Notes on Coxeter Transformations and the McKay Correspondence

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.

Includes supplementary material: sn.pub/extras

Autorentext

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
Preliminaries.- The Jordan normal form of the Coxeter transformation.- Eigenvalues, splitting formulas and diagrams Tp,q,r.- R. Steinberg's theorem, B. Kostant's construction.- The affine Coxeter transformation.