J-M. Maillard

Integrable Coxeter groups

Abstract We describe the construction of a class of mappings in projective space C PN for any N. These mappings are non-linear representations of Coxeter groups by birational and therefore almost everywhere defined and invertible transformations. We give specific examples of the construction and exhibit algebraic invariants. The class of mappings we consider has a variety of behaviours according to the number of independent invariants. We introduce the notion of integrability of a group of mappings. The concept is related to the nation of integrability in the realm of statistical physics and field theory as will appear elsewhere. There is a natural set of deformation parameters of our mappings, allowing for a study of their stability. We comment on the algebraic structures we are handling.

Rational mappings, arborescent iterations, and the symmetries of integrability

We describe a class of nonlinear birational representations of groups generated by a finite number of involutions. These groups are symmetries of the Yang-Baxter equations and their higher-dimensional generalizations. They provide discrete dynamical systems with a variety of behaviors, from chaotic to integrable, according to the number of invariants of the representation.

Infinite discrete symmetry group for the Yang-Baxter equations. Vertex models

Abstract We show that the Yang-Baxter equations for two-dimensional vertex models admit as a group of symmetry the infinite discrete group A 2 (1) . The existence of this symmetry explains the presence of a spectral parameter in solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetries. Although generalizing very naturally the previous one, this is a much bigger hyperbolic Coxeter group. We indicate how this symmetry should be used to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of a family of three-dimensional vertex models.

Growth-complexity spectrum of some discrete dynamical systems

We first study the iteration of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3 3 matrices, and consider the degree d.n/ of the numerators (or denominators) of the corresponding successive rational expressions for the nth iterate. The growth of this degree is (generically) exponential with n: d.n/ ’ n . is called the growth complexity. We introduce a semi-numerical analysis which enables to compute these growth complexities for all the 9! possible birational transformations. These growth complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their growth complexities. ©1999 Elsevier Science B.V. All rights reserved.

Factorization properties of birational mappings

We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive permutations.

Quasi integrability of the sixteen-vertex model

Abstract We analyze the symmetries of the sixteen-vertex model. We prove the existence of a natural parametrization of the parameter space of the model by elliptic curves, grounding the inversion trick for the exact calculation of the partition function. We proceed with a “pre-Bethe-ansatz” system of equations whose analysis produces an algebraic modular invariant and yields candidates for criticality and disorder conditions.

Integrable mappings and polynomial growth

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q × q matrices: the inversion of the q × q matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for q = 4, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.

Higher dimensional mappings

Abstract We present a new class of mappings acting on many variables, and depending on many parameters. These mappings are nonlinear (birational) representations of discrete groups generated by involutions, having their origin in the theory of integrable models in statistical mechanics. Various quantities of statistical mechanics present automorphy properties under the action of these groups, which appear thus as a generalization to several complex variables of the fundamental group for Riemann surface. They enjoy many remarkable properties, and we give a preliminary study of these mappings.

ALMOST INTEGRABLE MAPPINGS

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. Generically, the iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. Nevertheless some orbits of the parameter space lie on (transcendental) curves. These transformations act on fifteen (or q2 − 1) variables, however one can associate to them remarkably simple non-linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings, which could provide interesting tools to analyze weak chaos.

DETERMINANTAL IDENTITIES ON INTEGRABLE MAPPINGS

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on q×q matrices: the inversion of the q×q matrix and an (involutive) permutation of the entries of the matrix. In a case where the permutation is a particular elementary transposition of two entries, it is shown that the iteration of this group of birational transformations yield algebraic elliptic curves in the parameter space associated with the (homogeneous) entries of the matrix. It is also 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 with biquadratic relations. In fact, thes...