McKay Correspondence

Finite subgroups of SU(2) ↔ ADE Dynkin diagrams ↔ Affine Lie algebras

Dynkin diagram / McKay quiver
Orbit of (1,0) ∈ ℂ² under group action
ADE TypeΓ ⊂ SU(2)Order |Γ|Singularity ℂ²/ΓAffine Lie algebraCoxeter # h
McKay's observation (1980): For each irrep ρᵢ of Γ, tensor with the 2D defining rep V: V⊗ρᵢ = ⊕ aᵢⱼ ρⱼ. The matrix (aᵢⱼ) is exactly the adjacency matrix of the corresponding affine Dynkin diagram.
Resolution: The minimal resolution of ℂ²/Γ has exceptional divisors arranged as the Dynkin diagram — each exceptional ℙ¹ corresponds to a non-trivial irrep.
ADE=ADE: Same diagrams classify simple Lie algebras, surface singularities, finite reflection groups, and binary polyhedral groups.