Stekolshchik Notes on Coxeter Transformations and the McKay Correspondence

1. Auflage 2008
ISBN: 978-3-540-77399-3
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 240 Seiten

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.

Autoren/Hrsg.

Stekolshchik, Rafael

Inhaltsverzeichnis

1;Summary;5
2;Contents;9
3;List of Figures;15
4;List of Tables;17
5;List of Notions;19
6;1 Introduction;21
6.1;1.1 The three historical aspects of the Coxeter transformation;21
6.2;1.2 A brief review of this work;23
6.3;1.3 The spectrum and the Jordan form;26
6.4;1.4 Splitting formulas and the diagrams;29
6.5;1.5 Coxeter transformations and the McKay correspondence;33
6.6;1.6 The a.ne Coxeter transformation;36
6.7;1.7 The regular representations of quivers;39
7;2 Preliminaries;43
7.1;2.1 The Cartan matrix and the Tits form;43
7.2;2.2 Representations of quivers;58
7.3;2.3 The Poincare series;66
8;3 The Jordan normal form of the Coxeter transformation;71
8.1;3.1 The Cartan matrix and the Coxeter transformation;71
8.2;3.2 An application of the Perron-Frobenius theorem;76
8.3;3.3 The basis of eigenvectors and a theorem on the Jordan form;81
9;4 Eigenvalues, splitting formulas and diagrams;87
9.1;4.1 The eigenvalues of the a.ne Coxeter transformation are roots of unity;87
9.2;4.2 Bibliographical notes on the spectrum of the Coxeter transformation;91
9.3;4.3 Splitting and gluing formulas for the characteristic polynomial;94
9.4;4.4 Formulas of the characteristic polynomials for the diagrams Tp,q,r;100
10;5 R. Steinberg’s theorem, B. Kostant’s construction;115
10.1;5.1 R. Steinberg’s theorem and a (p, q, r) mystery;115
10.2;5.2 The characteristic polynomials for the Dynkin diagrams;119
10.3;5.3 A generalization of R. Steinberg’s theorem;122
10.4;5.4 The Kostant generating function and Poincare series;125
10.5;5.5 The orbit structure of the Coxeter transformation;136
11;6 The affine Coxeter transformation;149
11.1;6.1 The Weyl Group and the affine Weyl group;149
11.2;6.2 R. Steinberg’s theorem again;157
11.3;6.3 The defect;168
12;A The McKay correspondence and the Slodowy correspondence;175
12.1;A.1 Finite subgroups of SU(2) and SO(3, R);175
12.2;A.2 The generators and relations in polyhedral groups;176
12.3;A.3 The Kleinian singularities and the Du Val resolution;178
12.4;A.4 The McKay correspondence;180
12.5;A.5 The Slodowy generalization of the McKay correspondence;181
12.6;A.6 The characters of the binary polyhedral groups;199
13;B Regularity conditions for representations of quivers;203
13.1;B.1 The Coxeter functors and regularity conditions;203
13.2;B.2 The necessary regularity conditions for diagrams with indefinite Tits form;208
13.3;B.3 Transforming elements and suficient regularity conditions;211
13.4;B.4 Examples of regularity conditions;217
14;C Miscellanea;223
14.1;C.1 The triangle groups and Hurwitz groups;223
14.2;C.2 The algebraic integers;224
14.3;C.3 The Perron-Frobenius Theorem;226
14.4;C.4 The Schwartz inequality;227
14.5;C.5 The complex projective line and stereographic projection;228
14.6;C.6 The prime spectrum, the coordinate ring, the orbit space;230
14.7;C.7 Fixed and anti-fixed points of the Coxeter transformation;235
15;References;241
16;Index;253