2002/09/12 by Alexei Borodin, Borodin, Alexei · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #math-ph #math.CA #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.math/0209144
AMSTeX, 37 pages
openalex publication_date 2002/09/12 · arxiv created 2002/10/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and study isomonodromy transformations of the matrix linear difference equation Y(z+1)=A(z)Y(z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Zm(n+1)-1 on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Schlesinger equations and the Schlesinger transformations known in the isomonodromy theory, can be obtained as limits of our action in two different limit regimes. Similarly to the continuous case, for m=n=2 the difference Schlesinger equations and their q-analogs yield discrete Painleve equations; examples include dPII, dPIV, dPV, and q-PVI.