Refinement of Interval Approximations for Fully Commutative Quivers

TL;DR

This paper introduces a new framework for analyzing fully commutative quivers in multiparameter persistent homology, improving invariant approximation and computational efficiency for complex topological data.

Contribution

It proposes an enhanced, holistic approach to quiver representation analysis, including a novel invariant for infinite-type cases and efficient decomposition methods using one-parameter persistence.

Findings

Efficient indecomposable decomposition for finite-type quivers.

New invariant for detecting persistence in the second parameter.

Effective analysis of material topology using the proposed toolkit.

Abstract

A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative quiver's representation via synthesizing interpretations obtained from intervals. Additionally, it provides a mechanism to tune the balance between approximation resolution and computational complexity. This framework is evaluated on commutative ladders of both finite-type and infinite-type. For the former, we discover an efficient method for the indecomposable decomposition leveraging solely one-parameter persistent homology. For the latter, we introduce a new invariant that reveals persistence in the second parameter by connecting two standard persistence diagrams using interval approximations. We subsequently present several models for constructing commutative ladder filtrations, offering fresh insights into random filtrations and demonstrating our toolkit's effectiveness in analyzing the topology of materials.