Weyl Reflections and Cartan Integers in Anyon Fusion Rules — E8 Intelligence Research

FINDING: Weyl group reflections and Cartan integers form the algebraic backbone of non-Abelian anyon fusion rules, linking root system geometry to quantum group representations in topological quantum computing. | MATH: Weyl group \(W\) generated by reflections \(s_i\) with Cartan integers \(a_{ij} = 2 \langle \alpha_i, \alpha_j \rangle / \langle \alpha_i, \alpha_i \rangle\); fusion rules for anyons correspond to tensor product decompositions of quantum group representations at roots of unity, e.g., \( \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c \) where \(N_{ab}^c\) are fusion coefficients derived from quantum \(R\)-matrix and \(q\)-deformed Clebsch-Gordan series. | CONNECTION: Root system lattices (e.g., \(A_n, D_n, E_6, E_7, E_8\)) exhibit crystallographic symmetries; Cartan integers are integers 0,1,2,3; the golden ratio \(\phi = 1.618\) appears in quantum dimensions for \(SU(2)_k\) anyons (e.g., Fibonacci anyon quantum dimension \(d = \phi\)), and base-60 emerges in modular data Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com