  For any symmorphic magnetic space group $G$, it is proven that topological
band insulators with vanishing first Chern numbers cannot have a groundstate
composed of a single, energetically-isolated band. This no-go statement implies
that the minimal dimension of tight-binding Hamiltonians for such topological
insulators is four if the groundstate is stable to addition of trivial bands,
and three if the groundstate is unstable. A sure-fire recipe is provided to
design models for Chern and unstable topological insulators by splitting
elementary band representations; this recipe, combined with
recently-constructed Bilbao tables on such representations, can be systematized
for mass identification of topological materials. All results follow from our
theorem which applies to any single, isolated energy band of a $G$-symmetric
Schr\"odinger-type or tight-binding Hamiltonian: for such bands, being
topologically trivial is equivalent to being a band representation of $G$.
