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\(\tilde{A}\) and \(\tilde{D}\) type cluster algebras: triangulated surfaces and friezes

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Abstract

By viewing \(\tilde{A}\) and \(\tilde{D}\) type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze pattern (or bipartite belt) or (ii) the periodic quantities previously found for the cluster map associated with these frieze patterns. We show that these cluster variables form friezes which are precisely the ones found by Assem–Dupont by applying the cluster character to the associated cluster category.

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Acknowledgements

The author thanks Andy Hone, Rei Inoue and Philipp Lampe for helpful discussions and advice. This paper was improved thanks to advice from anonymous reviewers. This research was carried out while the author was a recipient of a Japan Society for the Promotion of Science (JSPS) postdoctoral fellowship and was supported by JSPS KAKENHI Grant Number 21F20788.

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  1. Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba, Japan

    Joe Pallister

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Pallister, J. \(\tilde{A}\) and \(\tilde{D}\) type cluster algebras: triangulated surfaces and friezes. J Algebr Comb 56, 1163–1202 (2022). https://doi.org/10.1007/s10801-022-01152-z

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