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We consider the Coxeter transformation in the context of the McKay correspondence, representations of quivers, and Poincaré series.
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
In the study of representations
Using the Jordan form of the Coxeter transformation we prove a criterion of V.~Dlab and C.~M.~Ringel of regularity of quiver representations, consider necessary and sufficient conditions of this criterion for extended Dynkin diagrams and for diagrams with indefinite Tits form.
We prove one more McKay's observation concerning the Kostant generating functions
A connection between fixed and anti-fixed points of the powers of the Coxeter transformations and Chebyshev polynomials of the first and second kind is established.