  We study the variational perturbations for the mean-field solution of an
interacting spinor system with underlying rotational symmetries. An approach
based upon the Majorana stellar representation for mixed states and group
theory is introduced to this end. The method reduces significantly the unknown
degrees of freedom of the perturbation, allowing us a simplified and direct
exploration on emergent physical phenomena. We apply it to characterize the
phases of a spin-1 Bose-Einstein condensate and to study the behavior of these
phases with entropy. The spin-2 phase diagram was also investigated within the
Hartree-Fock approximation, where a non-linear deviation of the cyclic-nematic
phase boundary with temperature is predicted.
