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McKay's correspondence, Auslander's theorem, and reflection groups

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Associative rings and algebras; Algebraic geometry; Algebraic structures

video/mp4

Author affiliation: University of Michigan

2017-03-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60930

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/