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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 [Formula: see text] symmetry. By constructing the action of [Formula: see text] as a subgroup of [Formula: see text], i.e. the symmetry group of P VI , we show how to relate [Formula: see text] to the symmetry group of the lattice. Moreover, by using translations in [Formula: see text], we explain the odd-even structure appearing in previously known explicit formulae in terms of the τ function.

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