A relationship connecting finite group representation data for certain linear actions (notably finite subgroups of SL(2,C)) with geometric invariants of quotient singularities and their resolutions, often realized by matching the representation graph to intersection or Dynkin-type graphs and extended to derived-category and string-theoretic contexts.

Principle

Discrete symmetry of a linear action is reflected in the geometry of the quotient: irreducible representations and their tensor relations produce a combinatorial graph (the McKay graph) that matches the configuration of exceptional cycles or Dynkin diagrams appearing in minimal/crepant resolutions of the singularity.

Demonstration

For a finite subgroup G ⊂ SL(2,C), the McKay graph built from the representation ring is the affine ADE Dynkin diagram associated to the minimal resolution of C^2/G; nodes correspond to irreducible representations and exceptional curves, and adjacency encodes tensoring with the standard 2-dimensional representation.

Misapplication

Applying the naive two-dimensional McKay dictionary in higher dimensions or for groups not satisfying special determinant/crepant hypotheses without invoking derived categories, stacks, or additional hypotheses leads to incorrect identifications between representations and geometric features.

Consequence

Properly applied, the correspondence links representation theory to geometry: it predicts intersection patterns of exceptional divisors, guides constructions of resolutions and equivalences of derived categories, and informs invariants like orbifold cohomology and stringy Hodge numbers.

Reversal

Instead of starting from a group and predicting geometry, one can begin with the geometry of a resolution and recover constraints on possible symmetry groups; reversing emphasizes reconstruction of group-theoretic data from geometric invariants.

Boundary

Classic McKay is for finite subgroups of SL(2,C); higher-dimensional generalizations require crepant resolutions, derived-category enhancements, or stack-theoretic formulations and may fail or need modification when these are absent.

Semantic Tension

There is tension between the original combinatorial ADE identification in dimension two and broader derived/crepant generalizations: the simple node↔curve dictionary competes with more sophisticated derived equivalences and with purely representation-theoretic invariants like character tables.

Synthesis

The McKay correspondence ties finite-group representation data to the geometry of quotient singularities: in its classic form the McKay graph reproduces Dynkin diagrams of resolutions, and in extended forms it forges deep equivalences between representation theory, geometry and homological invariants.