[Paper Review] McKay correspondence

Miles Reid|arXiv (Cornell University)|Feb 25, 1997

Homotopy and Cohomology in Algebraic Topology76 citations

TL;DR

This paper formulates the McKay correspondence as a natural bijection between irreducible representations of a finite subgroup $G \subset \operatorname{SL}(n,\mathbb{C})$ and homology classes of a crepant resolution $Y$ of the quotient singularity $\mathbb{C}^n/G$, establishing a deep link between representation theory and algebraic geometry. The key contribution is a conjectural tautological framework—supported by examples and Nakamura’s $G$-Hilbert scheme—for realizing this correspondence via toric geometry and stringy invariants, particularly in dimension three.

ABSTRACT

This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how the McKay correspondence for finite subgroups of SL(n,C) relates to mirror symmetry. The main aim is to give numerical examples of how the 2 McKay correspondences (1) representations of G cohomology of resolution (2) conjugacy classes of G homology must work, and to restate my 1992 Conjecture as a tautology, like cohomology or K-theory of projective space. Another aim is to give an introduction to Nakamura's results on the Hilbert scheme of G-clusters, following his preprints and his many helpful explanations. This is partly based on joint work with Y. Ito, and has benefited from encouragement and invaluable suggestions of S. Mukai.

Motivation & Objective

• To establish a natural correspondence between irreducible representations of a finite group $G \subset \operatorname{SL}(n,\mathbb{C})$ and homology classes of a crepant resolution $Y$ of $\mathbb{C}^n/G$.
• To provide a geometric and cohomological foundation for the McKay correspondence in higher dimensions, especially $n=3$, beyond the known 2D case.
• To connect the McKay correspondence with mirror symmetry and string theory via Vafa’s stringy Euler characteristic and the conjecture that $e_{\text{string}}(M,G) = e(Y)$.
• To propose that the McKay correspondence can be derived as a tautological statement in cohomology or K-theory, akin to the structure of $\mathbb{P}^n$.
• To explore the role of Nakamura’s $G$-Hilbert scheme as a distinguished crepant resolution, especially in 3-folds, and its realization as a toric variety.

Proposed method

• Use of the GSp–V sheaves (tautological sheaves) as a key tool to relate representation theory to cohomology of $Y$.
• Application of Nakamura’s $G$-Hilbert scheme construction to define a canonical crepant resolution $Y = G\text{-Hilb}$, especially for $n=3$, replacing non-unique minimal models.
• Employment of toric geometry and triangulations of the junior simplex to describe the resolution $Y$ as a toric variety with explicit fan structures.
• Use of Newton polygons and lattice point analysis to classify $G$-clusters and describe the geometry of the resolution via tripods and jigsaw-like patching of triants.
• Adoption of the stringy Euler characteristic formula $e_{\text{string}}(M,G) = \sum_{H \subset G} e(X_H) \cdot \#\{\text{conj. classes in } H\}$ to link topological invariants to group character theory.
• Construction of the McKay quiver and its fundamental domain to encode the homological structure of $Y$, with explicit equations for $G$-clusters in local charts.

Experimental results

Research questions

• RQ1 Can the McKay correspondence be formulated as a tautological equivalence between the representation theory of $G$ and the homology of a crepant resolution $Y$?
• RQ2 Does Nakamura’s $G$-Hilbert scheme provide a canonical crepant resolution in dimension $n=3$, resolving the non-uniqueness issues of minimal models?
• RQ3 How does the stringy Euler characteristic $e_{\text{string}}(M,G)$ relate to the Euler characteristic of the crepant resolution $Y$?
• RQ4 What is the geometric meaning of the two distinct patching rules (‘up’ and ‘down’ triads) in the toric description of $G$-Hilb?
• RQ5 Can the McKay correspondence be derived from a deeper principle, such as $G$-mirror symmetry, in the zero-dimensional limit of Calabi–Yau geometry?

Key findings

• The McKay correspondence is conjectured to hold as a tautological equivalence between irreducible representations of $G$ and a basis of $H^*(Y,\mathbb{Z})$, with compatibility to cup product and duality.
• For $n=3$, a weak version of the conjugacy class correspondence (2) is proven in [IR], suggesting a path toward full generalization.
• The $G$-Hilbert scheme provides a distinguished crepant resolution of $\mathbb{C}^3/G$, resolving the non-uniqueness problem of minimal models in 3-folds.
• The resolution $Y = G\text{-Hilb}$ is realized as a toric variety via triangulation of the junior simplex, with explicit fan structures encoding the geometry.
• The geometry of $G$-clusters is encoded in Newton polygons (tripods), with two distinct patching rules corresponding to $\uparrow$ and $\downarrow$ configurations, related by layer-shaving operations.
• The fundamental domain of the McKay quiver for $\frac{1}{37}(1,5,31)$ yields explicit equations such as $x^4 = \lambda y^2 z$, $y^4 = \mu x z^3$, $z^5 = \nu x^2 y$, describing the local structure of the resolution.