Geometric Analysis of Reductions from Schlesinger Transformations to Difference Painlev\'e Equations

TL;DR

This paper explores geometric reductions from discrete Schlesinger transformations to specific difference Painlevé equations, detailing the computation of initial conditions and the role of geometry in understanding these reductions.

Contribution

It provides explicit geometric methods for reducing Schlesinger transformations to difference Painlevé equations and analyzes their initial condition spaces.

Findings

Explicit descriptions of initial condition spaces for the equations.

Demonstration of geometric methods in understanding reductions.

Comparison of different instances of difference Painlevé equations.

Abstract

We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlev\'e equations: difference Painlev\'e equation d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painlev\'e equation d-$P\left({A}_{1}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.