Rafael Stekolshchik (auth.)9783540773986, 3540773983
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.
Table of contents :
Front Matter….Pages I-XX
Introduction….Pages 1-21
Preliminaries….Pages 23-50
The Jordan normal form of the Coxeter transformation….Pages 51-66
Eigenvalues, splitting formulas and diagrams T p,q,r ….Pages 67-93
R. Steinberg’s theorem, B. Kostant’s construction….Pages 95-127
The affine Coxeter transformation….Pages 129-153
Back Matter….Pages 155-239
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