Academic Journal
Okamoto's symmetry on the representation space of the sixth Painlevé equation.
| Τίτλος: | Okamoto's symmetry on the representation space of the sixth Painlevé equation. |
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| Συγγραφείς: | Martello, Davide Dal |
| Πηγή: | IMRN: International Mathematics Research Notices; Jun2026, Vol. 2026 Issue 11, p1-31, 31p |
| Θεματικοί όροι: | Painlevé equations, Symmetry, Mathematical convolutions, Mutations (Algebra), Representation theory, Teichmüller spaces, Differential equations |
| Περίληψη: | The sixth Painlevé equation (|$P_{V\!I}$|) admits dual isomonodromy representations of type 2D Fuchsian and 3D Birkhoff. Taking the multiplicative middle convolution of a Teichmüller |$\mathcal{X}$| -coordinatization for the Fuchsian monodromy group, we give Okamoto's symmetry |$w_{2}$| of |$P_{V\!I}$| 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 maps, each one realizing |$w_{2}$| , a unified diagrammatic description in purely convolutional terms. [ABSTRACT FROM AUTHOR] |
| Copyright of IMRN: International Mathematics Research Notices is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Βάση Δεδομένων: | Complementary Index |
| FullText | Text: Availability: 0 CustomLinks: – Url: https://resolver.ebsco.com/c/fiv2js/result?sid=EBSCO:edb&genre=article&issn=10737928&ISBN=&volume=2026&issue=11&date=20260601&spage=1&pages=1-31&title=IMRN: International Mathematics Research Notices&atitle=Okamoto%27s%20symmetry%20on%20the%20representation%20space%20of%20the%20sixth%20Painlev%C3%A9%20equation.&aulast=Martello%2C%20Davide%20Dal&id=DOI:10.1093/imrn/rnag110 Name: Full Text Finder (for New FTF UI) (ns324271) Category: fullText Text: Full Text Finder MouseOverText: Full Text Finder |
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| Items | – Name: Title Label: Title Group: Ti Data: Okamoto's symmetry on the representation space of the sixth Painlevé equation. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Martello%2C+Davide+Dal%22">Martello, Davide Dal</searchLink> – Name: TitleSource Label: Source Group: Src Data: IMRN: International Mathematics Research Notices; Jun2026, Vol. 2026 Issue 11, p1-31, 31p – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Painlevé+equations%22">Painlevé equations</searchLink><br /><searchLink fieldCode="DE" term="%22Symmetry%22">Symmetry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+convolutions%22">Mathematical convolutions</searchLink><br /><searchLink fieldCode="DE" term="%22Mutations+%28Algebra%29%22">Mutations (Algebra)</searchLink><br /><searchLink fieldCode="DE" term="%22Representation+theory%22">Representation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Teichmüller+spaces%22">Teichmüller spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+equations%22">Differential equations</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The sixth Painlevé equation (|$P_{V\!I}$|) admits dual isomonodromy representations of type 2D Fuchsian and 3D Birkhoff. Taking the multiplicative middle convolution of a Teichmüller |$\mathcal{X}$| -coordinatization for the Fuchsian monodromy group, we give Okamoto's symmetry |$w_{2}$| of |$P_{V\!I}$| 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 maps, each one realizing |$w_{2}$| , a unified diagrammatic description in purely convolutional terms. [ABSTRACT FROM AUTHOR] – Name: Abstract Label: Group: Ab Data: <i>Copyright of IMRN: International Mathematics Research Notices is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1093/imrn/rnag110 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 31 StartPage: 1 Subjects: – SubjectFull: Painlevé equations Type: general – SubjectFull: Symmetry Type: general – SubjectFull: Mathematical convolutions Type: general – SubjectFull: Mutations (Algebra) Type: general – SubjectFull: Representation theory Type: general – SubjectFull: Teichmüller spaces Type: general – SubjectFull: Differential equations Type: general Titles: – TitleFull: Okamoto's symmetry on the representation space of the sixth Painlevé equation. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Martello, Davide Dal IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 10737928 Numbering: – Type: volume Value: 2026 – Type: issue Value: 11 Titles: – TitleFull: IMRN: International Mathematics Research Notices Type: main |
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