Cluster mutation-periodic quivers and associated Laurent sequences

TL;DR

This paper classifies certain mutation-periodic quivers and explores their associated nonlinear Laurent recurrences, revealing new integrable maps, connections to Pell equations, and links to quiver gauge theories.

Contribution

It provides a classification of mutation-periodic quivers, introduces new families of nonlinear recurrences with the Laurent property, and connects these to integrable systems and gauge theories.

Findings

Identified quivers with higher periodicity under mutation.

Discovered new nonlinear recurrences with the Laurent property.

Linked some recurrences to solutions of Pell equations.

Abstract

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale. Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.