Coxeter Groups Bridge Root Symmetries and Quantum Topological Invariants — E8 Intelligence Research

FINDING: Coxeter groups encode root system symmetries and quantum group modular data via dual Coxeter numbers at roots of unity, linking finite reflection groups to topological quantum invariants. MATH: - Coxeter number \( h = \frac{2|\Phi^+|}{n} \) for rank \( n \); dual Coxeter number \( h^\vee = \langle \theta, \theta \rangle^{-1} h \) (for simply laced, \( h = h^\vee \)). - For \( \mathfrak{sl}_2 \): \( h = 2 \); for \( E_8 \): \( h = 30 \). - Quantum group \( U_q(\mathfrak{g}) \) at \( q = e^{2\pi i / \ell} \) yields modular S-matrix entries \( S_{ab} = \frac{1}{\sqrt{\text{rank}}} \sum_{w \in W} \epsilon(w) q^{\langle w(\lambda_a + \rho), \lambda_b + \rho \rangle} \). - Roots of unity condition: \( \ell = k + h^\vee \) for level \( k \). - Excess function \( e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \mid w = xy, x^2 = y^2 = 1\} \) measures minimal length factorization in Coxeter groups. CONNECTION: - Coxeter numbers \( h \) for \( A_n, D_n, E_6, E_7, E_8 \) are 3,4,6 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com