Local characterizations for decomposability of 2-parameter persistence modules

TL;DR

This paper explores local conditions for decomposing 2-parameter persistence modules into indecomposables, focusing on rectangle modules, and establishes the largest subclass with such characterizations.

Contribution

It provides a local characterization for decomposing rectangle modules in 2-parameter persistence, identifying the largest class with this property.

Findings

Rectangle modules admit a local decomposition characterization.

The class of rectangle modules is the largest with this property.

Not all interval modules have such local characterizations.

Abstract

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.