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

Abstract

We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group, in which an intrinsic $S_3$ 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 $\mathbb{C}\ell(10)$, while the three-generation structure arises from an embedded discrete $S_3$ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this $S_3$ 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.