Okamoto's symmetry on the representation space of the sixth Painlevé equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909405473144832 |
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| author | Martello, D. Dal |
| author_facet | Martello, D. Dal |
| contents | The sixth Painlevé equation (PVI) admits dual isomonodromy representations of type $2$-dimensional Fuchsian and $3$-dimensional Birkhoff. Taking the multiplicative middle convolution of a higher Teichmüller 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Okamoto's symmetry on the representation space of the sixth Painlevé equation Martello, D. Dal Mathematical Physics 34M35, 34C14, 13F60 The sixth Painlevé equation (PVI) admits dual isomonodromy representations of type $2$-dimensional Fuchsian and $3$-dimensional Birkhoff. Taking the multiplicative middle convolution of a higher Teichmüller 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. |
| title | Okamoto's symmetry on the representation space of the sixth Painlevé equation |
| topic | Mathematical Physics 34M35, 34C14, 13F60 |
| url | https://arxiv.org/abs/2411.17397 |