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

2017/05/31 by Nalini Joshi, Kenji Kajiwara, Tetsu Masuda +2 · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1098/rspa.2017.0312

18 pages, 3 figures

arxiv created 2017/10/20 · arxiv updated 2018/02/07

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 \widetildeW(3A₁⁽¹⁾) symmetry. By constructing the action of \widetildeW(3A₁⁽¹⁾) as a subgroup of \widetildeW(D₄⁽¹⁾), i.e., the symmetry group of P_{\rm VI}, we show how to relate \widetildeW(D₄⁽¹⁾) to the symmetry group of the lattice. Moreover, by using translations in \widetildeW(3A₁⁽¹⁾), we explain the odd-even structure appearing in previously known explicit formulas in terms of the τ function.

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