Richard Kane

Reflection Groups and Invariant Theory

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• CMS Books in Mathematics
• Verlag: Springer, Berlin
• Softcover reprint of the original 1st ed. 2001
• Seitenzahl: 392
• Erscheinungstermin: 9. Oktober 2011
• Englisch
• Abmessung: 235mm x 155mm x 22mm
• Gewicht: 606g
• ISBN-13: 9781441931948
• ISBN-10: 1441931945
• Artikelnr.: 32220619

Autorenporträt

Preliminary Text. Reflection groups and invarian theory is a branch of the intersection between geometry and algebra. It has a deep and elegant theory which is both of great interest in its own right and also related to other significant areas of mathematics. This 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.

Inhaltsangabe

I Reflection groups.- 1 Euclidean reflection groups.- 2 Root systems.- 3 Fundamental systems.- 4 Length.- 5 Parabolic subgroups.- II Coxeter groups.- 6 Reflection groups and Coxeter systems.- 7 Bilinear forms of Coxeter systems.- 8 Classification of Coxeter systems and reflection groups.- III Weyl groups.- 9 Weyl groups.- 10 The Classification of crystallographic root systems.- 11 Affine Weyl groups.- 12 Subroot systems.- 13 Formal identities.- IV Pseudo-reflection groups.- 14 Pseudo-reflections.- 15 Classifications of pseudo-reflection groups.- V Rings of invariants.- 16 The ring of invariants.- 17 Poincaré series.- 18 Nonmodular invariants of pseudo-reflection groups.- 19 Modular invariants of pseudo-reflection groups.- VI Skew invariants.- 20 Skew invariants.- 21 The Jacobian.- 22 The extended ring of invariants.- VII Rings of covariants.- 23 Poincaré series for the ring of covariants.- 24 Representations of pseudo-reflection groups.- 25 Harmonic elements.- 26 Harmonics and reflection groups.- VIII Conjugacy classes.- 27 Involutions.- 28 Elementary equivalences.- 29 Coxeter elements.- 30 Minimal decompositions.- IX Eigenvalues.- 31 Eigenvalues for reflection groups.- 32 Eigenvalues for regular elements.- 33 Ring of invariants and eigenvalues.- 34 Properties of regular elements.- Appendices.- A Rings and modules.- B Group actions and representation theory.- C Quadratic forms.- D Lie algebras.- References.

Rezensionen

From the reviews: CANADIAN MATHEMATICAL SOCIETY NOTES "The former group are interested in Lie algebras and algebraic groups while the latter group includes commutative algebraists and topologists. This is the first book I¿ve read that is aimed at both audiences, providing a clear exposition of a truly beautiful area of mathematics. It is probably fair to say there isn¿t much new in the book, except to find all the relevant material in the one place, well-organized and well-explained¿This is a very lovely book to read. I found the pace of the book to be perfect. The approach follows the development of the subject in the literature closely ¿ this is how it was discovered, more or less. It appears to my eye to be accessible to a first-year graduate student. It is not overly terse, nor overly wordy and there are many examples." ZENTRALBLATT MATH "The book is certainly a good source of information mainly in the more advanced areas of the subject. The references compile nearly a hundred articles and books." MATHEMATICAL REVIEWS "The book is a very accessible introduction to a wonderful part of mathematics that has many applications¿it can be pointed out that the approach taken in this book includes examples of the use of Galois theory, Cohen Macaulay properties, Hopf algebras, representation theory and basic algebraic geometry. This is a nice way to see applications of these areas¿The book would certainly be a good choice to teach the ideas in the above list, as the book flows very well. Each chapter is well motivated and summarized in its introduction."