Thursday, February 14, 2013

1302.2972 (Anton Dzhamay et al.)

Discrete Hamiltonian Structure of Schlesinger Transformations    [PDF]

Anton Dzhamay, Hidetaka Sakai, Tomoyuki Takenawa
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 Painlev\'e equations; this is also the main theme behind our work. In this paper we show how to write an elementary Schlesinger transformation as a discrete Hamiltonian system w.r.t. the standard symplectic structure on the space of Fuchsian systems. We then show how Schlesinger transformations reduce to discrete Painlev\'e equations by considering two explicit examples, d-$P(D_{4}^{(1)})$ (or difference Painlev\'e V) and d-$P(A_{2}^{(1)*})$.
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