Unitary reflection groups / Gustav I. Lehrer, Donald E. Taylor.

Format:

Book

Author/Creator:

Lehrer, Gus, 1947-

Contributor:

Taylor, Donald E. (Donald Ewen), 1945-

Series:

Australian Mathematical Society lecture series ; 20.

Australian Mathematical Society lecture series ; 20

Language:

English

Subjects (All):

Group theory--Reflections.

Group theory.

Mathematics.

Physical Description:

1 online resource (304 p.)

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2009.

Language Note:

English

Summary:

A complete and clear account of the classification of unitary reflection groups, which arise naturally in many areas of mathematics.

Contents:

Cover; Unitary Reflection Groups; Series Page; Title; Copyright; Contents; Introduction; 1. Overview of this book; 2. Some detail concerning the content; 3. Acknowledgements; 4. Leitfaden; CHAPTER 1 Preliminaries; 1. Hermitian forms; 2. Reflections; 3. Groups; 4. Modules and representations; 5. Irreducible unitary reflection groups; 6. Cartan matrices; 7. The field of definition; Exercises; CHAPTER 2 The groups G(m, p, n); 1. Primitivity and imprimitivity; 2. Wreath products and monomial representations; 3. Properties of the groups G(m, p, n); 4. The imprimitive unitary reflection groups

5. Imprimitive subgroups of primitive reflection groups6. Root systems for G(m, p, n); 7. Generators for G(m, p, n); 8. Invariant polynomials for G(m, p, n); Exercises; CHAPTER 3 Polynomial invariants; 1. Tensor and symmetric algebras; 2. The algebra of invariants; 3. Invariants of a finite group; 4. The action of a reflection; 5. The Shephard-Todd-Chevalley Theorem; 6. The coinvariant algebra; Exercises; CHAPTER 4 Poincaré series and characterisations of reflection groups; 1. Poincaré series; 2. Exterior and symmetric algebras and Molien's Theorem

3. A characterisation of finite reflection groups4. Exponents; Exercises; CHAPTER 5 Quaternions and the finite subgroups of SU2(C); 1. The quaternions; 2. The groups O3 (R) and O4 (R); 3. The groups SU2 (C) and U2 (C); 4. The finite subgroups of the quaternions; 5. The finite subgroups of SO3 (R) and SU2 (C); 6. Quaternions, reflections and root systems; Exercises; CHAPTER 6 Finite unitary reflection groups of rank two; 1. The primitive reflection subgroups of U2 (C); 2. The reflection groups of type T; 3. The reflection groups of type O; 4. The reflection groups of type I

5. Cartan matrices and the ring of definition6. Invariants; Exercises; CHAPTER 7 Line systems; 1. Bounds on line systems; 2. Star-closed Euclidean line systems; 3. Reflections and star-closed line systems; 4. Extensions of line systems; 5. Line systems for imprimitive reflection groups; 6. Line systems for primitive reflection groups; 7. The Goethals-Seidel decomposition for 3-systems; 8. Extensions of Dn(2) and Dn(3); 9. Further structure of line systems in Cn; 10. Extensions of Euclidean line systems; 11. Extensions of An, En and Kn in Cn; 12. Extensions of 4-systems; Exercises

CHAPTER 8 The Shephard and Todd classification1. Outline of the classification; 2. Blichfeldt's Theorem; 3. Consequences of Blichfeldt's Theorem; 4. Extensions of 5-systems; 5. Line systems and reflections of order three; 6. Extensions of ternary 6-systems; 7. The classification; 8. Root systems and the ring of definition; 9. Reduction modulo p; 10. Identification of the primitive reflection groups; Exercises; CHAPTER 9 The orbit map, harmonic polynomials and semi-invariants; 1. The orbit map; 2. Skew invariants and the Jacobian; 3. The rank of the Jacobian; 4. Semi-invariants

5. Differential operators

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

ISBN:

1-107-19342-7

9786612393150

0-511-64710-7

1-282-39315-4

0-511-60416-5

0-511-60338-X

0-511-65118-X

0-511-60260-X

OCLC:

609845649