Fundamental fermion representations in generalisations of the SU(5) grand unified gauge theory

Generalisations of the SU(5) grand unified gauge theory are discussed. It is assumed that the gauge group is simple, and that the theory is both anomaly free and asymptotically free. All possible fundamental fermion representations are determined given that the fermions are massless at the unification level, but acquire mass at the level of the exact SU(3) × U(1) symmetry. No a priori restriction to standard colour SU(3) representations is made. It is found that E₆, SO(10) and SU(n) with n ? 5 are the only acceptable gauge groups. Standard colour solutions are legion, but dull, incorporating at the SU(5) level, nothing other than p generations or copies of the familiar representation $\text{10}\text{+}\text{5}$. Exotic colour solutions are sparse, but interesting. Nine of these, all associated with SO(10), can accomodate those quarks and leptons currently thought to be fundamental, along with such things as colour sextets and octets, as well as doubly charged leptons.