A generating function approach to new representation stability phenomena in orbit configuration spaces

[1911.02125] A generating function approach to new representation stability phenomena in orbit configuration spaces Skip to main content Search Submit Donate Log in Mathematics > Algebraic Topology arXiv:1911.02125 (math) [Submitted on 5 Nov 2019 ( v1 ), last revised 20 Apr 2020 (this version, v2)] Title: A generating function approach to new representation stability phenomena in orbit configuration spaces Authors: Christin Bibby , Nir Gadish View PDF HTML (experimental) Abstract: As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces: using the notion of twisted commutative algebras, which essentially categorify exponential generating functions. This idea allows for a factorization of the orbit configuration space "generating function" into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it reveals a sequence of primary, secondary, and higher representation stability phenomena. Based on this, we give a simple geometric technique for identifying new stabilization actions with finiteness properties, which we use to unify and generalize known stability results. As a first new application of our methods, we establish secondary and higher stability for configuration spaces on $i$-acyclic spaces. For another application, we describe a natural filtration by which one observes a filtered representation stability phenomenon in configuration spaces on graphs. Comments: 45 pages; minor changes to exposition Subjects: Algebraic Topology (math.AT) ; Combinatorics (math.CO); Representation Theory (math.RT) Cite as: arXiv:1911.02125 [math.AT] (or arXiv:1911.02125v2 [math.AT] for this version) https://doi.org/10.48550/arXiv.1911.02125 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Trans. Amer. Math. Soc. Ser. B 10 (2023), 241-287 Related DOI : https://doi.org/10.1090/btran/130 Focus to learn more DOI(s) linking to related resources Submission history From: Christin Bibby [ view email ] [v1] Tue, 5 Nov 2019 22:53:21 UTC (49 KB) [v2] Mon, 20 Apr 2020 21:10:42 UTC (51 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AT < prev | next > new | recent | 2019-11 Change to browse by: math math.CO math.RT 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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