Three Fermion Generations from Point-Group Symmetry in a Discrete Lattice Model

Abstract:This work presents a group-theoretic result: the symmetric cube of the T_{1u} vector representation of the O_h point group decomposes into exactly three irreducible representation types—A_{2u}, 2T_{1u}, and T_{2u}. The derivation uses only character theory, orthogonal projection, and symmetric tensor product decomposition, and is verified by Burnside's theorem and independent dimension counting. No free parameters are involved. The number of irreducible representation types coincides with the number of observed fermion generations in the Standard Model.