McKay's correspondence, Auslander's theorem, and reflection groups
Eleonore Faber · Open Collections · 2017
This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. Let $G$ be a finite subgroup of $GL(n,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $K$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with a singular locus of codimension 1. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of $D$ as a quotient of the skew group ring $A=G*S$. We will explain this construction, which gives a new view on Knörrer's periodicity theorem for matrix factorizations and allows to extend Auslander's theorem about the algebraic version of the McKay correspondence to reflection groups.