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Register a new userPoint configurations, Cremona transformations and the elliptic difference Painleve equation
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A theoretical foundation for a generalization of the elliptic difference Painleve equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $tau$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $tau$-functions on the lattice.
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Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve 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 Painleve 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 Painleve equations. The first example, d-$Pbig(D_{4}^{(1)}big)$ (or difference Painleve V), corresponds to Backlund transformations for continuous $P_{text{VI}}$. The second example, d-$Pbig(A_{2}^{(1)*}big)$ (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.
An interpolation problem related to the elliptic Painleve equation is formulated and solved. A simple form of the elliptic Painleve equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.
A Lax formalism for the elliptic Painleve equation is presented. The construction is based on the geometry of the curves on ${mathbb P}^1times{mathbb P}^1$ and described in terms of the point configurations.
We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painleve equations: difference Painleve equation d-$Pleft({A}_{2}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painleve equation d-$Pleft({A}_{1}^{(1)*}right)$ 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.
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