Pure Mathematics Colloquium: Eleonore Faber
Symmetries and Singularities via the McKay correspondence
The classification of finite subgroups of SO(3) is well known: these are either cyclic or dihedral groups or one of the symmetry groups of the Platonic solids. In the 19th century, Felix Klein investigated the orbit spaces of those groups and their double covers, the so-called binary polyhedral groups. This investigation is at the origin of singularity theory.Quite surprisingly, in 1979, John McKay found a direct relationship between the resolution of the singularities of the orbit spaces and the representation theory of the finite group one starts from.This "classical McKay correspondence" is manifested, in particular, by the ubiquitious Coxeter-Dynkin diagrams.In this talk I will first review the history of this fascinating result, and then give an outlook on recent joint work with Ragnar-Olaf Buchweitz and Colin Ingalls about a McKay correspondence for finite reflection groups in GL(n,C).
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