Notes on Coxeter transformations and the McKay correspondence
By: R. Stekolshchik
Series: Springer monographs in mathematics; Springer monographs in mathematics
Publisher: Springer
Published: 2008
Place of publication: Berlin
Language: eng
Number of pages: 239
ISBN-13: 9783540773986; 9783540773993
Subjects: Finite groups; Coxeter-Gruppe; Cartan-Matrix; Transformations (Mathematics); Poincaré-Reihe; Representations of algebras
Description: 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.
Notes: Includes bibliographical references (p. [221]-231) and index.