  We consider insulating phases of cold spin-1 bosonic particles with
antiferromagnetic interactions, such as Na-23, in optical lattices. We show
that spin-exchange interactions give rise to several distinct phases, which
differ in their spin correlations. In two- and three- dimensional lattices
insulating phases with odd number of particles per site are always nematic. For
insulating states with even number of particles per site there is always a
spin-singlet phase and there may also be a first-order transition into the
nematic phase. The nematic phase breaks spin rotational symmetry, but preserves
time reversal symmetry, and has gapless spin-wave excitations. The spin singlet
phase does not break spin symmetry and has a gap to all excitations. In
one-dimensional lattices insulating phases with odd number of particles per
site always have a regime where translational symmetry is broken and the ground
state is dimerized. We discuss signatures of various phases in Bragg scattering
and time-of-flight measurements.
