Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras

Man Wai Cheung, Elizabeth Kelley, Gregg Musiker

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)615-691
Number of pages77
JournalAnnals of Combinatorics
Issue number3
StatePublished - Sep 2023

Bibliographical note

Funding 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.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.


  • Cluster algebras
  • Generalized cluster algebras
  • Scattering diagrams
  • Theta functions


Dive into the research topics of 'Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras'. Together they form a unique fingerprint.

Cite this