Geometric description of a 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 W ~ ( 3 A 1 ( 1 ) ) symmetry. By constructing the action of W ~ ( 3 A 1 ( 1 ) ) as a subgroup of W ~ ( D 4 ( 1 ) ) , i.e. the symmetry group of P VI , we show how to relate W ~ ( D 4 ( 1 ) ) to the symmetry group of the lattice. Moreover, by using translations in W ~ ( 3 A 1 ( 1 ) ) , we explain the odd–even structure appearing in previously known explicit formulae in terms of the τ function.