Commutation Relations and Discrete Garnier Systems

Christopher M. Ormerod, Eric M. Rains
2016 Symmetry, Integrability and Geometry: Methods and Applications  
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
more » ... 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.
doi:10.3842/sigma.2016.110 fatcat:2uruah6oxzacziumvkyft3qrqe