| Introduction | p. 1 |
| Overview of this book | p. 1 |
| Some detail concerning the content | p. 4 |
| Acknowledgements | p. 5 |
| Leitfaden | p. 5 |
| Preliminaries | p. 7 |
| Hermitian forms | p. 7 |
| Reflections | p. 9 |
| Groups | p. 12 |
| Modules and representations | p. 13 |
| Irreducible unitary reflection groups | p. 15 |
| Cartan matrices | p. 17 |
| The field of definition | p. 19 |
| Exercises | p. 21 |
| The groups G(m, p, n) | p. 23 |
| Primitivity and imprimitivity | p. 23 |
| Wreath products and monomial representations | p. 24 |
| Properties of the groups G(m, p, n) | p. 25 |
| The imprimitive unitary reflection groups | p. 27 |
| Imprimitive subgroups of primitive reflection groups | p. 32 |
| Root systems for G(m, p, n) | p. 34 |
| Generators for G(m, p, n) | p. 35 |
| Invariant polynomials for G(m, p, n) | p. 36 |
| Exercises | p. 37 |
| Polynomial invariants | p. 39 |
| Tensor and symmetric algebras | p. 39 |
| The algebra of invariants | p. 41 |
| Invariants of a finite group | p. 42 |
| The action of a reflection | p. 46 |
| The Shephard-Todd-Chevalley Theorem | p. 46 |
| The coinvariant algebra | p. 51 |
| Exercises | p. 53 |
| Poincaré series and characterisations of reflection groups | p. 54 |
| Poincaré series | p. 54 |
| Exterior and symmetric algebras and Molien's Theorem | p. 56 |
| A characterisation of finite reflection groups | p. 61 |
| Exponents | p. 63 |
| Exercises | p. 65 |
| Quaternions and the finite subgroups of SU2(C) | p. 66 |
| The quaternions | p. 67 |
| The group O3(R) and O4(R) | p. 69 |
| The groups SU2(C) and U2(C) | p. 71 |
| The finite subgroups of the quaternions | p. 72 |
| The finite subgroups of SO3(R) and SU2(C) | p. 77 |
| Quaternions, reflections and root systems | p. 79 |
| Exercises | p. 83 |
| Finite unitary reflection groups of rank two | p. 84 |
| The primitive reflection subgroups of U2(C) | p. 84 |
| The reflection groups of type T | p. 85 |
| The reflection groups of type O | p. 87 |
| The reflection groups of type I | p. 89 |
| Cartan matrices and the ring of definition | p. 90 |
| Invariants | p. 93 |
| Exercises | p. 98 |
| Line systems | p. 99 |
| Bounds on line systems | p. 99 |
| Star-closed Euclidean line systems | p. 100 |
| Reflections and star-closed line systems | p. 101 |
| Extensions of line systems | p. 103 |
| Line systems for imprimitive reflection groups | p. 104 |
| Line systems for primitive reflection groups | p. 105 |
| The Goethals-Seidel decomposition for 3-systems | p. 111 |
| Extensions of Dn(2)and Dn(3) | p. 115 |
| Further structure of line systems in Cn | p. 119 |
| Extensions of Euclidean line systems | p. 120 |
| Extensions of An, En and Kn in Cn | p. 125 |
| Extensions of 4-systems | p. 127 |
| Exercises | p. 133 |
| The Shephard and Todd classification | p. 137 |
| Outline of the classification | p. 137 |
| Blichfeldt's Theorem | p. 138 |
| Consequences of Blichfeldt's Theorem | p. 140 |
| Extensions of 5-systems | p. 142 |
| Line systems and reflections of order three | p. 146 |
| Extensions of ternary 6-systems | p. 149 |
| The classification | p. 151 |
| Root systems and the ring of definition | p. 153 |
| Reduction modulo p | p. 155 |
| Identification of the primitive reflection groups | p. 157 |
| Exercises | p. 168 |
| The orbit map, harmonic polynomials and semi-invariants | p. 171 |
| The orbit map | p. 171 |
| Skew invariants and the Jacobian | p. 172 |
| The rank of the Jacobian | p. 174 |
| Semi-invariants | p. 176 |
| Differential operators | p. 179 |
| The space of G-harmonic polynomials | p. 183 |
| Steinberg's fixed point theorem | p. 186 |
| Exercises | p. 189 |
| Covariants and related polynomial identities | p. 191 |
| The space of covariants | p. 191 |
| Gutkin's Theorem | p. 194 |
| Differential invariants | p. 198 |
| Some special cases of covariants | p. 199 |
| Two-variable Poincare series and specialisations | p. 201 |
| Exercises | p. 206 |
| Eigenspace theory and reflection subquotients | p. 208 |
| Basic affine algebraic geometry | p. 208 |
| Eigenspaces of elements of reflection groups | p. 212 |
| Reflection subquotients of unitary reflection groups | p. 213 |
| Regular elements | p. 215 |
| Properties of the reflection subquotients | p. 218 |
| Eigenvalues of pseudoregular elements | p. 222 |
| Reflection cosets and twisted invariant theory | p. 228 |
| Reflection cosets | p. 228 |
| Twisted invariant theory | p. 229 |
| Eigenspace theory for reflection cosets | p. 231 |
| Subquotients and centralisers | p. 237 |
| Parabolic subgroups and the coinvariant algebra | p. 239 |
| Duality groups | p. 242 |
| Exercises | p. 244 |
| Some background in commutative algebra | p. 246 |
| Forms over finite fields | p. 250 |
| Basic definitions | p. 250 |
| Witt's Theorem | p. 251 |
| The Wall form, the spinor norm and Dickson's invariant | p. 251 |
| Order formulae | p. 252 |
| Reflections in finite orthogonal groups | p. 253 |
| Applications and further reading | p. 255 |
| The space of regular elements | p. 255 |
| Fundamental groups, braid groups, presentations | p. 258 |
| Hecke algebras | p. 261 |
| Reductive groups over finite fields | p. 266 |
| Tables | p. 271 |
| The primitive unitary reflection groups | p. 272 |
| Degrees and codegrees | p. 274 |
| Cartan matrices | p. 276 |
| Maximal subsystems | p. 277 |
| Reflection cosets | p. 277 |
| Bibliography | p. 279 |
| Index of notation | p. 289 |
| Index | p. 291 |
| Table of Contents provided by Ingram. All Rights Reserved. |
