Finite Reflection Groups

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Springer Science & Business Media, Jun 20, 1996 - Mathematics - 136 pages
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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.
 

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Contents

PRELIMINARIES
1
12 GROUP THEORY
3
FINITE GROUPS IN TWO AND THREE DIMENSIONS
5
22 FINITE GROUPS IN TWO DIMENSIONS
7
23 ORTHOGONAL TRANSFORMATIONS IN THREE DIMENSIONS
9
24 FINITE ROTATION GROUPS IN THREE DIMENSIONS
11
25 FINITE GROUPS IN THREE DIMENSIONS
18
26 CRYSTALLOGRAPHIC GROUPS
21
53 CONSTRUCTION OF IRREDUCIBLE COXETER GROUPS
65
54 ORDERS OF IRREDUCIBLE COXETER GROUPS
77
Exercises
80
GENERATORS AND RELATIONS FOR COXETER GROUPS
83
Exercises
101
INVARIANTS
104
72 POLYNOMIAL FUNCTIONS
105
73 INVARIANTS
107

Exercises
22
FUNDAMENTAL REGIONS
27
Exercises
32
COXETER GROUPS
34
42 FUNDAMENTAL REGIONS FOR COXETER GROUPS
43
Exercises
50
CLASSIFICATION OF COXETER GROUPS
53
52 THE CRYSTALLOGRAPHIC CONDITION
63
74 THE MOLIEN SERIES
112
Exercises
120
POSTLUDE
124
CRYSTALLOGRAPHIC POINT GROUPS
127
REFERENCES
129
INDEX
131
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