  We investigate the structure of the spectrum of antiferromagnetically coupled
spin-1 bosons on a square lattice using degenerate perturbation theory and
exact diagonalizations of finite clusters. We show that the superfluid phase
develops an Anderson tower of states typical of nematic long-range order with
broken SU(2) symmetry.We further show that this order persists into the Mott
insulating phase down to zero hopping for one boson per site, and down to a
critical hopping for two bosons per site, in agreement with mean-field and
Quantum Monte Carlo results. The connection with the transition between a
fragmented condensate and a polar one in a single trap is briefly discussed.
