  Spinor Bose Einstein condensates (BEC) can be realized nowadays using
different atomic species of several spin values, offering unprecedented
opportunities to scrutinize the underlying physics of its spin phase domains
and of its quantum phase transitions. At sufficient low temperatures, lower
than the critical temperature, a fraction of thermally excited atoms of the
condensate can still interact with the whole system leading to spin-dependent
interactions that can modify the nature of its phase domains. In this work, we
characterize the thermal fraction of atoms of a spinorial BEC of general
spin-$f$ value, provided that its ground state lies in a given spin phase with
rotational symmetry. To that end, we use the Hartree-Fock approximation and a
method based on the Majorana stellar representation for mixed quantum states
and symmetry arguments. We consider the spin phases with usual point group
symmetries, including those with some exotic phases associated to the platonic
solids. The method leads to useful analytical expressions of the eigenspectrum
of the thermal cloud allowing us to study the admissible regions and multipolar
magnetic moments of the spin phases as a function of the temperature for
general spin values.
