论文标题
$ a^{（1）} _ n $的种子突变中的周期性，线性化和集成性
Periodicity, linearizability and integrability in seed mutations of type $A^{(1)}_N$
论文作者
论文摘要
在由特定初始种子引起的种子突变网络中，目的是从初始种子发出的适当路径，并注意到在路径中交换矩阵的周期性，每个路径的周期性分配给了$ a^{（1）} _ n $的类型$ a^{（1）_ n $的通用cartan矩阵。然后对沿路径的种子突变的动态特性进行了深入研究，该属性被深入研究为$ a^{（1）} _ n $。分配给路径形式的系数某些$ n $单元，这些单元与种子突变下的周期性$ n $具有周期性，并能够获得系数的一般条款。分配给类型$ a^{（1）} _ n $的群集变量还形成了某些$ n $ laurent多项式，具有与系数生成的单一元素相同的周期性。这些劳伦（Laurent）多项式导致沿路径群集突变得出的足够数量的保守量。此外，由于具有周期性的劳伦（Laurent）多项式，动力学系统是非自主性线性化的，并且其一般解决方案是具体构建的。因此，沿$ a^{（1）} _ n $类型路径的种子突变具有离散的集成性。
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.
