Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlev\'e Equations

TL;DR

This paper explores factorizations of rational matrix functions with simple poles, linking them to discrete isomonodromic transformations and difference Painlevé equations, and introduces a coordinate system based on residues and eigenvectors.

Contribution

It introduces a new coordinate system for rational matrix functions based on residues and eigenvectors, connecting factorizations to isomonodromic transformations and difference Painlevé equations.

Findings

Coordinate system based on residues and eigenvectors simplifies analysis.

Discrete Euler-Lagrange equations describe isomonodromic transformations.

Difference Painlevé equations relate residues of matrices and their inverses.

Abstract

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlev\'e equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D. Arinkin and A. Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.