Frieze varieties are invariant under Coxeter mutation
Frieze varieties are invariant under Coxeter mutation
复制标题
饰带品种在 Coxeter 突变下保持不变
DOI：
10.1090/conm/761/15310
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Schiffler, Ralf
中科院分区：
文献类型：
--
作者：
Igusa, Kiyoshi;Schiffler, Ralf
我们定义了由Lee， Li, Mills， Seceleanu和第二作者引入的frieze品种的广义版本。广义frieze变种是由一个无环颤振和该颤振的簇代数中簇变量的一般特化所决定的代数变种。当这种专门化为（1，…）时，获得原始的横条品种。, 1)。主要的结果是，一个广义的粗纱品种是由该品种的任何成分的任何一般元素决定的。我们还表明，“考克斯特突变”循环排列这些成分。特别地，这表明在一般点的Coxeter突变下，frieze变异是不变的。本文包含了许多使用我们称之为不变洛朗多项式的新技术生成的例子。我们证明了一个颤振突变的对称给出了这样一个不变有理函数。
We define a generalized version of the frieze variety introduced by Lee, Li, Mills, Seceleanu and the second author. The generalized frieze variety is an algebraic variety determined by an acyclic quiver and a generic specialization of cluster variables in the cluster algebra for this quiver. The original frieze variety is obtained when this specialization is (1, . . . , 1). The main result is that a generalized frieze variety is determined by any generic element of any component of that variety. We also show that the "Coxeter mutation" cyclically permutes these components. In particular, this shows that the frieze variety is invariant under the Coxeter mutation at a generic point. The paper contains many examples which are generated using a new technique which we call an invariant Laurent polynomial. We show that a symmetry of a mutation of a quiver gives such an invariant rational function.