Half-periodicity of Zamolodchikov periodic cluster algebras

Abstract

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite mutations whose exchange relations form a discrete dynamical system. Periodicity of this system is known as Zamolodchikov periodicity. In our previous work we have classified all Zamolodchikov periodic cluster algebras, but behavior halfway through the period was still unknown. This so-called half-periodicity was conjectured by Kuniba--Nakanishi--Suzuki for $Y$-systems of finite type Cartan matrices, and was proved by Inoue--Iyama--Keller--Kuniba--Nakanishi for tensor products of two simply-laced Dynkin diagrams. In this paper, we prove that for any Zamolodchikov periodic cluster algebra, the form at the half-period is a permutation of the cluster variables of order at most two.