Hyperbolic coxeter groups, symmetry group invariants for lattice models in statistical mechanics and the Tutte-Beraha numbers

The symmetry groups, generated by the inversion relations of lattice models of statistical mechanics, are analysed for vertex models and for the standard scalar Potts model with two and three site interactions on triangular lattices. These groups are generated by three inversion relations and are noticeably generically very large ones: hyperbolic groups. Various situations for which the representations of these groups degenerate into smaller ones, hopefully compatible with integrability, are considered. For instance, the group becomes smaller for q-state Potts models 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. This analysis provides nice birational representations of hyperbolic Coxeter groups. Remarkable varieties breaking the symmetry of the lattice are seen to occur specifically for the Tutte-Beraha numbers. A detailed analysis of these Potts models is performed for q = 3. In particular, the algebraic varieties corresponding to conditions for the symmetry group to be finite order are carefully examined. Finally, specifically for the Tutte-Beraha numbers, the introduction of algebraic group invariants is discussed in detail for q = 3 in order to get closed expressions for the spontaneous magnetization of the edge Potts models.