Spin models with an eigenvalue of small multiplicity

Spin models were introduced by Jones as statistical mechanical models which give link invariants in knot theory. In this paper we study spin models having an eigenvalue of multiplicity one or two. For a spin model S with an eigenvalue of multiplicity m = 1, 2, we shall show that S splits into a direct product S = S1 × S2 with S1 the Ising model (when m = 1) or a cyclic model (when m = 2) under some additional conditions which are essentially needed. We shall give some examples which do not satisfy these additional conditions and which do not split into a direct product with the Ising (or cyclic) model. In the proof we use a new method to determine the Boltzmann weights which enables us to localize the star-triangle relation.