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Cluster Algebras and Binary Subwords

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Abstract

This paper establishes a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion w can be associated to a piecewise-linear poset P and a snake graph G. We construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti. We then present bijections from the subwords of w to the antichains of P and to the perfect matchings of G.

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Acknowledgements

This project was inspired by conversations with M. Stipulanti and L. Tarsissi during the conference “Sage Days 82 : Women in Sage”, held January 2017 in Paris, funded by the OpenDreamKit project. We thank the organizers (J. Balakrishnan, V. Pons, and J. Striker) for creating such a productive environment. We are also grateful for helpful conversations with E. Barnard, R. Schiffler, K. Serhiyenko, and E. Yıldırım. Finally, we thank the anonymous reviewers whose suggestions helped improve and clarify this paper. Most of this work was completed while the second author was an Assistant Research Professor at University of Connecticut.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-1009, USA

    Rachel Bailey

  2. Department of Mathematics, University of Oklahoma, Norman, OK, 73019-3103, USA

    Emily Gunawan

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  1. Rachel Bailey
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  2. Emily Gunawan
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Correspondence to Emily Gunawan.

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Both authors were supported by the University of Connecticut. E.G. was supported by the NSF grant DMS-1254567.

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Bailey, R., Gunawan, E. Cluster Algebras and Binary Subwords. Order 39, 55–69 (2022). https://doi.org/10.1007/s11083-021-09562-7

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