DISCRETE SYMMETRY GROUPS OF VERTEX MODELS IN STATISTICAL-MECHANICS

We analyze discrete symmetry groups of vertex models in lattice statis tical mechanics represented as groups of birational transformations. T hey can be seen as generated by involutions corresponding respectively to two kinds of transformations on q x q matrices: the inversion of t he q x q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterati ons of these transformations is a precious tool in the study of lattic e models in statistical mechanics. This approach enables one to analyz e two-dimensional q(4)-state vertex models as simply as three-dimensio nal vertex models, or higher-dimensional vertex models. Various exampl es of birational symmetries of vertex models are analyzed. A particula r emphasis is devoted to a three-dimensional vertex model, the 64-stat e cubic vertex model, which exhibits a polynomial growth of the comple xity of the calculations. A subcase of this general model is seen to y ield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growt h of the calculations reduces to a polynomial growth when the model be comes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transf ormations yielding algebraic surfaces, or higher-dimensional algebraic varieties.