  We study Hubbard models for ultracold bosonic or fermionic atoms loaded into
an optical lattice. The atoms carry a high spin $F>1/2$, and interact on site
via strong repulsive Van der Waals forces. Making convenient rearrangements of
the interaction terms, and exploiting their symmetry properties, we derive low
energy effective models with nearest-neighbor interactions, and their
properties. We apply our method to $F=3/2$, and 5/2 fermions on two-dimensional
square lattice at quarter, and 1/6 fillings, respectively, and investigate
mean-field equations for repulsive couplings. We find for $F=3/2$ fermions that
the plaquette state appearing in the highly symmetric SU(4) case does not
require fine tuning, and is stable in an extended region of the phase diagram.
This phase competes with an SU(2) flux state, that is always suppressed for
repulsive interactions in absence of external magnetic field. The SU(2) flux
state has, however, lower energy than the plaquette phase, and stabilizes in
the presence of weak applied magnetic field. For $F=5/2$ fermions a similar
SU(2) plaquette phase is found to be the ground state without external magnetic
field.
