Coxeter Groups Link Root Symmetries to Quantum S-Matrices — E8 Intelligence Research

FINDING: Coxeter groups encode root-system symmetries that directly determine quantum group modular S-matrices at roots of unity, linking finite reflection groups to topological quantum field theories. MATH: - Coxeter number \ (h = 2mm-2 \) for dihedral groups; for simple Lie algebras, \ (h = ₈ aᵢ^ \) (dual Coxeter number). - Dual Coxeter number \ (h^ = 2 ₌₀ₗ, _₌₀ₗ ₌₈₍, ₌₈₍ \) for simply laced types \ (Aₙ, Dₙ, E₆, E₇, E₈ \). - Quantum group \ (Uq (g) \) at \ (q = e^2 i / \) yields modular S-matrix entries: \ S = 1|C| ₖ ₖ (w) \, q^2 w (+), + \ where \ (W \) is the Weyl group (a Coxeter group), \ (\) the Weyl vector, \ (= k + h^ \). - Roots of unity condition: \ (q^h^ = -1 \) for certain modular categories (e. g. , Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com