  Lattice symmetries are central to the characterization of electronic
topology. Recently, it was shown that Green's function eigenvectors form a
representation of the space group. This formulation has allowed the
identification of gapless topological states even when quasiparticles are
absent. Here we demonstrate the profundity of the framework in the extreme
case, when interactions lead to a Mott insulator, through a solvable model with
long-range interactions. We find that both Mott poles and zeros are subject to
the symmetry constraints, and relate the symmetry-enforced spectral crossings
to degeneracies of the original non-interacting eigenstates. Our results lead
to new understandings of topological quantum materials and highlight the
utility of interacting Green's functions toward their symmetry-based design.
