Half-periodicity of Zamolodchikov periodic cluster algebras

TL;DR

This paper proves that in Zamolodchikov periodic cluster algebras, the cluster variables at half-period form a permutation of the initial variables, confirming a long-standing conjecture and extending previous results.

Contribution

It establishes the half-period behavior of all Zamolodchikov periodic cluster algebras, showing the variables form a permutation of order at most two.

Findings

Half-period variables are a permutation of initial variables.

Confirmed the conjecture for all Zamolodchikov periodic cluster algebras.

Extended previous results beyond tensor products of Dynkin diagrams.

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.