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

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