Periodic Cluster Mutations and Related Integrable Maps

Abstract: One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the "Laurent phenomenon"). There are many nonlinear recurrences which exhibit the Laurent phenomenon and thus unexpectedly generate integer sequences. The mutation of a typical quiver will not generate a recurrence, but rather an erratic sequence of exchange relations. How do we "design" a quiver which gives rise to a given recurrence? A key role is played by the concept of "periodic cluster mutation", introduced in 2009. Each recurrence corresponds to a finite dimensional map. In the context of cluster mutations, these are called "cluster maps". What properties do cluster maps have? Are they integrable in some standard sense? In this review I describe how integrable maps arise in the context of cluster mutations. I first explain the concept of "periodic cluster mutation", giving some classification results. I then give a review of what is meant by an integrable map and apply this to cluster maps. Two classes of integrable maps are related to interesting monodromy problems, which generate interesting Poisson algebras of functions, used to prove complete integrability and a linearisation. A connections to the Hirota-Miwa equation is explained.