Modelling three fermion generations with S₃ family symmetry within ℂℓ(8)

Abstract We present a model of three fermion generations with SU (3) × U (1) gauge symmetry constructed from the complex Clifford algebra ℂ ℓ (8), within which the discrete group S 3 acts as a family symmetry. ℂ ℓ (8) corresponds to the algebra of complex linear maps from the (complexification of the) Cayley-Dickson algebra of sedenions, 𝕊, to itself. The automorphism group of 𝕊 is G 2 × S 3 . We interpret S 3 , suitably embedded into ℂ ℓ (8), as a family symmetry. The gauge symmetry SU (3) × U (1) is invariant under S 3 . First-generation states are represented in terms of two even ℂ ℓ (8) semi-spinors, obtained from two minimal left ideals, related to each other via the order-two S 3 symmetry. The remaining two generations are obtained by applying the S 3 symmetry of order-three to the first generation, resulting in three linearly independent generations.