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.
Springer Monographs in Mathematics R. Stekolshchik Notes on Coxeter Transformations and the McKay Correspondence Rafael Stekolshchik Str. Kehilat Klivlend 7 Tel-Aviv Israel
[email protected] ISBN 978-3-540-77398-6 e-ISBN 978-3-540-77399-3 DOI 10.1007/978-3-540-77399-3 Springer Monographs in Mathematics ISSN 1439-7382 Library of Congress Control Number: 2007941499 Mathematics Subject Classification (2000): 20F55, 15A18, 17B20, 16G20 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication