Cluster algebraic interpretation of infinite friezes

Abstract

Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.Date: June 1, 2017. 2010 Mathematics Subject Classification. 13F60 (primary), 05C70, 05E15 (secondary).

Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers

“…Similar looking recurrence relations involving bracelets but for arcs that have both endpoints on the same boundary component were found in[3, Theorem 2.5] and[18, Theorem 5.4].…”

Growth behaviour of periodic tame friezes

“…Corollary 5.4 now enables us to prove the following Proposition 5.6, which characterises all one-, two-, three-, four-and six-periodic solutions of the recursion (10). In particular, we are able to identify all possible periodic solutions of (10) for the integer values r 1 " 0,˘1,˘2.…”

“…Theorem 5.2 is a interesting result on its own, since it seems to serve as a starting point for a characterisation of all possible finite friezes. Caldero and Chapoton [5] established a connection between finite friezes of positive integers and cluster algebras of type A. Baur and Marsh [1] subsequently extended this idea by producing modified (branched) finite frieze patterns associated to cluster algebras of type D. Following the initial appearance of the present article, Gunawan, Musiker and Vogel [10] have initiated the study of the connection between periodic infinite friezes and cluster algebras of type D (punctured disc case) and affine type A (annulus case). By considering infinite friezes whose entries are cluster algebra elements, they obtain geometric and cluster algebraic interpretations of a number of our main ideas and results.…”

“…The following theorem exhibits an interesting relationship between any real-valued sequence pr k q kPN 0 satisfying (10) and the specific associated sequence ps k q kPN 0 , whose initial values are s 0 " 2 and s 1 " r 1 , and which also satisfies (10). Namely, the elements of the latter sequence appear as coefficients in generalised recursion formulas involving the elements of the former.…”

“…(EG) (Exponential Growth) If r 1 ą 2 and r 0 P R, then the behaviour of the twostage recursion (10) follows from the one-stage recursion (11) with the extension rule (12). In particular, Lemma B.1 implies that the recursion (10) grows exponentially in the following sense: For every 0 ă ε ă S´1 with S given in (20), i.e.…”