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Schlesinger transformations are algebraic transformations of a Fuchsiansystem that preserve its monodromy representation and act on the characteristicindices of the system by integral shifts. One of the important reasons to studysuch transformations is the relationship between Schlesinger transformationsand discrete Painlev\'e equations; this is also the main theme behind our work.We derive \emph{discrete Schlesinger evolution equations} describing discretedynamical systems generated by elementary Schlesinger transformations and givetheir discrete Hamiltonian description w.r.t.~the standard symplectic structureon the space of Fuchsian systems. As an application, we compute explicitly twoexamples of reduction from Schlesinger transformations to difference Painlev\'eequations. The first example, d-$P\big(D_{4}^{(1)}\big)$ (or differencePainlev\'e V), corresponds to B\"acklund transformations for continuous$P_{\text{VI}}$. The second example, d-$P\big(A_{2}^{(1)*}\big)$ (with thesymmetry group $E_{6}^{(1)}$), is purely discrete. We also describe the roleplayed by the geometry of the Okamoto space of initial conditions in comparingdifferent equations of the same type.

Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlev\'e Equations