Linearly recursive sequences and Dynkin diagrams

TL;DR

This paper explores the connection between linear recursions, Dynkin diagrams, and cluster algebra friezes, revealing that certain integer sequences arise from these diagrams and are characterized by linear recursions and rationality.

Contribution

It establishes a characterization of sequences associated with Dynkin and Euclidean diagrams as solutions to linear recursions and introduces $SL_2$-tilings as a key tool.

Findings

Sequences from Dynkin diagrams satisfy linear recursions.

Associated sequences are $\ extbf{\mathbb N}$-rational.

$SL_2$-tilings are useful in understanding these structures.

Abstract

Motivated by a construction in the theory of cluster algebras (Fomin and Zelevinsky), one associates to each acyclic directed graph a family of sequences of natural integers, one for each vertex; this construction is called a {\em frieze}; these sequences are given by nonlinear recursions (with division), and the fact that they are integers is a consequence of the Laurent phenomenon of Fomin and Zelevinsky. If the sequences satisfy a linear recursion with constant coefficients, then the graph must be a Dynkin diagram or an extended Dynkin diagram, with an acyclic orientation. The converse also holds: the sequences of the frieze associated to an oriented Dynkin or Euclidean diagram satisfy linear recursions, and are even $\mathbb N$-rational. One uses in the proof objects called $SL_2$-{\em tilings of the plane}, which are fillings of the discrete plane such that each adjacent 2 by 2 minor is equal to 1. These objects, which have applications in the theory of cluster algebras, are interesting for themselves. Some problems, conjectures and exercises are given.