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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

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Springer Journals
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2367-1726
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2367-1734
DOI
10.1007/s41468-022-00111-8
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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 Topology – Springer Journals

Published: Sep 1, 2023

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

References

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    Alpert, H; Kahle, M; MacPherson, R
  • Min-type Morse theory for configuration spaces of hard spheres
    Baryshnikov, Y; Bubenik, P; Kahle, M
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    Church, T; Ellenberg, JS; Farb, B; Nagpal, R
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    Church, T; Ellenberg, JS; Farb, B
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    Ramos, E
  • Configuration spaces of graphs with certain permitted collisions
    Ramos, E
  • Gröbner methods for representations of combinatorial categorie
    Sam, S; Snowden, A