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 \( h = \frac{2m}{m-2} \) for dihedral groups; for simple Lie algebras, \( h = \sum_{i} a_i^\vee \) (dual Coxeter number). - Dual Coxeter number \( h^\vee = \frac{2 \langle \alpha_{\text{max}}, \alpha_{\text{max}} \rangle}{\langle \alpha_{\text{min}}, \alpha_{\text{min}} \rangle} \) for simply laced types \( A_n, D_n, E_6, E_7, E_8 \). - Quantum group \( U_q(\mathfrak{g}) \) at \( q = e^{2\pi i / \ell} \) yields modular S-matrix entries: \[ S_{\lambda \mu} = \frac{1}{\sqrt{|C|}} \sum_{w \in W} \epsilon(w) \, q^{2 \langle w(\lambda+\rho), \mu+\rho \rangle} \] where \( W \) is the Weyl group (a Coxeter group), \( \rho \) the Weyl vector, \( \ell = k + h^\vee \). - Roots of unity condition: \( q^{h^\vee} = -1 \) for certain modular categories (e.g., Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com