Birational mappings and matrix subalgebra from the chiral Potts model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models with stable patterns and signed patterns, we give general results which allow us to find all chiral q -state spin-edge Potts models when the number of states q is a prime or the square of a prime, as well as several q -dependent family of models. We also prove the absence of monocolor stable signed pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated with these lattice spin-edge models show complexity reduction. In particular, we recover a one-parameter family of integrable transformations, for which we give a matrix representation when the parameter has a suitable value.