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.
Original language | English |
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Pages (from-to) | 1145-1226 |
Number of pages | 82 |
Journal | International Mathematics Research Notices |
Volume | 2019 |
Issue number | 4 |
Early online date | 17 Jul 2017 |
DOIs | |
Publication status | Published - Feb 2019 |
Externally published | Yes |
Funding
The authors were supported by NSF grant DMS-10001637; I.C. was also supported by EPSRC grant number EP/K026364/1, UK, and by the University of Leicester; and R.S. was also supported by the NSF grants DMS-1254567, DMS-1101377, and by the University of Connecticut.
Funders | Funder number |
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National Science Foundation | DMS-10001637 |
Directorate for Mathematical and Physical Sciences | 1254567, 1101377 |
University of Connecticut | |
Engineering and Physical Sciences Research Council | EP/K026364/1 |
University of Leicester | DMS-1254567, DMS-1101377 |