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Discrete Schlesinger Transformations and Difference Painlev Equations

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We study a discrete version of isomonodromic deformations of Fuchsian systems, called Schlesinger transformations, and their reductions to discrete Painlev equations. We obtain an explicit formula for the generating function of elementary Schlesinger transformations in terms of the coordinates on the so-called decomposition space associated to the Fuchsian system and interpret it as a discrete Hamiltonian of our dynamic. We then consider some explicit examples of reductions of such transformations to discrete Painlev equations. Using the birational geometry of rational surfaces associated to these equations, we compare the form of the equations that correspond to the elementary Schlesinger transformations to standard form of the equations of the same type.

This talk is part of the Isaac Newton Institute Seminar Series series.

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