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

在2021年4月至2022年3月的研究中，研究人员发表了两篇论文，题为“~A和~D型簇代数：三角曲面和friezes”和“Period 2 quivers及其T和y系统”。其中的第一篇已经提交给了《代数组合学杂志》，并且已经完成了修正。在第一篇论文中，我们能够对仿射~A和~D型簇代数的簇变量进行分类。这是通过将这些聚类变量视为适当表面的三角形弧线来完成的，这是[Fomin， Shapiro和Thurston 2008]首创的方法。通过首先将[Fordy and Hone 2014]和[Pallister 2020]中研究的t系统集群变量和周期量识别为弧（集群变量），我们能够根据这些识别所有其他集群变量。我们还能够证明这些簇变量可以被排列成friezes。我们的结果与通过这些颤振的表示理论得到的类似结果一致。在第二篇论文中，我们感兴趣的是找到所有周期为2的振动。这些颤栗，如果突变两次，看起来基本上是一样的。这是一个很困难的问题，相反，我们发现所有的周期二颤振的顶点数量都很低，而顶点的数量却惊人地多。然后，我们使用[Nakanishi 2011]的结果来编写这些颤振的T-和Y-系统，这里是2递归关系的系统。我们看了一些有周期量的有趣例子。

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.