  $U(n)$ is a semi-direct product group that is characterized by non-trivial
homomorphisms mapping $U(1)$ into the automorphism group of $SU(n)$. For
$U(3)$, there are three non-trivial 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. By employing a
$\mathrm{mod}\,3$ permutation of the vector space carrying the defining
representation induced by a ``large'' gauge transformation, 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.
