The Three Faces of U(3)

U(n) is a semi-direct product group characterized by nontrivial homomorphisms mapping U(1) into the automorphism group of SU(n). For U(3), there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of U(3) Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a mod3 permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U(3) quantum numbers. One can interpret this as three species of matter fields in the defining representation.