Okamoto's symmetry on the representation space of the sixth Painlev\'e equation

TL;DR

This paper provides a monodromic realization of Okamoto's symmetry for the sixth Painlevé equation using cluster mutations and geometric triangulations, linking it to middle convolution and dual representations.

Contribution

It introduces a novel monodromic interpretation of Okamoto's symmetry via cluster mutations and geometric models, unifying various representations of PVI.

Findings

Explicit mutation formula in geometric terms

Unified diagrammatic description of symmetry maps

Connection between middle convolution and monodromic realizations

Abstract

The sixth Painlev\'e equation (PVI) admits dual isomonodromy representations of type $2$-dimensional Fuchsian and $3$-dimensional Birkhoff. Taking the multiplicative middle convolution of a higher Teichm\"uller coordinatization for the Fuchsian monodromy group, we give Okamoto's symmetry $w_2$ of PVI a monodromic realization in the language of cluster $\mathcal{X}$-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 $w_2$-related maps a unified diagrammatic description in purely convolutional terms.