Full-parameter discrete Painlev\'e systems from non-translational Cremona isometries

TL;DR

This paper constructs full-parameter discrete Painlevé systems from non-translational Cremona isometries within affine Weyl groups, expanding the understanding of integrable equations associated with rational surfaces.

Contribution

It demonstrates how non-translation elements of infinite order in symmetry groups can generate discrete Painlevé equations with maximal parameters, broadening the scope of integrable systems derived from surface symmetries.

Findings

Constructed examples of full-parameter discrete Painlevé equations from non-translation symmetries.

Proved the integrability of these new difference equations.

Generalized previous special cases to a broader class of integrable systems.

Abstract

Since the classification of discrete Painlev\'e equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list. For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries. Translation elements of this symmetry group define discrete Painlev\'e equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlev\'e-type through projective reduction, these have fewer than the maximal number of free parameters corresponding to their surface type. We show that difference equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlev\'e equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature, which we generalise to a full-parameter integrable difference equation, given by the Cremona action of a non-translation element of the extended affine Weyl group $\widetilde{W}(D_4^{(1)})$ on a family of generic $D_4^{(1)}$- surfaces.