Periodicity, linearizability, and integrability in seed mutations of type A N (1)

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 exchange matrices in the path each of which is assigned to the generalized Cartan matrix of type AN(1). Then, the dynamical property of seed mutations along the path, which is referred to as of type AN(1), is intensively investigated. The coefficients assigned to the path form certain N monomials that possess periodicity with period N under seed mutations and enable us to obtain the general terms of the coefficients. The cluster variables assigned to the path of type AN(1) also form certain N Laurent polynomials possessing the same periodicity as the monomials generated by the coefficients. These Laurent polynomials lead to a sufficient number of conserved quantities of the dynamical system derived from 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, seed mutations along the path of type AN(1) exhibit discrete integrability.