[Paper Review] The GL(2) McKay Correspondence

Michael Wemyss|arXiv (Cornell University)|Sep 11, 2008

Algebraic structures and combinatorial models8 references5 citations

TL;DR

This paper establishes the GL(2, C) McKay correspondence by showing that the quiver of the reconstruction algebra—derived from endomorphism rings of special Cohen-Macaulay modules—can be combinatorially reconstructed from the dual graph of the minimal resolution of any affine complete rational surface singularity. The key result is a 1-1 correspondence between the quiver structure and the resolution's dual graph with self-intersection numbers, extending McKay’s classical correspondence beyond SL(2, C) 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).

Motivation & Objective

• . The paper aims to resolve the geometric ambiguity in skew group rings for non-Gorenstein surface quotient singularities by introducing a smaller, more informative algebra.
• It addresses the challenge of extracting geometric data—specifically the dual graph of the minimal resolution—from the representation theory of finite subgroups of GL(2, C).
• The objective is to generalize McKay’s classical correspondence, originally valid only for finite subgroups of SL(2, C), to all finite subgroups of GL(2, C).
• It seeks to establish a derived equivalence between the minimal resolution and the derived category of a finite-dimensional algebra whose quiver is determined by the resolution’s dual graph.
• The paper aims to clarify the homological structure of the reconstruction algebra, particularly its global dimension and projective dimension of simple modules.

Proposed method

• . The method uses Riemann-Roch and Serre duality to compute Ext groups, enabling a geometric proof of quiver and relation structure.
• It applies Wunram’s theory of special Cohen-Macaulay modules to define the reconstruction algebra as the endomorphism ring of the sum of indecomposable special CM modules over a rational surface singularity.
• The quiver is reconstructed combinatorially from the dual graph of the minimal resolution, with self-intersection numbers and continued fraction expansions of rational singularities guiding the construction.
• For each type of rational singularity (A, D, E, T, O, I), the paper provides explicit rules to build the quiver from the dual graph, including vertex additions and arrow multiplicities based on continued fraction coefficients.
• The proof leverages Bridgeland and Van den Bergh’s work on noncommutative resolutions and derived categories to show that the derived category of the resolution is equivalent to the derived category of the reconstruction algebra.
• The global dimension of the reconstruction algebra is computed via homological algebra, distinguishing between Gorenstein and non-Gorenstein cases.

Experimental results

Research questions

• RQ1 . Can the quiver of the reconstruction algebra be determined purely combinatorially from the dual graph of the minimal resolution of an affine complete rational surface singularity?
• RQ2 Does the endomorphism ring of the special CM modules over C[[x,y]]^G reconstruct the dual graph of the minimal resolution of C^2/G for any finite small subgroup G ≤ GL(2,C)?
• RQ3 What is the global dimension of the reconstruction algebra, and how does it depend on the Gorenstein property of the singularity?
• RQ4 How does the projective dimension of simple modules in the reconstruction algebra reflect the geometry of the resolution, particularly for non-Gorenstein singularities?
• RQ5 Can the derived category of the minimal resolution be realized as the derived category of a finite-dimensional algebra constructed from the representation theory of G?nowledgement.

Key findings

• . The quiver of the reconstruction algebra can be computed combinatorially from the dual graph of the minimal resolution, labeled with self-intersection numbers, for any affine complete rational surface singularity.
• For finite subgroups G ≤ GL(2,C), the special representations of G reconstruct the dual graph of the minimal resolution of C^2/G via the quiver of End_{C[[x,y]]^G}(⊕ρ special(ρ⊗C[[x,y]])^G), with the trivial representation vertex removed.
• The global dimension of the reconstruction algebra is 2 if the singularity is Gorenstein, and 3 otherwise.
• In the non-Gorenstein case, the simple module corresponding to the special vertex ⋆ has projective dimension 3, while all others have projective dimension 2; this asymmetry breaks the homologically homogeneous property.
• The derived category of the minimal resolution ˆX is equivalent to the derived category of the reconstruction algebra A.
• For each type of rational singularity (A, D, T, O, I), explicit combinatorial rules are provided to construct the quiver from the continued fraction expansion of the singularity, with arrow multiplicities determined by coefficients αi > 2 or >3.