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 Poincare 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. "

Here is a key text on the subject of representation theory in finite groups. The pages of this excellent little book, prepared by Rafael Stekolshchik, contain a number of new proofs relating to Coxeter Transformations and the McKay Correspondence. They include ideas and formulae from a number of luminaries including J. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, as well as material from Coxeter and McKay themselves. Many other authors have material published here too.