Notes on Coxeter Transformations and the McKay Correspondence (eBook)

Rafael Stekolshchik (Autor)

eBook Download: PDF

2008 | 1. Auflage
240 Seiten
Springer-Verlag
9783540773993 (ISBN)

Infos

One of the beautiful results in the representation theory of the finite groups is McKay's theorem on a correspondence between representations of the binary polyhedral group of SU(2) and vertices of an extended simply-laced Dynkin diagram.

The Coxeter transformation is the main tool in the proof of the McKay correspondence, and is closely interrelated with the Cartan matrix and Poincaré series. The Coxeter functors constructed by Bernstein, Gelfand and Ponomarev plays a distinguished role in the representation theory of quivers.

On these pages, the ideas and formulas due to J. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, H.S.M. Coxeter, V. Dlab and C.M. Ringel, V. Kac, J. McKay, T.A. Springer, B. Kostant, P. Slodowy, R. Steinberg, W. Ebeling and several other authors, as well as the author and his colleagues from Subbotin's seminar, are presented in detail. Several proofs seem to be new.

Autor

1980 - 1991, CAM (Center of Automation and Metrology), Academy of Sciences of Moldova, Project leader of experimental data processing. Research and development of programs and mathematical tools for Academy of Sciences of Moldova,  999 – 2007, ECI Telecom  (Electronics Corporation of Israel), Israel, Project leader in the Network Management department.Research and development of algorithmes in the field of Communications and Big Systems.

Inhalt

Summary 5
Contents 9
List of Figures 15
List of Tables 17
List of Notions 19
1 Introduction 21
1.1 The three historical aspects of the Coxeter transformation 21
1.2 A brief review of this work 23
1.3 The spectrum and the Jordan form 26
1.4 Splitting formulas and the diagrams 29
1.5 Coxeter transformations and the McKay correspondence 33
1.6 The a.ne Coxeter transformation 36
1.7 The regular representations of quivers 39
2 Preliminaries 43
2.1 The Cartan matrix and the Tits form 43
2.2 Representations of quivers 58
2.3 The Poincare series 66
3 The Jordan normal form of the Coxeter transformation 71
3.1 The Cartan matrix and the Coxeter transformation 71
3.2 An application of the Perron-Frobenius theorem 76
3.3 The basis of eigenvectors and a theorem on the Jordan form 81
4 Eigenvalues, splitting formulas and diagrams 87
4.1 The eigenvalues of the a.ne Coxeter transformation are roots of unity 87
4.2 Bibliographical notes on the spectrum of the Coxeter transformation 91
4.3 Splitting and gluing formulas for the characteristic polynomial 94
4.4 Formulas of the characteristic polynomials for the diagrams Tp,q,r 100
5 R. Steinberg’s theorem, B. Kostant’s construction 115
5.1 R. Steinberg’s theorem and a (p, q, r) mystery 115
5.2 The characteristic polynomials for the Dynkin diagrams 119
5.3 A generalization of R. Steinberg’s theorem 122
5.4 The Kostant generating function and Poincare series 125
5.5 The orbit structure of the Coxeter transformation 136
6 The affine Coxeter transformation 149
6.1 The Weyl Group and the affine Weyl group 149
6.2 R. Steinberg’s theorem again 157
6.3 The defect 168
A The McKay correspondence and the Slodowy correspondence 175
A.1 Finite subgroups of SU(2) and SO(3, R) 175
A.2 The generators and relations in polyhedral groups 176
A.3 The Kleinian singularities and the Du Val resolution 178
A.4 The McKay correspondence 180
A.5 The Slodowy generalization of the McKay correspondence 181
A.6 The characters of the binary polyhedral groups 199
B Regularity conditions for representations of quivers 203
B.1 The Coxeter functors and regularity conditions 203
B.2 The necessary regularity conditions for diagrams with indefinite Tits form 208
B.3 Transforming elements and suficient regularity conditions 211
B.4 Examples of regularity conditions 217
C Miscellanea 223
C.1 The triangle groups and Hurwitz groups 223
C.2 The algebraic integers 224
C.3 The Perron-Frobenius Theorem 226
C.4 The Schwartz inequality 227
C.5 The complex projective line and stereographic projection 228
C.6 The prime spectrum, the coordinate ring, the orbit space 230
C.7 Fixed and anti-fixed points of the Coxeter transformation 235
References 241
Index 253

Produkt-Details

• Erscheint lt. Verlag: 18.1.2008
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XX, 240 p. 28 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: Cartan matrix • Coxeter transformation • Dynkin diagram • eigenvalue • Matrix • McKay correspondence • Poincare series • Representation Theory
• ISBN-13: 9783540773993 / 9783540773993