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

Canakci, I.; Schiffler, R.

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
• Public URL: https://durham-repository.worktribe.com/output/1424305
• Related Public URLs: http://arxiv.org/abs/1506.01742