Commutation Relations and Discrete Garnier Systems

TL;DR

This paper introduces four classes of discrete systems that serve as analogues to the Garnier system, derived from isomonodromic deformations of linear difference equations with factored Lax matrices, highlighting their symmetry properties.

Contribution

It provides a new framework for discrete Garnier systems based on commutation relations and reparameterization, connecting them to discrete Painlevé equations.

Findings

Four classes of discrete Garnier systems identified.

Reparameterization in terms of image and kernel vectors.

Symmetry condition linked to discrete Painlevé equations.

Abstract

We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a symmetry condition on the associated linear problems that only appears as a necessary feature of the Lax pairs for the least degenerate discrete Painlev\'e equations.