We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.
Bibliographical noteFunding Information:
M. Cheung was partially supported by NSF Grant DMS-1854512. G. Musiker was partially supported by NSF Grant DMS-1745638. E. Kelley was partially supported by NSF Grants DMS-1745638 and DMS-1937241. The authors are grateful for the hospitality of RIMS in 2019 in Kyoto, Japan, during the “Cluster Algebras 2019” workshop, where this collaboration began, and to the anonymous reviewers whose careful reading and insightful comments improved the exposition of this paper. On behalf of all authors, the corresponding author states that the authors have no conflicts of interest.
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
- Cluster algebras
- Generalized cluster algebras
- Scattering diagrams
- Theta functions