Decomposition of Certain Representations Into A Direct Sum of Indecomposable Representations

Abstract

Representations of quivers and posets are produced by persistent homology. It is possible to decompose such representations into direct sums of indecomposable representations. The indecomposable representations of quivers and posets that arise from one dimensional persistent homology are well understood. However, the same is not true for multidimensional persistent homology. This thesis finds such a list of indecomposable representations for a very simple case of a poset that can arise from multidimensional persistent homology, and proves that it is possible to decompose a representation of such a poset into a direct sum of these indecomposables.