Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review
Abstract
The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article reviews an algebraic framework in which an intrinsic S3 family symmetry relates three gauge-equivalent fermion sectors. The framework is motivated by division- and Clifford-algebra constructions in which the complex octonions and ℂℓ(6) organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley--Dickson sequence, the sedenions provide an intrinsic S3 automorphism structure, while their complexified left-multiplication operators generate an associative algebra isomorphic to ℂℓ(8). In the ℂℓ(8) formulation, the order-three family action generates three linearly independent fermion sectors while leaving a single su(3)C⊕u(1)em gauge algebra. Extension to ℂℓ(10) incorporates a single su(3)C⊕su(2)L⊕u(1)Y gauge algebra, with each generation including a right-handed neutrino sterile under the SM gauge interactions. We place the construction in context by comparing it with representative algebraic and family-symmetry approaches to three generations, including triality-based proposals, and clarify why the family symmetry used here is distinct from standard Spin(8) triality despite the common appearance of ℂℓ(8) and S3. The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family-symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.
Related
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BibTeX
@misc{gresnigt2026sedenions,
title = {{Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review}},
author = {Niels Gresnigt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10569},
eprint = {2609.10569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10569},
}
RIS
TY - UNPB
TI - Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review
AU - Niels Gresnigt
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.10569
UR - https://arxiv.org/abs/2609.10569
AB - The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article reviews an algebraic framework in which an intrinsic S_3 family symmetry relates three gauge-equivalent fermion sectors. The framework is motivated by division- and Clifford-algebra constructions in which the complex octonions and ℂℓ(6) organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley--Dickson sequence, the sedenions provide an intrinsic S_3 automorphism structure, while their complexified left-multiplication operators generate an associative algebra isomorphic to ℂℓ(8). In the ℂℓ(8) formulation, the order-three family action generates three linearly independent fermion sectors while leaving a single su(3)_C⊕u(1)_em gauge algebra. Extension to ℂℓ(10) incorporates a single su(3)_C⊕su(2)_L⊕u(1)_Y gauge algebra, with each generation including a right-handed neutrino sterile under the SM gauge interactions. We place the construction in context by comparing it with representative algebraic and family-symmetry approaches to three generations, including triality-based proposals, and clarify why the family symmetry used here is distinct from standard Spin(8) triality despite the common appearance of ℂℓ(8) and S_3. The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family-symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.
ER -