{
  "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: \\( 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\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bwt8tqw4axbgx.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23587162",
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  "id": "bwt8tqw4axbgx",
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  "landing_url": "https://zenodo.org/records/23587162",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bwt8tqw4axbgx.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T12:21:51Z",
      "native_id": "23587162",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23587162"
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  ],
  "subjects": [],
  "title": "A₁ Root System and Weyl Group Z₂ in SU(2) Tensor Product Decomposition — E8 Intelligence Research",
  "title_text": "A₁ Root System and Weyl Group Z₂ in SU(2) Tensor Product Decomposition — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-06T12:21:51Z",
  "url": "https://research.draconitic.com/paper/bwt8tqw4axbgx",
  "venue": "Zenodo",
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}