论文标题
在类型$ \ mathbb {a} $和$ \ tilde {\ mathbb {a}} $中
On Clusters and Exceptional Sets in Types $\mathbb{A}$ and $\tilde{\mathbb{A}}$
论文作者
论文摘要
在本文中，我们首先研究$ \ tilde {\ mathbb {a}} $的类型群集，通过将它们收集到有限数量的无限家庭中，由dehn twist twists twists twist，并表明这些家族被加泰罗尼亚人的数字计数。我们还强调了用于研究簇的Annuli图与用于研究类型$ \ tilde {\ Mathbb {a}} $的异常集的Annuli图之间的相似性和差异。然后，我们首先表明特殊集合中相对投影率的概念是很好的定义，我们将重点放在模块的特殊集合（集合）上。我们通过计算具有直方向的类型$ \ mathbb {a} $ Quivers类型的特殊表示形式的数量，并使用它来计算具有直接方向的类型$ \ tilde {\ tilde {\ tilde {\ mathbb {a}} $的典型家族的数量。
In this paper we first study clusters in type $\tilde{\mathbb{A}}$ by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type $\tilde{\mathbb{A}}$. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type $\mathbb{A}$ quivers with straight orientation and using this to count the number of families of exceptional sets of type $\tilde{\mathbb{A}}$ with straight orientation.
