Completely-decomposable subcategories of quiver representations

TL;DR

This paper develops a framework for identifying and decomposing subcategories of multi-parameter quiver representations, facilitating computational analysis of complex persistence modules in topological data analysis.

Contribution

It introduces a method to restrict to subcategories of quiver representations that are computationally distinguishable, aiding in the decomposition of multi-parameter persistence modules.

Findings

Framework for subcategory restriction in quiver representations

Enhanced decomposition methods for multi-parameter persistence modules

Potential for improved computational topological data analysis

Abstract

When filtering a topological space by a single parameter, the theory of quiver representations provides a complete framework for decomposing the resulting persistence module to obtain its barcode. This is achieved by interpreting the persistence module as a representation of a Type $\mathbb{A}_n$ quiver. The complexity increases significantly when filtering by two or more parameters. In particular, multi-parameter persistence typically yields tame or wild type quivers whose indecomposable representations are more complicated to describe and for which arbitrary representations are much more difficult to decompose. The theme of this work is to provide a framework for restricting to subcategories of quiver representations whose objects can be distinguished from one another through computational means.