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In the network of seed mutations arising from a certain initial seed, anappropriate path emanating from the initial seed is intendedly chosen, noticingperiodicity of the exchange matrices in the path each of which is assigned tothe generalized Cartan matrix of type $A^{(1)}_N$. Then dynamical property ofthe 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 mutationsand enable to obtain the general terms of the coefficients. The clustervariables assigned to the path of type $A^{(1)}_N$ also form certain $N$Laurent polynomials possessing the same periodicity as the monomials generatedby the coefficients. These Laurent polynomials lead to sufficiently number ofconserved quantities of the dynamical system derived from the cluster mutationsalong the path. Furthermore, by virtue of the Laurent polynomials withperiodicity, the dynamical system is non-autonomously linearized and itsgeneral solution is concretely constructed. Thus the seed mutations along thepath of type $A^{(1)}_N$ exhibit discrete integrability.

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