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

Abstract

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: j₁ ⊗ j₂ = ⨁_{J=|j₁−j₂|}^j₁+j₂ J (step 1). The A₁ root system has one positive root α, Cartan integers ⟨ α,α^∨⟩ = 2, Weyl group W(A₁) = ℤ₂ acting by reflection s_α(λ) = λ − ⟨ λ,α^∨⟩ α. Clebsch–Gordan coefficients are the structure constants of this decomposition, satisfying orthogonality ∑_m₁,m₂ C^JM_{j₁m₁j₂m₂} C^J'M'_{j₁m₁j₂m₂} = δ_JJ'δ_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

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BibTeX

@misc{caldin2026a₁,
  title = {{A₁ Root System and Weyl Group Z₂ in SU(2) Tensor Product Decomposition — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23587162},
  url = {https://zenodo.org/records/23587162},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A₁ Root System and Weyl Group Z₂ in SU(2) Tensor Product Decomposition — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/23
PB  - Zenodo
DO  - 10.5281/zenodo.23587162
UR  - https://zenodo.org/records/23587162
AB  - 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 C^JM_j_1m_1j_2m_2 C^J'M'_j_1m_1j_2m_2 = δ_JJ'δ_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
ER  -

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