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Unitary Reflection Groups [Mīkstie vāki]

(University of Sydney), (University of Sydney)
  • Formāts: Paperback / softback, 302 pages, height x width x depth: 228x152x16 mm, weight: 440 g, Worked examples or Exercises; 12 Tables, unspecified
  • Sērija : Australian Mathematical Society Lecture Series
  • Izdošanas datums: 13-Aug-2009
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521749891
  • ISBN-13: 9780521749893
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  • Mīkstie vāki
  • Cena: 108,03 €
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Unitary Reflection GroupsPalielināt
  • Formāts: Paperback / softback, 302 pages, height x width x depth: 228x152x16 mm, weight: 440 g, Worked examples or Exercises; 12 Tables, unspecified
  • Sērija : Australian Mathematical Society Lecture Series
  • Izdošanas datums: 13-Aug-2009
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521749891
  • ISBN-13: 9780521749893
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521749893.html
Keywords:
A complete and clear account of the classification of unitary reflection groups, which arise naturally in many areas of mathematics.

A complex reflection is a linear transformation which fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or arrangement of mirrors. This book gives a complete classification of all groups of transformations of n-dimensional complex space which are generated by complex reflections, using the method of line systems. In particular: irreducible groups are studied in detail, and are identified with finite linear groups; reflection subgroups of reflection groups are completely classified; the theory of eigenspaces of elements of reflection groups is discussed fully; an appendix outlines links to representation theory, topology and mathematical physics. Containing over 100 exercises ranging in difficulty from elementary to research level, this book is ideal for honors and graduate students, or for researchers in algebra, topology and mathematical physics.

Papildus informācija

A complete and clear account of the classification of unitary reflection groups, which arise naturally in many areas of mathematics.
Introduction 1(1)
Overview of this book
1(3)
Some detail concerning the content
4(1)
Acknowledgements
5(1)
Leitfaden
5(2)
Preliminaries
7(16)
Hermitian forms
7(2)
Reflections
9(3)
Groups
12(1)
Modules and representations
13(2)
Irreducible unitary reflection groups
15(2)
Cartan matrices
17(2)
The field of definition
19(4)
Exercises
21(2)
The groups G(m, p, n)
23(16)
Primitivity and imprimitivity
23(1)
Wreath products and monomial representations
24(1)
Properties of the groups G(m, p, n)
25(2)
The imprimitive unitary reflection groups
27(5)
Imprimitive subgroups of primitive reflection groups
32(2)
Root systems for G(m, p, n)
34(1)
Generators for G(m, p, n)
35(1)
Invariant polynomials for G(m, p, n)
36(3)
Exercises
37(2)
Polynomial invariants
39(15)
Tensor and symmetric algebras
39(2)
The algebra of invariants
41(1)
Invariants of a finite group
42(4)
The action of a reflection
46(1)
The Shephard-Todd-Chevalley Theorem
46(5)
The coinvariant algebra
51(3)
Exercises
53(1)
Poincare series and characterisations of reflection groups
54(12)
Poincare series
54(2)
Exterior and symmetric algebras and Molien's Theorem
56(5)
A characterisation of finite reflection groups
61(2)
Exponents
63(3)
Exercises
65(1)
Quaternions and the finite subgroups of SU2 (C)
66(18)
The quaternions
67(2)
The groups O3 (R) and O4 (R)
69(2)
The groups SU2 (C) and U2 (C)
71(1)
The finite subgroups of the quaternions
72(5)
The finite subgroups of SO3 (R) and SU2 (C)
77(2)
Quaternions, reflections and root systems
79(5)
Exercises
83(1)
Finite unitary reflection groups of rank two
84(15)
The primitive reflection subgroups of U2 (C)
84(1)
The reflection groups of type T
85(2)
The reflection groups of type O
87(2)
The reflection groups of type I
89(1)
Cartan matrices and the ring of definition
90(3)
Invariants
93(6)
Exercises
98(1)
Line systems
99(38)
Bounds on line systems
99(1)
Star-closed Euclidean line systems
100(1)
Reflections and star-closed line systems
101(2)
Extensions of line systems
103(1)
Line systems for imprimitive reflection groups
104(1)
Line systems for primitive reflection groups
105(6)
The Goethals-Seidel decomposition for 3-systems
111(4)
Extensions of D(2)n and D(3)n
115(4)
Further structure of line systems in Cn
119(1)
Extensions of Euclidean line systems
120(5)
Extensions of An, εn and κn in Cn
125(2)
Extensions of 4-systems
127(10)
Exercises
133(4)
The Shephard and Todd classification
137(34)
Outline of the classification
137(1)
Blichfeldt's Theorem
138(2)
Consequences of Blichfeldt's Theorem
140(2)
Extensions of 5-systems
142(4)
Line systems and reflections of order three
146(3)
Extensions of ternary 6-systems
149(2)
The classification
151(2)
Root systems and the ring of definition
153(2)
Reduction modulo p
155(2)
Identification of the primitive reflection groups
157(14)
Exercises
168(3)
The orbit map, harmonic polynomials and semi-invariants
171(20)
The orbit map
171(1)
Skew invariants and the Jacobian
172(2)
The rank of the Jacobian
174(2)
Semi-invariants
176(3)
Differential operators
179(4)
The space of G-harmonic polynomials
183(3)
Steinberg's fixed point theorem
186(5)
Exercises
189(2)
Covariants and related polynomial identities
191(17)
The space of covariants
191(3)
Gutkin's Theorem
194(4)
Differential invariants
198(1)
Some special cases of covariants
199(2)
Two-variable Poincare series and specialisations
201(7)
Exercises
206(2)
Eigenspace theory and reflection subquotients
208(20)
Basic affine algebraic geometry
208(4)
Eigenspaces of elements of reflection groups
212(1)
Reflection subquotients of unitary reflection groups
213(2)
Regular elements
215(3)
Properties of the reflection subquotients
218(4)
Eigenvalues of pseudoregular elements
222(6)
Reflection cosets and twisted invariant theory
228(18)
Reflection cosets
228(1)
Twisted invariant theory
229(2)
Eigenspace theory for reflection cosets
231(6)
Subquotients and centralisers
237(2)
Parabolic subgroups and the coinvariant algebra
239(3)
Duality groups
242(4)
Exercises
244(2)
Appendix A. Some background in commutative algebra
246(4)
Appendix B. Forms over finite fields
250(5)
Basic definitions
250(1)
Witt's Theorem
251(1)
The Wall form, the spinor norm and Dickson's invariant
251(1)
Order formulae
252(1)
Reflections in finite orthogonal groups
253(2)
Appendix C. Applications and further reading
255(16)
The space of regular elements
255(3)
Fundamental groups, braid groups, presentations
258(3)
Hecke algebras
261(5)
Reductive groups over finite fields
266(5)
Appendix D. Tables
271(8)
The primitive unitary reflection groups
272(2)
Degrees and codegrees
274(2)
Cartan matrices
276(1)
Maximal subsystems
277(1)
Reflection cosets
277(2)
Bibliography 279(10)
Index of notation 289(2)
Index 291
Gustav I. Lehrer is a Professor in the School of Mathematics and Statistics at the University of Sydney. Donald E. Taylor is an Associate Professor in the School of Mathematics and Statistics at the University of Sydney.