A₁ Root System and Weyl Group Z₂ in SU(2) Tensor Product Decomposition — E8 Intelligence Research

FINDING: SU(2) tensor product decomposition is governed by the A₁ root system and its Weyl group Z₂, with Clebsch–Gordan coefficients encoding the reduction of tensor products into irreducibles. | MATH: For su(2), irreducible representations labeled by spin j (dimension 2j+1). Tensor product rule: \( j_1 ⊗ j_2 = J=|j_1-j_2|j_1+j_2 J \) (step 1). The A₁ root system has one positive root α, Cartan integers \( α,α^ = 2 \), Weyl group \( W(A_1) = Z_2 \) acting by reflection \( s_α(λ) = λ - λ,α^ α \). Clebsch–Gordan coefficients are the structure constants of this decomposition, satisfying orthogonality \( ∑m_1,m_2 CJMj_1m_1j_2m_2 CJ'M'j_1m_1j_2m_2 = δJJ'δMM' \). Cartan decomposition of su(N) (arXiv:quant-ph/0603190) generalizes this via conjugate partitions and quotient algebras. | CONNECTION: The A₁ root system is the simplest crystallographic ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com