Periodicity, linearizability and integrability in seed mutations of type $A^{(1)}_N$

TL;DR

This paper investigates seed mutations of type A^{(1)}_N, revealing their periodicity, integrability, and explicit solutions through the study of associated Laurent polynomials and conserved quantities.

Contribution

It introduces a framework connecting seed mutations, periodicity, and integrability, providing explicit solutions for the dynamical systems involved.

Findings

Seed mutations exhibit N-periodicity in coefficients and cluster variables.

The system admits conserved quantities derived from Laurent polynomials.

The dynamical system is non-autonomously linearizable with explicit solutions.

Abstract

In the network of seed mutations arising from a certain initial seed, an appropriate path emanating from the initial seed is intendedly chosen, noticing periodicity of the exchange matrices in the path each of which is assigned to the generalized Cartan matrix of type $A^{(1)}_N$. Then dynamical property of the seed mutations along the path, which is referred to as of type $A^{(1)}_N$, is intensively investigated. The coefficients assigned to the path form certain $N$ monomials that posses periodicity with period $N$ under the seed mutations and enable to obtain the general terms of the coefficients. The cluster variables assigned to the path of type $A^{(1)}_N$ also form certain $N$ Laurent polynomials possessing the same periodicity as the monomials generated by the coefficients. These Laurent polynomials lead to sufficiently number of conserved quantities of the dynamical system derived from the cluster mutations along the path. Furthermore, by virtue of the Laurent polynomials with periodicity, the dynamical system is non-autonomously linearized and its general solution is concretely constructed. Thus the seed mutations along the path of type $A^{(1)}_N$ exhibit discrete integrability.