Ebook: Reflection Groups and Invariant Theory

Reflection Groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.
Richard Kane is a professor of mathematics at the University of Western Ontario. His research interests are algebra and algebraic topology. Professor Kane is a former President of the Canadian Mathematical Society.

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Reflection Groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.
Richard Kane is a professor of mathematics at the University of Western Ontario. His research interests are algebra and algebraic topology. Professor Kane is a former President of the Canadian Mathematical Society.

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Reflection Groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.
Richard Kane is a professor of mathematics at the University of Western Ontario. His research interests are algebra and algebraic topology. Professor Kane is a former President of the Canadian Mathematical Society.
Content:
Front Matter....Pages i-ix
Introduction: Reflection groups and invariant theory....Pages 1-3
Front Matter....Pages 5-5
Euclidean reflection groups....Pages 6-24
Root systems....Pages 25-34
Fundamental systems....Pages 35-44
Length....Pages 45-56
Parabolic subgroups....Pages 57-63
Front Matter....Pages 65-65
Reflection groups and Coxeter systems....Pages 66-74
Bilinear forms of Coxeter systems....Pages 75-80
Classification of Coxeter systems and reflection groups....Pages 81-96
Front Matter....Pages 97-97
Weyl groups....Pages 98-108
The Classification of crystallographic root systems....Pages 109-117
Affine Weyl groups....Pages 118-134
Subroot systems....Pages 135-143
Formal identities....Pages 144-151
Front Matter....Pages 153-153
Pseudo-reflections....Pages 154-160
Classifications of pseudo-reflection groups....Pages 161-167
Front Matter....Pages 169-169
The ring of invariants....Pages 170-179
Poincar? series....Pages 180-190
Nonmodular invariants of pseudo-reflection groups....Pages 191-201
Modular invariants of pseudo-reflection groups....Pages 202-211
Front Matter....Pages 213-213
Skew invariants....Pages 214-220
 The Jacobian....Pages 221-228
The extended ring of invariants....Pages 229-234
Front Matter....Pages 235-235
Poincar? series for the ring of covariants....Pages 236-246
Representations of pseudo-reflection groups....Pages 247-255
Harmonic elements....Pages 256-262
Harmonics and reflection groups....Pages 263-278
Front Matter....Pages 279-279
Involutions....Pages 280-289
Elementary equivalences....Pages 290-298
Coxeter elements....Pages 299-310
Minimal decompositions....Pages 311-317
Front Matter....Pages 319-319
Eigenvalues for reflection groups....Pages 320-324
Eigenvalues for regular elements....Pages 325-333
Ring of invariants and eigenvalues....Pages 334-340
Properties of regular elements....Pages 341-348
Back Matter....Pages 349-379

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Reflection Groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.
Richard Kane is a professor of mathematics at the University of Western Ontario. His research interests are algebra and algebraic topology. Professor Kane is a former President of the Canadian Mathematical Society.
Content:
Front Matter....Pages i-ix
Introduction: Reflection groups and invariant theory....Pages 1-3
Front Matter....Pages 5-5
Euclidean reflection groups....Pages 6-24
Root systems....Pages 25-34
Fundamental systems....Pages 35-44
Length....Pages 45-56
Parabolic subgroups....Pages 57-63
Front Matter....Pages 65-65
Reflection groups and Coxeter systems....Pages 66-74
Bilinear forms of Coxeter systems....Pages 75-80
Classification of Coxeter systems and reflection groups....Pages 81-96
Front Matter....Pages 97-97
Weyl groups....Pages 98-108
The Classification of crystallographic root systems....Pages 109-117
Affine Weyl groups....Pages 118-134
Subroot systems....Pages 135-143
Formal identities....Pages 144-151
Front Matter....Pages 153-153
Pseudo-reflections....Pages 154-160
Classifications of pseudo-reflection groups....Pages 161-167
Front Matter....Pages 169-169
The ring of invariants....Pages 170-179
Poincar? series....Pages 180-190
Nonmodular invariants of pseudo-reflection groups....Pages 191-201
Modular invariants of pseudo-reflection groups....Pages 202-211
Front Matter....Pages 213-213
Skew invariants....Pages 214-220
 The Jacobian....Pages 221-228
The extended ring of invariants....Pages 229-234
Front Matter....Pages 235-235
Poincar? series for the ring of covariants....Pages 236-246
Representations of pseudo-reflection groups....Pages 247-255
Harmonic elements....Pages 256-262
Harmonics and reflection groups....Pages 263-278
Front Matter....Pages 279-279
Involutions....Pages 280-289
Elementary equivalences....Pages 290-298
Coxeter elements....Pages 299-310
Minimal decompositions....Pages 311-317
Front Matter....Pages 319-319
Eigenvalues for reflection groups....Pages 320-324
Eigenvalues for regular elements....Pages 325-333
Ring of invariants and eigenvalues....Pages 334-340
Properties of regular elements....Pages 341-348
Back Matter....Pages 349-379
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