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 m odels of statisticaI 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 g roups. Various situations for which the representations of these group s degenerate into smaller ones, hopefully compatible with integrabilit y, are considered. For instance, the group becomes smaller for q-state Potts models for particular values of q, the so-called Tutte-Beraha n umbers. For this model, algebraic varieties, including the known ferro magnetic critical variety, happen to be invariant under such large gro ups of symmetries. This analysis provides nice birational representati ons of hyperbolic Coxeter groups. Remarkable varieties breaking the sy mmetry of the lattice are seen to occur specifically for the Tutte-Ber aha numbers. A detailed analysis of these Potts models is performed fo r q = 3. In particular, the algebraic varieties corresponding to condi tions 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 ord er to get closed expressions for the spontaneous magnetization of the edge Potts models.