Harder-Narasimhan Filtrations of Persistence Modules

Abstract: The Harder-Narasimhan type of a quiver representation is a discrete invariant parameterised by a real-valued function (called a central charge) defined on the vertices of the quiver. In this paper, we investigate the strength and limitations of Harder-Narasimhan types for several families of quiver representations which arise in the study of persistence modules. We introduce the skyscraper invariant, which amalgamates the HN types along central charges supported at single vertices, and generalise the rank invariant from multiparameter persistence modules to arbitrary quiver representations. Our four main results are as follows: (1) we show that the skyscraper invariant is strictly finer than the rank invariant in full generality, (2) we characterise the set of complete central charges for zigzag (and hence, ordinary) persistence modules, (3) we extend the preceding characterisation to rectangle-decomposable multiparameter persistence modules of arbitrary dimension; and finally, (4) we show that although no single central charge is complete for nestfree ladder persistence modules, a finite set of central charges is complete.

• The paper introduces the skyscraper invariant to extend the conventional rank invariant, capturing richer structural details in persistence modules.
• It categorically characterizes complete central charges for zigzag and ordinary persistence modules, enhancing classification in persistent homology.
• The study classifies central charges for rectangle-decomposable and nestfree ladder persistence modules, providing insights for improved topological data algorithms.

Insights into Harder-Narasimhan Filtrations of Persistence Modules

The paper "Harder-Narasimhan Filtrations of Persistence Modules" explores the application of the Harder-Narasimhan (HN) filtration, a concept traditionally used in the stability analysis of complex vector bundles and quiver representations, to the field of persistence modules. The authors introduce a new concept termed the skyscraper invariant and offer a broader generalization of the rank invariant within the scope of multiparameter persistence modules and arbitrary quiver representations.

Central Contributions

1. Skyscraper Invariant: By leveraging the Harder-Narasimhan types of quiver representations, the authors introduce the skyscraper invariant, which extends beyond the conventional rank invariant. They demonstrate that the skyscraper invariant captures richer structural details and provides a more discriminative measure than the rank invariant, especially in the context of multidimensional persistence modules.
2. Characterization of Complete Central Charges: The paper provides a categorical characterization of complete central charges for zigzag and ordinary persistence modules. These findings contribute to a refined understanding of how quiver representations associated with persistence modules can be effectively characterized, augmenting the classification schemes used in persistent homology.
3. Exploration of Rectangle-Decomposable Modules: The analysis extends to two-dimensional multiparameter persistence modules, particularly focusing on rectangle-decomposable modules. The results in this area suggest a classification of complete central charges that are independent of dimension, providing insights into their applications in higher-dimensional data settings.
4. Nestfree Ladder Persistence Modules: An intriguing aspect of the study is the extension of central charge classification to nestfree ladder persistence modules. The paper claims that although no single central charge is complete for these modules, a finite collection can provide completeness. This finding enriches the existing literature on ladder persistence modules by identifying a set of central charges that can adequately capture their structure.

Theoretical and Practical Implications

The theoretical implications of this research provide a foundational understanding that bridges algebraic topology, representation theory, and data analysis. By advancing the discriminative power of invariants derived from quiver representations, the study enhances the toolbox available for computational topology, particularly in data-intensive fields such as machine learning and computational geometry.

Practically, this paper has significant implications for topological data analysis (TDA). The results have the potential to improve algorithms for processing high-dimensional data, enabling better noise reduction and feature extraction strategies in large datasets. Further, the advent of a finer invariant (the skyscraper invariant) paves the way for more nuanced analysis of data structures that are commonly encountered in applications ranging from computational biology to sensor networks.

Future Directions

While this paper addresses several pivotal issues, it simultaneously opens new avenues for research. Subsequent work could focus on the development and implementation of efficient algorithms that utilize these enhanced invariants in real-time data processing scenarios. Moreover, further exploration into the properties and applicability of these techniques to other classes of data structures would be valuable. For instance, investigating the role of these invariants in the dynamic setting of data streams could extend their utility.

In conclusion, this paper represents a thoughtful expansion of the role of Harder-Narasimhan filtrations within the field of persistence modules. It provides both a theoretical framework and a set of practical tools that enrich the ongoing dialogue between topology, algebra, and data science.