Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras

Man Wai Cheung; Elizabeth Kelley; Gregg Musiker

doi:10.1007/s00026-022-00623-1

Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras

Man Wai Cheung, Elizabeth Kelley, Gregg Musiker

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 language	English (US)
Journal	Annals of Combinatorics
DOIs	https://doi.org/10.1007/s00026-022-00623-1
State	Accepted/In press - 2022

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.

Keywords

Cluster algebras
Generalized cluster algebras
Scattering diagrams
Theta functions

Publisher link

10.1007/s00026-022-00623-1

Cite this

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abstract = "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.",

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AU - Kelley, Elizabeth

AU - Musiker, Gregg

N1 - 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.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Cluster algebras

KW - Generalized cluster algebras

KW - Scattering diagrams

KW - Theta functions

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Cluster Scattering Diagrams and Theta Functions for Reciprocal Generalized Cluster Algebras

Abstract

Bibliographical note

Keywords

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