Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 110, 50 pages      arXiv:1601.06179      https://doi.org/10.3842/SIGMA.2016.110

Commutation Relations and Discrete Garnier Systems

Christopher M. Ormerod a and Eric M. Rains b
a) University of Maine, Department of Mathemaitcs & Statistics, 5752 Neville Hall, Room 322, Orono, ME 04469, USA
b) California Institute of Technology, Mathematics 253-37, Pasadena, CA 91125, USA

Received March 30, 2016, in final form October 30, 2016; Published online November 08, 2016

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 Painlevé equations.

Key words: integrable systems; difference equations; Lax pairs; discrete isomonodromy.

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