Decomposing zero-dimensional persistent homology over rooted tree quivers

[2411.19319] Decomposing zero-dimensional persistent homology over rooted tree quivers Skip to main content Search Submit Donate Log in Mathematics > Representation Theory arXiv:2411.19319 (math) [Submitted on 28 Nov 2024 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Decomposing zero-dimensional persistent homology over rooted tree quivers Authors: Riju Bindua , Thomas Brüstle , Luis Scoccola View PDF HTML (experimental) Abstract: Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is fundamental in persistence theory, where it is known as persistent homology, and where the category is typically a poset. Persistence theory is particularly successful when the poset is a finite linearly ordered set, owing to the fact that in this case its category of representations is of finite type. We show that when the poset is a rooted tree poset (a poset with a maximum and whose Hasse diagram is a tree) the additive closure of the category of representations obtainable as zero-dimensional persistent homology is of finite type, and give a quadratic-time algorithm for decomposition into indecomposables. In doing this, we give an algebraic characterization of the additive closure in terms of Ringel's tree modules, and show that its indecomposable objects are the reduced representations of Kinser. Comments: 28 pages, 10 figures, 6 tables Subjects: Representation Theory (math.RT) ; Computational Geometry (cs.CG); Algebraic Topology (math.AT) Cite as: arXiv:2411.19319 [math.RT] (or arXiv:2411.19319v2 [math.RT] for this version) https://doi.org/10.48550/arXiv.2411.19319 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Riju Bindua [ view email ] [v1] Thu, 28 Nov 2024 18:46:35 UTC (270 KB) [v2] Thu, 27 Aug 2026 02:47:23 UTC (778 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2024-11 Change to browse by: cs cs.CG math math.AT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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