 3 Fundamental Regions.- 4 Coxeter Groups.- 5 Classification of Coxeter Groups.- 6 Generators and Relations for Coxeter Groups.- 7 Invariants.- Postlude.- Crystallographic Point Groups.- References.

Additional information

NLS9781441930729
9781441930729
1441930728
Finite Reflection Groups by L.C. Grove
New
Paperback
Springer-Verlag New York Inc.
20101203
136
N/A
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