Decomposition Methods for Representations of Quivers appearing in Topological Data Analysis

This paper describes methods to compute the decomposition type of representationsof some quivers that appear in topological data analysis. Specically, these are theAn-quivers, and the quivers CL(f) and CL(f; f). A background is given to explain theappearance of these quivers in the data analysis setting. The theoretical backgroundcovers the basic category theoretical concepts used to dene decomposition of repre-sentations, as well as the necessary tools used to nd indecomposable representations.A brief review of how these tools can be applied on An-quivers is given. The mainresults in the paper are the matrices presented at the ends of sections 4 and 5, whichgive linear relations between vectors representing the decomposition types of repre-sentations of these quivers and vectors of numerical invariants of the representations.These invariants are simply the dimensions of some subspaces of the vector spaces inthe representation, and are therefore easily calculated. Finally, more general laddersand grids are considered, and it it proven that the only grids of nite representationtype are the ladders with at most four rungs