TY - JOUR

T1 - Factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference Painlevé equations

AU - Dzhamay, Anton

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=72449131585&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/42/45/454008

DO - 10.1088/1751-8113/42/45/454008

M3 - Article

AN - SCOPUS:72449131585

VL - 42

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 45

M1 - 454008

ER -