保突变的广义簇代数与Laurent突变不变量
Mutation-preserving generalized cluster algebras and Laurent mutation invariants
中文总结 AI 辅助
本文研究保突变的广义簇代数，分类其不可约类型，分析具该结构的马尔可夫型丢番图方程解的突变轨道，还证明了Chen-Li关于Laurent突变不变量的猜想。
中文摘要 AI 辅助
我们引入保突变的广义簇代数，对于这类代数，每个方向上的广义簇突变都与突变等价类中的种子无关。我们对所有具有该性质的不可约广义簇代数进行了分类。随后，我们研究了马尔可夫型丢番图方程$x^2+y^2+z^2+2yz=kxyz$，该方程具有保突变广义簇代数的结构。我们证明正整数解存在当且仅当$1\leq k\leq5$，并确定了这些解的所有突变轨道。特别地，当$k=1,3$时存在多个轨道，而当$k=2,4,5$时解各自形成单个轨道。我们还证明了Chen-Li提出的关于秩3簇代数（具有不可约符号等价交换矩阵）的Laurent突变不变量的猜想，表明每个Laurent突变不变量本质上都是对应基本不变量的多项式。
英文摘要
We introduce mutation-preserving generalized cluster algebras, for which the generalized cluster mutation in each direction is independent of the seed in the mutation equivalence class. We classify all irreducible generalized cluster algebras with this property. Then, a Markov-type Diophantine equation $x^2+y^2+z^2+2yz=kxyz$ is studied, which has a structure of the mutation-preserving generalized cluster algebra. We prove that positive integer solutions exist if and only if $1\leq k\leq 5$ and determine all their orbits under the associated generalized cluster mutation groups. In particular, multiple orbits occur for each $k=1,3$, whereas the solutions form a single orbit for each $ k=2,4,5$. We then classify all generalized Markov Laurent mutation invariants. As an application, a conjecture proposed by Chen-Li is proved, showing that every Laurent mutation invariant of irreducible sign-equivalent cluster algebras is essentially a polynomial in the corresponding basic invariant.
