Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlevé Equations

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 Painlevé 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.

9 pages; minor typos fixed, journal reference added

References in corpus (6)

• Yang-Baxter maps and integrable dynamics
• Vector bundles and Lax equations on algebraic curves
• Yang-Baxter maps: dynamical point of view
• Tau-function of discrete isomonodromy transformations and probability
• Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem
• Quadratic algebras and integrable chains

Cited by in corpus (1)

• Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlevé Equations