The McKay Correspondence on Kleinian Singularities

Date: Thursday 4 August 2022

Abstract

A finite subgroup G of SU(2) produces a surface singularity by its action on the complex space C^2, called a Kleinian singularity. We can study this singularity by taking its minimal resolution; that is, by constructing a smooth surface that is isomorphic to the singular surface away from the singular point, and that replaces the singular point with a collection of projective lines. Taking the intersection graph of these lines gives a graph with an ADE classification.

The action of G on C^2 also produces a different graph, called the McKay graph, which encodes information about the irreducible representations of G. In 1980, John McKay observed that these two graphs - the McKay graph and the intersection graph - are isomorphic (up to the removal of a specific vertex) for all finite subgroups G of SU(2). This gives us a connection between the geometry of the singularity (given by the intersection graph on its minimal resolution) and the algebraic structure of G (given by the McKay graph). In this talk I will present the construction of the two graphs in the case that G is cyclic, and demonstrate the isomorphism between the two graphs.