Abstract: This work presents a group-theoretic result: the symmetric cube of the T₁ₔ vector representation of the Oₕ point group decomposes into exactly three irreducible representation types—A₂ₔ, 2T₁ₔ, and T₂ₔ. 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.
卓冰 蒋 (Fri,) studied this question.