Hyperbolic Coxeter groups for triangular Potts models

Abstract

The symmetry groups, generated by the inversion relations of lattice models of statistical mechanics on triangular lattices, are analysed for vertex models and for the standard scalar Potts model with two- and three-site interactions. These groups are generated by three inversion relations and are seen to be generically very large ones: hyperbolic groups. Two situations for which the representations of these groups degenerate into smaller ones, hopefully compatible with integrability, are considered. The first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left- or right-hand side of a Yang-Baxter relation. In this case the representation of the group is isomorphic, up to a semi-direct product by a finite group, to Z*Z. The second reduction for q-state Ports models occurs for particular values of q, the so-called Tutte-Beraha numbers. For this model, algebraic varieties, including the known ferromagnetic critical variety, happen to be invariant under such large groups of symmetries. As a byproduct, this analysis provides nice birational representations of hyperbolic Coxeter groups.