线性递归序列与Dynkin图
Linearly recursive sequences and Dynkin diagrams
摘要：
受簇代数理论（Fomin和Zelevinsky）中的一个构造的启发，人们将每个非循环有向图关联到一个自然整数序列族，每个顶点一个;这种构造称为{ \em frieze};这些序列由非线性递归（带除法）给出，它们是整数的事实是Fomin和Zelevinsky的Laurent现象的结果。如果序列满足常系数线性递归，则该图必须是具有非循环方向的Dynkin图或扩展Dynkin图。反之亦然：与有向Dynkin图或欧几里德图相关联的中楣的序列满足线性递归，并且甚至是$\mathbb N$ -有理数。在证明中使用称为$SL_2$ -{ \em tilings of the plane}的对象，它们是离散平面的填充，使得每个相邻的2乘2子面等于1。这些对象在簇代数理论中有应用，它们本身就很有趣。给出了一些问题、解答和练习。
英文摘要：
Motivated by a construction in the theory of cluster algebras (Fomin and Zelevinsky), one associates to each acyclic directed graph a family of sequences of natural integers, one for each vertex; this construction is called a {\em frieze}; these sequences are given by nonlinear recursions (with division), and the fact that they are integers is a consequence of the Laurent phenomenon of Fomin and Zelevinsky. If the sequences satisfy a linear recursion with constant coefficients, then the graph must be a Dynkin diagram or an extended Dynkin diagram, with an acyclic orientation. The converse also holds: the sequences of the frieze associated to an oriented Dynkin or Euclidean diagram satisfy linear recursions, and are even $\mathbb N$-rational. One uses in the proof objects called $SL_2$-{\em tilings of the plane}, which are fillings of the discrete plane such that each adjacent 2 by 2 minor is equal to 1. These objects, which have applications in the theory of cluster algebras, are interesting for themselves. Some problems, conjectures and exercises are given.
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