Notes on Coxeter Transformations and the McKay Correspondence [electronic resource] / by Rafael Stekolshchik.
Contributor(s): SpringerLink (Online service)Material type: TextSeries: Springer Monographs in Mathematics: Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2008Description: XX, 240 p. 28 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783540773993Subject(s): Mathematics | Algebra | Commutative algebra | Commutative rings | Group theory | Topological groups | Lie groups | Functional analysis | Mathematics | Algebra | Functional Analysis | Commutative Rings and Algebras | Topological Groups, Lie Groups | Group Theory and GeneralizationsAdditional physical formats: Printed edition:: No titleDDC classification: 512 LOC classification: QA150-272Online resources: Click here to access online
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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.
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.