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: \( j_1 \otimes j_2 = \bigoplus_{J=|j_1-j_2|}^{j_1+j_2} J \) (step 1). The A₁ root system has one positive root α, Cartan integers \( \langle \alpha,\alpha^\vee\rangle = 2 \), Weyl group \( W(A_1) = \mathbb{Z}_2 \) acting by reflection \( s_\alpha(\lambda) = \lambda - \langle \lambda,\alpha^\vee\rangle \alpha \). Clebsch–Gordan coefficients are the structure constants of this decomposition, satisfying orthogonality \( \sum_{m_1,m_2} C^{JM}_{j_1m_1j_2m_2} C^{J'M'}_{j_1m_1j_2m_2} = \delta_{JJ'}\delta_{MM'} \). 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