Geometric description of discrete power function associated with the sixth Painlevé equation

Abstract

In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $\tilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $\tilde{W}(3A_1^{(1)})$ as a subgroup of $\tilde{W}(D_4^{(1)})$, i.e., the symmetry group of P${}_{\mathrm{VI}}$, we show how to relate $\tilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $\tilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $\tau$ function.

arXiv comment18 pages, 3 figures

References in corpus (7)

• Classification of integrable equations on quad-graphs. The consistency approach
• Affine Weyl groups, discrete dynamical systems and Painleve equations
• Geometric Aspects of Painlevé Equations
• Classification of 3D consistent quad-equations
• Lattice equations arising from discrete Painlevé systems (I): $(A_2+A_1{)}^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$ cases
• Bilinearization and Special Solutions to the Discrete Schwarzian KdV Equation
• Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations