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Affine cluster algebras as dynamical systems, surface triangulations, quiver representations and friezes

仿射簇代数作为动力系统、表面三角测量、箭袋表示和饰带

基本信息

批准号：
21F20788
负责人：
山崎玲
金额：
$ 0.19万
依托单位：
Chiba University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2021
资助国家：
日本
起止时间：
2021-04-28 至 2023-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21F20788/
关键词：
cluster algebras dynamical systems friezes T- and Y-systems representation theory

项目摘要

The research of the year April 2021 - March 2022 has resulted in the two paperstitled ``~A and ~D type cluster algebras: Triangulated surfaces and friezes'' and ``Period 2 quivers and their T- and Y-systems''. The first of those has been submitted to the Journal of Algebraic Combinatorics, and the corrections have been completed.In the first paper we were able to classify cluster variables for cluster algebras of affine ~A and ~D type. This was done by looking at these cluster variables as arcs of triangulations for appropriate surfaces, a method pioneered in [Fomin, Shapiro and Thurston 2008]. By first identifying the T-system cluster variables and periodic quantities studied in [Fordy and Hone 2014] and [Pallister 2020] as arcs (cluster variables) we were able to identify all other cluster variables in terms of these. We were also able to prove that these cluster variables can be arranged as friezes. Our results agree with similar results found via the representation theory of these quivers. In the second paper we were interested in finding all period 2 quivers. These are quivers that, if mutated twice, look essentially the same. This turned out to be a difficult problem, instead we found all period two quivers with a low number of vertices, of which there are surprisingly many. We then used results of [Nakanishi 2011] to write the T- and Y- systems for these quivers, which here are systems of 2 recurrence relations. We looked at some interesting examples of these which had periodic quantities.

2021年4月至2022年3月的研究已经产生了两篇论文《~A和~D型簇代数：三角化曲面和褶皱》和《周期2箭图及其T-和Y-系》。在第一篇论文中，我们能够对仿射~A和~D类型的簇代数进行分类。这是通过将这些集群变量视为适当表面的三角剖分的弧线来完成的，这是[Fomin，Shapiro和瑟斯顿2008]首创的方法。通过首先将[Fordy and Hone 2014]和[Pallister 2020]中研究的T系统簇变量和周期量识别为圆弧(簇变量)，我们能够根据这些识别出所有其他的簇变量。我们还证明了这些星系团变量可以排列成片状。我们的结果与通过这些颤动的表象理论发现的类似结果一致。在第二篇论文中，我们感兴趣的是找出所有周期2的颤动。这些颤动如果突变两次，看起来基本上是一样的。这被证明是一个困难的问题，相反，我们发现所有的周期2箭图的顶点数都很少，令人惊讶的是，顶点数很多。然后我们使用[Nakanishi 2011]的结果来写出这些箭图的T-和Y-系统，这里是2个递归关系的系统。我们看了一些有趣的例子，这些例子都有周期量。