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

Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra C(8), by incorporating an unbroken U(1)ₑₘ gauge symmetry. The algebra C(8) is the multiplication algebra of the complexification of the Cayley-Dickson algebra of sedenions, S. Previous work represented three generations of fermions with SU(3)C colour symmetry, permuted by an S₃ symmetry of order-three. However, it proved difficult to find a $U(1)$ generator that assigns the correct electric charge to all states. By generalising the embedding of the order-three S₃ automorphism of S into C(8), we include an S₃-invariant $U(1)$ that correctly assigns electric charge. First generation states are represented in terms of two even C(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 C(8), the three generations are linearly independent, in contrast to earlier work.