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

In this paper, we consider the discrete power function associated with the sixth Painleve 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 $widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $widetilde{W}(3A_1^{(1)})$ as a subgroup of $widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{rm VI}$, we show how to relate $widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $tau$ function.