Inversion relations and symmetry groups for Potts models on the triangular lattice

Inversion relations are obtained for the standard scalar q-state triangular Potts model with two- and three-spin interactions, generalizing previously known results for two-spin interaction models. It is shown that these inversion relations generate a group of symmetries of the model which is naturally represented in terms of birational transformations in a four dimensional parameter space. This group of birational transformations is generically a very large one, namely a hyperbolic Coxeter group. In this framework of very large groups of symmetries, a remarkable situation pops out: the one for which q corresponds to Tutte-Beraha numbers.