Treatment of the exchange-interaction model by means of the symmetric group

The representation theory of the symmetric group is used to study the spin-S exchange-interaction model of ferromagnetism within the infinite-range approximation. The 2S order parameters are determined by the row lengths of the Young diagram that specifies the free-energy extrema. The set of solutions of the order-parameter equations has been fully explored. Stability analysis shows that one of the solutions represents the absolute minimum and describes the thermodynamically stable state. This solution coincides with the mean-field solution due to Chen et al. [Phys. Rev. B 46, 8323 (1992)]. The other nontrivial solutions correspond to saddle points in the free-energy surface, with consecutively increasing indices.