  We study the spin-1 bilinear-biquadratic model on the complete graph of N
sites, i.e., when each spin is interacting with every other spin with the same
strength. Because of its complete permutation invariance, this Hamiltonian can
be rewritten as the linear combination of the quadratic Casimir operators of
su(3) and su(2). Using group representation theory, we explicitly diagonalize
the Hamiltonian and map out the ground-state phase diagram of the model.
Furthermore, the complete energy spectrum, with degeneracies, is obtained
analytically for any number of sites.
