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

By Ilke Çanakçi, Ralf Schiffler. 2012

48 citations recorded by the source corpus.

We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used 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 labelling of the graphs. The most restrictive form of this ring is isomorphic to the ring $\mathbb{Z}[x,y]$ of polynomials in two variables over the integers. A more general form contains all cluster algebras of unpunctured surface type. The definition of the rings requires the snake graph calculus which is completed in this paper 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 the current paper is the completion of the explicit construction of these bijections.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e02bc1f81b93fec0a1b549b0a4356f41a055d3ce

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Citations in the corpus

Cluster categories for marked surfaces: punctured case (2013, background): “expect for few cases. • Musiker-Schiﬄer-Williams [37] constructed two canonical bases by two types of collections of curves. • Musiker-Schiﬄer-Williams [36], Musiker-Williams [38] and Canakci-Schiﬄer [9, 10, 11] gave combinatorial formulas for cluster variables and relations. • Mills [35] showed tha”
Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, methodology): “As an application of the computational tools developed in [3] and in the present paper, it is proved in [2] that the basis construction of [24] also applies to surfaces with non-empty boundary and with exactly one marked point.”