Cluster algebras and continued fractions

Çanakçı, İlke; Schiffler, Ralf

Abstract

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction we associate a snake graph such that the continued fraction is the quotient of the number of perfect matchings of and . We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

Citation

Çanakçı, İ., & Schiffler, R. (2018). Cluster algebras and continued fractions. Compositio Mathematica, 154(03), 565-593. https://doi.org/10.1112/s0010437x17007631

• Journal Article Type: Article
• Acceptance Date: Aug 24, 2017
• Online Publication Date: Dec 22, 2017
• Publication Date: Mar 31, 2018
• Deposit Date: Aug 10, 2018
• Publicly Available Date: Nov 27, 2019
• Journal: Compositio Mathematica
• Print ISSN: 0010-437X
• Electronic ISSN: 1570-5846
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 154
• Issue: 03
• Pages: 565-593
• DOI: https://doi.org/10.1112/s0010437x17007631
• Public URL: https://durham-repository.worktribe.com/output/1323826
• Related Public URLs: https://eprint.ncl.ac.uk/pub_details2.aspx?pub_id=243310