Notes on Coxeter Transformations and the McKay correspondence

Abstract

We study in detail the Jordan forms of the Coxeter transformations and prove shearing formulas due to Subbotin and Sumin for the characteristic polynomials of the Coxeter transformations. Using shearing formulas we calculate characteristic polynomials of the Coxeter transformation for the diagrams T_{2,3,r}, T_{3,3,r}, T_{2,4,r}, prove J. S. Frame's formulas, and generalize R. Steinberg's theorem on the spectrum of the affine Coxeter transformation for the multiply-laced diagrams. This theorem is the key statement in R. Steinberg's proof of the McKay correspondence. B. Kostant's construction appears in the context of the McKay correspondence and gives a way to obtain multiplicities of indecomposable representations \rho_i of the binary polyhedral group G in the decomposition of \pi_n|G. In the case of multiply-laced graphs, instead of indecomposable representations \rho_i we use restricted representations and induced representations of G introduced by P. Slodowy. Using B. Kostant's construction we generalize to the case of multiply-laced graphs W. Ebeling's theorem which connects the Poincare series and the Coxeter transformations. Using the Jordan form of the Coxeter transformation we prove a criterion of V. Dlab and C. M. Ringel of regularity of quiver representations.