Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry

2026/01/09 by Niels Gresnigt · 1 voice · 4 citations
Physics and Astronomy · #physics.gen-ph

paper · pdf

arxiv published 2026/01/09 · arxiv updated 2026/04/13

Abstract

We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model SU(3)_C× SU(2)_L× U(1)_Y gauge group, in which an intrinsic S₃ family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra ℂℓ(10), while the three-generation structure arises from an embedded discrete S₃ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this S₃ action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions.

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Cited by

• Higgs and Yukawa Structure in a Clifford Algebra Model with Three Generations and S₃ Family Symmetry

Discussions

• Niels Gresnigt: Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry https://arxiv.org/abs/2601.07857 https://arxiv.org/pdf/2601.07857 https://arxiv.org/html [bsky, 0 points, 0 comments]

Related

• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• Higgs and Yukawa Structure in a Clifford Algebra Model with Three Generations and S₃ Family Symmetry
• Clifford algebra implying three fermion generations revisited
• Generations: Three Prints, in Colour
• Three Generations and a Trio of Trialities