A geometric approach to tau-functions of difference Painlevé equations

Abstract

We present a unified description of birational representation of Weyl groups associated with T-shaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of tau-functions is given in terms of defining polynomials of certain curves. If the Dynkin diagram is of affine type ($E_6^{(1)}$, $E_7^{(1)}$ or $E_8^{(1)}$), our representation gives rise to the difference Painlevé equations.

arXiv comment12 pages

References in corpus (3)

• Riccati Solutions of Discrete Painlevé Equations with Weyl Group Symmetry of Type $E_8^{(1)}$
• Quivers and difference Painleve equations
• Universal character and q-difference Painlevé equations with affine Weyl groups

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• Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n
• On the lattice-geometry and birational group of the six-point multi-ratio equation
• Tropical Krichever construction for the non-periodic box and ball system
• Reduction of quad-equations consistent around a cuboctahedron I: additive case