A generating function approach to new representation stability phenomena in orbit configuration spaces
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- by Christin Bibby and Nir Gadish
- Trans. Amer. Math. Soc. Ser. B 10 (2023), 241-287
- DOI: https://doi.org/10.1090/btran/130
- Published electronically: February 9, 2023
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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 algebras in 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 suggests a sequence of primary, secondary, and higher representation stability phenomena. Based on this, we give a simple geometric recipe for identifying new stabilization actions with finiteness properties in some cases, which we use to unify and generalize known stability results. We demonstrate our method by characterizing 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.References
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Bibliographic Information
- Christin Bibby
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Address at time of publication: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- MR Author ID: 1157407
- ORCID: 0000-0003-3840-6199
- Email: bibby@math.lsu.edu
- Nir Gadish
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 1211998
- ORCID: 0000-0003-4479-0537
- Email: gadish@umich.edu
- Received by editor(s): November 1, 2021
- Received by editor(s) in revised form: July 22, 2022
- Published electronically: February 9, 2023
- Additional Notes: The second author was supported by NSF Grant No. DMS-1902762
- © Copyright 2023 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 10 (2023), 241-287
- MSC (2020): Primary 55R80; Secondary 20C30, 05E18, 14L30
- DOI: https://doi.org/10.1090/btran/130
- MathSciNet review: 4546785