Binary Icosahedral Group, H3, and E8: McKay Correspondence via Golden Ratio — E8 Intelligence Research

Abstract

FINDING: The binary icosahedral group 2I is a finite subgroup of SU(2) whose character table and representation theory are deeply linked to the Coxeter group H3 (symmetry of the icosahedron) and the E8 lattice via the McKay correspondence and the golden ratio φ. MATH: - Coxeter group H3: order 120, generated by reflections in 3D, root system with simple roots at angles π/2, π/3, π/5. - Binary icosahedral group 2I: order 120, double cover of the icosahedral rotation group (order 60). - McKay correspondence: The extended Dynkin diagram of E8 (affine E8) is the McKay graph of 2I. The dimensions of irreducible representations of 2I are: 1, 2, 3, 4, 5, 6, 4, 2, 1 — summing to 120. The Cartan matrix of E8 is encoded in the tensor product decomposition of these representations. - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the character values of 2I: e.g., χ(5D) = φ, χ(5D*) = φ⁻¹ = φ-1 ≈ 0.618. - E8 lattice: root system of 240 vectors, Coxeter number 30, theta series with modular Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com