Dynkin Functions and Its Applications (2020)
Dynkin functions were introduced by Ringel as a tool to investigate combinatorial properties of hereditary artin algebras. According to Ringel, a Dynkin function consists of four sequences associated to An, Bn, Cn, Dn and five single values associated to the diagrams E6, E7, E8, F4 and G2. He also proposes to create an On-line Encyclopedia of Dynkin functions (OEDF) with the same purposes as the famous OEIS. Dynkin functions arise from the context of categorification of integer sequences, which according to Ringel and Fahr it means to consider suitable objects in a category instead of numbers of a given integer sequence. They gave a categorification of Fibonacci numbers by using the Gabriel's universal covering theory and the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver. For instance, if Λ denotes a hereditary artin algebra associated to a Dynkin diagram ∆n then r(∆n) the number of indecomposable modules, a(∆n) the number of antichains in mod Λ, and tn(∆n) the number of tilting modules are Dynkin functions. In particular, we are focused on the way that some Dynkin functions act on Dynkin diagrams of type An. In this work, we follow the ideas of Ringel regarding Dynkin functions by investigating the number of sections in the Auslander-Reiten quiver of algebras of finite representation type. Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An and it is shown an algebraic interpretation of frieze patterns as a direct sum of indecomposable objects of the category of Dyck paths. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin type An in terms of Dyck paths. At last but not least, it is introduced some Brauer configuration algebras such that the dimension of these algebras and its corresponding centers can be obtained via some combinatorial properties of the Catalan triangle. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.