Preprint 80/2006

Notes on Coxeter Transformations and the McKay correspondence

Rafael Stekolshchik

Contact the author: Please use for correspondence this email.
Submission date: 22. Aug. 2006
Pages: 208
published as:
Leites, D. A. and R. Stekolshchik (eds.): Notes on Coxeter transformations and the McKay correspondence
   New York : Springer, 2008. - XVIII, 239 p.
   (Springer monographs in mathematics ;)
   ISBN 978-3-540-77398-6 - ISBN 978-3-540-77399-3       
MSC-Numbers: 17B20
Keywords and phrases: Coxeter transformation, Mckay correspondence, Slodowy correspondence

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 formula26, 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. For every extended Dynkin diagram, the spectrum of the Coxeter transformation is easily obtained from R. Steinberg's theorem.

In the study of representations formula28 of SU(2), we extend B. Kostant's construction of a vector-valued generating function formula32. B. Kostant's construction appears in the context of the McKay correspondence and gives a way to obtain multiplicities of indecomposable representations formula34 of the binary polyhedral group G in the decomposition of formula38. In the case of multiply-laced graphs, instead of indecomposable representations formula34 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 Poincaré series formula44 and the Coxeter transformations. According to W. Ebeling's theorem


where formula46 is the characteristic polynomial of the Coxeter transformation and formula48 is the characteristic polynomial of the corresponding affine Coxeter transformation.

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 formula50:


where j runs over all vertices adjacent to i.

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.

18.10.2019, 02:13