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

Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standard symplectic structure on the space of Fuchsian systems. As an application, we compute explicitly two examples of reduction from Schlesinger transformations to difference Painlevé equations. The first example, d-$P\big(D_{4}^{(1)}\big)$ (or difference Painlevé V), corresponds to Bäcklund transformations for continuous $P_{\text{VI}}$. The second example, d-$P\big(A_{2}^{(1)*}\big)$ (with the symmetry group $E_{6}^{(1)}$), is purely discrete. We also describe the role played by the geometry of the Okamoto space of initial conditions in comparing different equations of the same type.