Wigner–Eckart Theorem: Factorization of Tensor Matrix Elements — E8 Intelligence Research

FINDING: The search results are dominated by pedagogical videos and a single arXiv paper on finite magnetic groups; no direct result for the specific arXiv:2007.03539 was returned. The core mathematical content is the Wigner–Eckart theorem itself: matrix elements of irreducible tensor operators factor into a Clebsch–Gordan coefficient (geometry/coupling) and a reduced matrix element (physics/dynamics). | MATH: ⟨j′ m′| T^(k)_q |j m⟩ = ⟨j k; m q | j′ m′⟩ · (1/√(2j′+1)) ⟨j′‖T^(k)‖j⟩. The Clebsch–Gordan coefficient encodes SU(2) coupling; the reduced matrix element is independent of m, m′, q. For finite magnetic groups (arXiv:0911.0276v1), the theorem generalizes to magnetic point groups, requiring projective representations and factor systems (cocycles) — the reduced matrix element becomes invariant under the group's corepresentations. | CONNECTION: The Clebsch–Gordan coefficients are intimately tied to SU(2) representation theory, whose weight spaces form the root system A₁. The 3j-symbo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com