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

TL;DR

This paper presents an algebraic model within Clifford algebra that naturally encodes three fermion generations and their Standard Model gauge symmetries, utilizing an embedded $S_3$ family symmetry.

Contribution

It introduces a novel Clifford algebra framework where three fermion generations are realized through an intrinsic $S_3$ symmetry without additional gauge bosons.

Findings

Reproduces Standard Model quantum numbers for three fermion generations.

Identifies gauge generators as elements commuting with the $S_3$ symmetry.

Models three generations within a single algebraic structure.

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.