导语

  

内容提要

  

    复反射是固定在超平面上每个点的线性变换，它类似于通过万花筒或镜子排列观看图像时所经历的转换。本书使用线性变换的方法对n维复空间中由复反射产生的所有变换组进行了完整的分类，对不可约群进行了详细的研究，对反射群的反射子群进行了完整的分类，充分讨论了反射群元素的本征空间理论。书中附录还概述了表示论、拓扑学和数学物理之间的联系。本书包含了100多个从简单到具有一定难度的练习题，适合大学师生、研究生及数学爱好者参考阅读，也适合代数、拓扑学和数学物理的研究人员参考阅读。

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. Caftan 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 groups
  6. 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. Poincare series and characterisations of reflection groups
  1. Poincare series
  2. Exterior and symmetric algebras and Molien's Theorem
  3. A characterisation of finite reflection groups
  4. Exponents
  Exercises
Chapter 5. Quaternions and the finite subgroups of SU2 (C)
  1. The quaternions
  2. The groups Oa (R) and 04 (R)
  3. The groups SU2 (C) and U2 (C)
  4. The finite subgroups of the quaternions
  5. The finite subgroups of S03 (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 definition
  6. Invariants
  Exercises
Chapter 7. Line systems
  1. Bounds online 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 D(2) and Dn(3)
  9. Further structure of line systems in Cn
  10. Extensions of Euclidean line systems
  11. Extensions of.An, gn and Kn in Cn
  12. Extensions of 4-systems
  Exercises
Chapter 8. The Shephard and Todd classification
  1. 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
  6. The space of G-harmonic polynomials
  7. Steinberg's fixed point theorem
  Exercises
Chapter 10. Covariants and related polynomial identities
  1. The space of covariants
  2. Gutkin's Theorem
  3. Differential invariants
  4. Some special cases of covariants
  5. Two-variable Poincar6 series and specialisations
  Exercises
Chapter 11. Eigenspace theory and reflection subquotients
  1. Basic affine algebraic geometry
  2. Eigenspaces of elements of reflection groups
  3. Reflection subquotients of unitary reflection groups
  4. Regular elements
  5. Properties of the reflection subquotients
  6. Eigenvalues of pseudoregular elements
Chapter 12. Reflection cosets and twisted invariant theory
  1. Reflection cosets
  2. Twisted invariant theory
  3. Eigenspace theory for reflection cosets
  4. Subquotients and centralisers
  5. Parabolic subgroups and the coinvariant algebra
  6. Duality groups
  Exercises
Appendix A. Some background in commutative algebra
Appendix B. Forms over finite fields
  1. Basic definitions
  2. Witt's Theorem
  3. The Wall form, the spinor norm and Dickson's invariant
  4. Order formulae
  5. Reflections in finite orthogonal groups
Appendix C. Applications and further reading
  1. The space of regular elements
  2. Fundamental groups, braid groups, presentations
  3. Hecke algebras
  4. Reductive groups over finite fields
Appendix D. Tables
  1. The primitive unitary reflection groups
  2. Degrees and codegrees
  3. Cartan matrices
  4. Maximal subsystems
  5. Reflection cosets
Bibliography
Index of notation
Index
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