McKay Correspondence: Binary Icosahedral Group Representations Map to Affine E8 Dynkin Diagram — E8 Intelligence Research

FINDING: The McKay correspondence establishes a bijection between irreducible representations of finite subgroups of SU(2) and the nodes of affine Dynkin diagrams, with the binary icosahedral group 2I mapping to the affine E8 diagram. | MATH: For Γ ⊂ SU(2) finite, irreducible reps {ρᵢ} ↔ nodes of affine Dynkin diagram. For binary icosahedral group 2I (order 120): 9 irreducible representations (dimensions 1,2,3,4,5,6,4,2,3) ↔ 9 nodes of affine E8 (E8⁽¹⁾ has 9 nodes: 8 + 1 affine node). McKay graph: tensor with 2-dim fundamental rep → adjacency matrix = Cartan matrix of affine E8. Coxeter number h(E8) = 30, |2I| = 120 = 4h. Exponents of E8: 1,7,11,13,17,19,23,29. | CONNECTION: The icosahedron's rotational symmetry group A5 (order 60) has character table dimensions {1,3,3,4,5}. Binary lift 2I → SU(2) double cover. The affine E8 Dynkin diagram encodes the tensor product decomposition of 2I representations. E8 root system: 240 roots in 8D, kissing number 240, densest lattice packing in 8D ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com