Okamoto’s symmetry on the representation space of the sixth Painlevé equation

The sixth Painlevé equation (PV\!I) admits dual isomonodromy representations of type 2D Fuchsian and 3D Birkhoff. Taking the multiplicative middle convolution of a Teichmüller X-coordinatization for the Fuchsian monodromy group, we give Okamoto’s symmetry w₂ of PV\!I a monodromic realization in the language of cluster X-mutations. The explicit mutation formula is encoded in dual geometric terms of colored equilateral triangulations and star-shaped fat graphs. Moreover, this realization has a known additive analogue through the middle convolution for Fuchsian systems, and dual formulations for both the Birkhoff representation and its Stokes data exist. We give this quadruple of maps, each one realizing w₂, a unified diagrammatic description in purely convolutional terms.