Octahedral Symmetry in SU(2) 3j-Symbols via A₁ Root System Decomposition — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The 3j-symbols of SU(2) encode the Clebsch–Gordan decomposition of the A₁ root system, with the adjoint representation (j=1) forming an octahedral (O_h) symmetry via the double cover Spin(3)→SO(3). | MATH: 3j-symbols: j1 & j2 & j3 m1 & m2 & m3 = (−1)j1−j2−m3(2j3+1)−1/2⟨ j1 m1; j2 m2 | j3 −m3⟩. Adjoint rep: j=1, generators ta = −iϵabc (structure constants of su(2) = A₁). Double cover: SU(2)/{± I} ≅ SO(3); octahedral group O ⊂ SO(3) lifts to binary octahedral group 2O ⊂ SU(2) of order 48. | CONNECTION: A₁ root system has two roots ±α with Cartan matrix [[2]] — its Weyl group is S2 (order 2), but the *adjoint* SU(2) action on su(2) gives the spherical harmonics Y1m (l=1), whose nodal surfaces form octahedral symmetry (the three axes). The 3j-symbols for j1=j2=j3=1 yield the structure constants \(\epsilon^{abc
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026octahedral,
title = {{Octahedral Symmetry in SU(2) 3j-Symbols via A₁ Root System Decomposition — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635280},
url = {https://zenodo.org/records/23635280},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Octahedral Symmetry in SU(2) 3j-Symbols via A₁ Root System Decomposition — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635280
UR - https://zenodo.org/records/23635280
AB - FINDING: The 3j-symbols of SU(2) encode the Clebsch–Gordan decomposition of the A₁ root system, with the adjoint representation (j=1) forming an octahedral (O_h) symmetry via the double cover Spin(3)→SO(3). | MATH: 3j-symbols: j_1 & j_2 & j_3 m_1 & m_2 & m_3 = (−1)^j_1−j_2−m_3(2j_3+1)^−1/2⟨ j_1 m_1; j_2 m_2 | j_3 −m_3⟩. Adjoint rep: j=1, generators t^a = −iϵ^abc (structure constants of su(2) = A₁). Double cover: SU(2)/{± I} ≅ SO(3); octahedral group O ⊂ SO(3) lifts to binary octahedral group 2O ⊂ SU(2) of order 48. | CONNECTION: A₁ root system has two roots ±α with Cartan matrix [[2]] — its Weyl group is S_2 (order 2), but the *adjoint* SU(2) action on su(2) gives the spherical harmonics Y_1m (l=1), whose nodal surfaces form octahedral symmetry (the three axes). The 3j-symbols for j_1=j_2=j_3=1 yield the structure constants \(\epsilon^{abc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -