Full-parameter discrete Painlevé systems from non-translational Cremona isometries

Abstract

Since the classification of discrete Painlevé 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 Painlevé 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 Painlevé-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 Painlevé 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 W (D (1) 4 ) on a family of generic D (1) 4 -surfaces.2

Two variations on ( A 3 × A 1 × A 1 ) (1) type discrete Painlevé equations

“…In particular, it was found that in this embedding the element φ which give rise to W(A 1 × A 3 ) (1) type Painlevé equation is not a translation in W(D (1) 5 ), whereas the element φ 2 is translational. Similar examples have been found in different contexts in the integrable system literature [9][10][11][12]. A common feature of these examples is that the element φ which gives rise to a discrete Painlevé equation is not translational, whereas some powers of φ is.…”

“…Quasi-translations arise under different guises in a variety of contexts in the theory of discrete integrable equations. For example, in 'symmetrisation' procedures of asymmetric discrete Painlevé equations known as projective reductions [11,12]; from reductions of partial difference equations [9]; or as elements that give rise to discrete equations which govern evolutions of Schramm's circle patterns [10]. Normalizer theory of Coxeter groups are extremely rich, not only it explains 'elements of quasi-translational' nature found in the study of Painlevé equations, it is also constructive.…”

Two variations on $(A_3\times A_1\times A_1)^{(1)}$ type discrete Painlevé equations

“…, whereas the element φ 2 is translational. Similar examples have been found in different contexts in the integrable system literature [6,15,1,5]. A common feature of these examples is that the element φ which gives rise to a discrete Painlevé equation is not translational, whereas some powers of φ is.…”

“…Quasi-translations arise under different guises in a variety of contexts in the theory of discrete integrable equations. For example, in "symmetrisation" procedures of asymmetric discrete Painlevé equations known as projective reductions [6,15]; from reductions of partial difference equations [1]; or as elements that give rise to discrete equations which govern evolutions of Schramm's circle patterns [5], these we plan to discuss in subsequent publications.…”

The sl2(R) coalgebra symmetry and the superintegrable discrete-time systems

“…In particular, we show that these discrete systems admit symmetries that do not appear explicitly in Sakai's classification. Such two-dimensional discrete integrable systems have been found in different contexts [1,6,7,24,29,44,46]. In [42], it was found that these symmetries arise as normalizers of certain parabolic subgroups of Sakai's Weyl groups using the normalizer theory of of Coxeter groups [5,21].…”