On the symmetric group action on rigid disks in a strip

On the symmetric group action on rigid disks in a strip In this paper we decompose the rational homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 as a direct sum of induced Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_{n}$$\end{document}-representations. In (Alpert in Generalized representation stability for disks in a strip and no-k-equal spaces, arXiv:2006.01240, 2020), Alpert proves that the kth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k^{\text {th}}$$\end{document}-integral homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 is an FIk+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_{k+1}$$\end{document}-module by studying certain operations on homology called “high-insertion maps.” The integral homology groups Hk(Conf(n,2))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{k}\big (\text {Conf}(n,2)\big )$$\end{document} are free abelian, and Alpert computes a basis for Hk(Conf(n,2))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{k}\big (\text {Conf}(n,2)\big )$$\end{document} as an abelian group. In this paper, we study the rational homology groups as Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_{n}$$\end{document}-representations. We find a new basis for H1(Conf(n,2);Q),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{1}\big (\text {Conf}(n,2);\mathbb {Q}\big ),$$\end{document} and use this, along with results of Ramos (Proc Am Math Soc 145(11):4647–4660, 2017), to give an explicit description of Hk(Conf(n,2);Q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{k}\big (\text {Conf}(n,2);\mathbb {Q}\big )$$\end{document} as a direct sum of induced Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_{n}$$\end{document}-representations arising from free FId\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_{d}$$\end{document}-modules. We use this decomposition to calculate the rank of the rational homology of the unordered configuration space of n open unit-diameter disks in the infinite strip of width 2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

On the symmetric group action on rigid disks in a strip

, Volume 7 (3) – Sep 1, 2023
46 pages

/lp/springer-journals/on-the-symmetric-group-action-on-rigid-disks-in-a-strip-NSga02tox0
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Springer Journals
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-022-00111-8
Publisher site
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Abstract

In this paper we decompose the rational homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 as a direct sum of induced Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_{n}$$\end{document}-representations. In (Alpert in Generalized representation stability for disks in a strip and no-k-equal spaces, arXiv:2006.01240, 2020), Alpert proves that the kth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k^{\text {th}}$$\end{document}-integral homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 is an FIk+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_{k+1}$$\end{document}-module by studying certain operations on homology called “high-insertion maps.” The integral homology groups Hk(Conf(n,2))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{k}\big (\text {Conf}(n,2)\big )$$\end{document} are free abelian, and Alpert computes a basis for Hk(Conf(n,2))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{k}\big (\text {Conf}(n,2)\big )$$\end{document} as an abelian group. In this paper, we study the rational homology groups as Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_{n}$$\end{document}-representations. We find a new basis for H1(Conf(n,2);Q),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{1}\big (\text {Conf}(n,2);\mathbb {Q}\big ),$$\end{document} and use this, along with results of Ramos (Proc Am Math Soc 145(11):4647–4660, 2017), to give an explicit description of Hk(Conf(n,2);Q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{k}\big (\text {Conf}(n,2);\mathbb {Q}\big )$$\end{document} as a direct sum of induced Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_{n}$$\end{document}-representations arising from free FId\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_{d}$$\end{document}-modules. We use this decomposition to calculate the rank of the rational homology of the unordered configuration space of n open unit-diameter disks in the infinite strip of width 2.

Journal

Journal of Applied and Computational TopologySpringer Journals

Published: Sep 1, 2023

Keywords: Configuration Space; Representation Stability; Representation Theory; Homological Stability; 55R80; 55R40; 18A25

References

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