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Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings

Canakci, I.; Schiffler, R.

Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings Thumbnail


Authors

I. Canakci

R. Schiffler



Abstract

We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras and that are interesting on their own right as combinatorially defined rings. The elements of these rings are residue classes of unions of certain labeled graphs that were instrumental to construct canonical bases in the theory of cluster algebras. We obtain several rings by varying the conditions on the structure as well as the labeling of the graphs. A general form of this ring contains all cluster algebras of unpunctured surface type. The definition of the rings requires the snake graph calculus which we also complete in this article building on two earlier articles on the subject. Identities in the snake ring correspond to bijections between the posets of perfect matchings of the graphs. One of the main results of this article is the completion of the explicit construction of these bijections.

Citation

Canakci, I., & Schiffler, R. (2019). Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings. International Mathematics Research Notices, 2019(4), 1145-1226. https://doi.org/10.1093/imrn/rnx157

Journal Article Type Article
Acceptance Date Jun 15, 2017
Online Publication Date Jul 17, 2017
Publication Date Feb 20, 2019
Deposit Date Dec 30, 2015
Publicly Available Date Mar 8, 2019
Journal International Mathematics Research Notices
Print ISSN 1073-7928
Electronic ISSN 1687-0247
Publisher Oxford University Press
Peer Reviewed Peer Reviewed
Volume 2019
Issue 4
Pages 1145-1226
DOI https://doi.org/10.1093/imrn/rnx157
Related Public URLs http://arxiv.org/abs/1506.01742

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Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in International mathematics research notices following peer review. The version of record Canakci, I. & Schiffler, R. (2019). Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings. International Mathematics Research Notices 2019(4): 1145-1226 is available online at: https://doi.org/10.1093/imrn/rnx157




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