Multispin interactions for multistate spin models

The multispin interactions for spin models with a transitive symmetry group and M states are considered. It is shown that the triangle-symmetric three-spin interactions depend only on the orbits in the set of all true triangles under the action of the symmetry group; this is analogous to the pair interaction case, which only depends on the orbits in the set of all edges. For four-spin interactions, no such property is shown to hold. The three-spin interactions for all spin models with a completely permissible symmetry group are listed for M 10. As an application, a dual transformation for the triangular lattice with a three-spin interaction in half of the triangles is studied; it is shown that this is formally the same as the dual transformation on the square lattice for a special type of group associated with the three-spin interaction. Solutions for the self-dual parts of the state space are outlined for a number of models, yielding information on the points, where a star-triangle transformation is exact.