We study, using quantum Monte Carlo (QMC) simulations, the ground state
properties of spin-1 bosons trapped in a square optical lattice. The phase
diagram is characterized by the mobility of the particles (Mott insulating or
superfluid phase) and by their magnetic properties. For ferromagnetic on-site
interactions, the whole phase diagram is ferromagnetic and the Mott
insulators-superfluid phase transitions are second order. For antiferromagnetic
on-site interactions, spin nematic order is found in the odd Mott lobes and in
the superfluid phase. Furthermore, the superfluid-insulator phase transition is
first or second order depending on whether the density in the Mott is even or
odd. Inside the even Mott lobes, we observe a singlet-to-nematic transition for
certain values of the interactions. This transition appears to be first order