  We consider many-body states of bosonic spinor atoms which, at the mean-field
level, can be characterized by a single-particle wave function. Such states
include BEC phases and insulating Mott states with one atom per site. We
describe and apply a classification scheme that makes explicit spin symmetries
of such states and enables one to naturally analyze their collective modes and
topological excitations. Quite generally, the method allows classification of a
spin F system as a polyhedron with 2F vertices. After discussing the general
formalism we apply it to the many-body states of bosons with hyperfine spins
two and three. For spin-two atoms we find the ferromagnetic state, a continuum
of nematic states, and a state having the symmetry of the point group of the
regular tetrahedron. For spin-three atoms we obtain similar ferromagnetic and
nematic phases as well as states having symmetries of various types of
polyhedra with six vertices: the hexagon, the pyramid with pentagonal base, the
prism, and the octahedron.
