## Finite Reflection GroupsChapter 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 |

129 | |

131 | |

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### Common terms and phrases

assume axes joining basis vectors called centered Chapter choose computations conclude convex Coxeter element Coxeter graph Coxeter group crystallographic condition cube define degree denote det(l dihedral group discussion dual basis eigenvalues entries Exercise f-base finite group finite reflection groups finite subgroup follows Functions fundamental reflections fundamental region geometrically graph G hence homogeneous hyperplane icosahedron induction inner product integer intersection invariant irreducible Coxeter groups isomorphic Lie algebras line segment linear algebra linear combination linear transformation marked graph matrix Molien series monomial negative nodes nonnegative Number Theory obtain orbits orthogonal transformations partial words path permutation group polynomial positive definite Proof Suppose Proposition 4.1.1 reflection groups region F regular polyhedron relation represented root system rotation group rotation subgroup subgraph subgroup of 0(V subset subspace symmetric symmetric group tetrahedron Theorem Topology unit vectors vector space write