DETERMINANTAL IDENTITIES ON INTEGRABLE MAPPINGS

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable ve rtex-models in lattice statistical mechanics. These involutions corres pond respectively to two kinds of transformations on q x q matrices: t he inversion of the q x q matrix and an (involutive) permutation of th e entries of the matrix. In a case where the permutation is a particul ar elementary transposition of two entries, it is shown that the itera tion of this group of birational transformations yield algebraic ellip tic curves in the parameter space associated with the (homogeneous) en tries of the matrix. It is shown that the successive iterated matrices do have remarkable factorization properties which yield introducing a series of canonical polynomials corresponding to the greatest common factor in the entries. These polynomials do satisfy a simple nonlinear recurrence which also yields algebraic elliptic curves, associated wi th biquadratic relations. In fact, these polynomials not only satisfy one recurrence but a whole hierarchy of recurrences. Remarkably these recurrences are universal: they are independent of q, the size of the matrices. This study provides examples of infinite dimensional integra ble mappings.