Algebraic realisation of three fermion generations with S3 family and unbroken gauge symmetry from Cℓ(8)

Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra Cℓ(8) by incorporating a U(1)_em gauge symmetry. The algebra Cℓ(8) corresponds to the algebra of complex linear maps from the (complexification of the) Cayley–Dickson algebra of sedenions, S, to itself. Previous work represented three generations of fermions with SU(3)_C colour symmetry permuted by an S₃ symmetry of order-three, but failed to include a U(1) generator that assigns the correct electric charge to all states. Furthermore, the three generations suffered from a degree of linear dependence between states. By generalising the embedding of the discrete group S₃, corresponding to automorphisms of S, into Cℓ(8), we include an S₃-invariant U(1) that correctly assigns electric charge. First-generation states are represented in terms of two even Cℓ(8) semi-spinors, obtained from two minimal left ideals, related to each other via the order-two S₃ symmetry. The remaining two generations are obtained by applying the S₃ symmetry of order-three to the first generation. In this model, the gauge symmetries, SU(3)_C×U(1)_em, are S₃-invariant and preserve the semi-spinors. As a result of the generalised embedding of the S₃ automorphisms of S into Cℓ(8), the three generations are now linearly independent.