Categorification of Some Integer Sequences and Its Applications (2020)
Categorification of real valued sequences, and in particular of integer sequences is a novel line of investigation in the theory of representation of algebras. In this theory introduced by Ringel and Fahr, numbers of a sequence are interpreted as invariants of objects of a given category. The categorification of the Fibonacci numbers via the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver is an example of this kind of identifications. In this thesis, we follow the ideas of Ringel and Fahr to categorify several integer sequences but instead of using the 3-Kronecker quiver, we deal with a kind of algebras introduced recently by Green and Schroll called Brauer configuration algebras. Relationships between these algebras, some matrix problems and rational knots are used to interpret numbers in some integer sequences as invariants of indecomposable modules over path algebras of the 2-Kronecker quiver and the four subspace quiver. The results enable us to define the message of a Brauer Configuration and labeled Brauer configurations in order to give an interpretation of the number of perfect matchings of snake graphs, the number of homological ideals of some Nakayama algebras, and the number of k-paths linking two fixed points (associated to the Lindström problem) in a quiver as specializations of indecomposable modules over suitable Brauer configuration algebras. Actually, this setting can be also used to define the Gutman index of a tree (or the trace norm of a digraph, which is a fundamental notion in the topological index theory), magic squares, and different parameters of traffic flow models in terms of this kind of algebras. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.