Coxeter Numbers as Keys to Quantum Group Symmetries — E8 Intelligence Research

FINDING: Coxeter groups provide the algebraic skeleton for root systems, Lie algebras, and quantum group symmetries, with the Coxeter number and dual Coxeter number governing modular S-matrix entries and fusion rules at roots of unity. MATH: - Coxeter number \( h \) = order of product of two generators in a Coxeter group; for \( A_n \), \( h = n+1 \). - Dual Coxeter number \( h^\vee \) = sum of positive roots' heights + 1; for \( A_n \), \( h^\vee = n+1 \). - At root of unity \( q = e^{2\pi i / k} \), the modular S-matrix of a quantum group \( U_q(\mathfrak{g}) \) has entries: \[ S_{\lambda\mu} = \text{const} \cdot \sum_{w \in W} \epsilon(w) \, q^{2(\lambda+\rho, w(\mu+\rho))} \] where \( \rho \) is the Weyl vector, \( W \) the Weyl group (a Coxeter group). - Key ratio: \( \frac{h^\vee}{h} \) appears in central charge \( c = \frac{k \dim \mathfrak{g}}{k + h^\vee} \) for WZW models. CONNECTION: - Root systems are crystallographic Coxeter groups; their Cartan matric Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com