TY - JOUR

T1 - Periodicity, linearizability, and integrability in seed mutations of type A N (1)

AU - Nobe, Atsushi

AU - Matsukidaira, Junta

N1 - Funding Information:
This work was partially supported by the JSPS KAKENHI (Grant No. 20K03692). The authors would like to thank the anonymous referee for careful reading of our manuscript and insightful comments and suggestions.
Publisher Copyright:
© 2021 Author(s).

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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.

AB - 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.

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U2 - 10.1063/5.0030007

DO - 10.1063/5.0030007

M3 - Article

AN - SCOPUS:85100028176

VL - 62

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

M1 - 013510

ER -