The GL(2) McKay Correspondence

TL;DR

This paper establishes a correspondence between the dual graph of minimal resolutions of affine rational surface singularities and the quiver of endomorphism rings of special CM modules, extending McKay's correspondence to all finite subgroups of GL(2,C).

Contribution

It introduces the reconstruction algebra linking minimal resolution graphs to endomorphism rings for all finite subgroups of GL(2,C), generalizing McKay's correspondence.

Findings

Dual graph corresponds to the quiver of the endomorphism ring.

Derived categories of the resolution and algebra are equivalent.

Extension of McKay correspondence to all finite subgroups of GL(2,C).

Abstract

In this paper we show that for any affine complete rational surface singularity there is a correspondence between the dual graph of the minimal resolution and the quiver of the endomorphism ring of the special CM modules. We thus call such an algebra the reconstruction algebra. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,C), it means that the endomorphism ring of the special CM C[[x,y]]^G modules can be used to build the dual graph of the minimal resolution of C^2/G, extending McKay's observation for finite subgroups of SL(2,C) to all finite subgroups of GL(2,C).