Abstract
To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero–Chapoton map defines a bijection between \(\tau \)-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.
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Acknowledgements
DLF received support from UNAM’s Dirección General de Asuntos del Personal Académico through its Programa de Apoyos para la Superación del Personal Académico, and from UNAM’s Physics Institute via a Cátedra Marcos Moshinsky. LM was supported by the Royal Society through the Newton International Fellowship NIF\R1\201849. The last part of the paper was completed during a visit of both authors to Sibylle Schroll at the Mathematisches Institut of the Universität zu Köln. We are grateful for the great working conditions and the hospitality. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Cluster Algebras and Representation Theory where work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1. We thank the Organizers of this programme, Karin Baur, Bethany Marsh, Ralf Schiffler and Sibylle Schroll for a wonderful working atmosphere. Part of this project was carried out during the junior trimester program New Trends in Representation Theory, held at the Hausdorff Research Institute for Mathematics, Bonn, and organized by Gustavo Jasso. We are grateful for the support and hospitality received.
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Open Access funding enabled and organized by Projekt DEAL. DLF received support from UNAM’s Dirección General de Asuntos del Personal Académico through its Programa de Apoyos para la Superación del Personal Académico, and from UNAM’s Physics Institute via a Cátedra Marcos Moshinsky. LM was supported by the Royal Society through the Newton International Fellowship NIF\R1\201849.
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Daniel Labardini-Fragoso and Lang Mou wrote the manuscript. All authors reviewed the manuscript.
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Labardini-Fragoso, D., Mou, L. Gentle Algebras Arising from Surfaces with Orbifold Points of Order 3, Part I: Scattering Diagrams. Algebr Represent Theor 27, 679–722 (2024). https://doi.org/10.1007/s10468-023-10233-x
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DOI: https://doi.org/10.1007/s10468-023-10233-x