A Review on the McKay Correspondence

The McKay correspondence is a deep and beautiful connection that links several areas of mathematics, including group theory, algebraic geometry, and representation theory. Discovered by John McKay in 1979, it has since been generalized and extensively studied. The original form of the correspondence relates finite subgroups of SL ( 2 , ℂ ) $$\mathrm {SL}(2,\mathbb {C})$$ to Kleinian singularities. Specifically, it establishes a bijection between the non-trivial irreducible representations of a finite subgroup Γ $$\varGamma $$ of SL ( 2 , ℂ ) $$\mathrm {SL}(2,\mathbb {C})$$ and the irreducible components of the exceptional divisor in the minimal resolution of the associated Kleinian singularity, X = ℂ 2 ∕ Γ $$X = \mathbb {C}^2 / \varGamma $$ . This chapter begins by reviewing the classical McKay correspondence for Kleinian singularities, then introduces the necessary definitions and results to describe the reinterpretation given by Artin and Verdier through the Riemann-Hilbert correspondence. In this reinterpretation, the McKay correspondence becomes a bijection between non-trivial indecomposable reflexive O X $$\mathcal {O}_X$$ -modules and the irreducible components of the exceptional divisor in the minimal resolution of X. Next, we explore the generalization of the McKay correspondence to rational singularities. We continue with the classification of reflexive modules of rank one over rational and minimally elliptic singularities. Finally, we examine the classification of special reflexive modules on Gorenstein singularities and discuss the deformation theory of full sheaves.