On the symmetric group action on rigid disks on a strip
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910032916905984 |
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| author | Wawrykow, Nicholas |
| author_facet | Wawrykow, Nicholas |
| contents | In this paper we decompose the rational homology of the ordered configuration space of $p$ open unit-diameter disks on the infinite strip of width $2$ as a direct sum of induced $S_{n}$-representations. Alpert proved that the $k^{\text{th}}$-integral homology of the ordered configuration space of $n$ open unit-diameter disks on the infinite strip of width $2$ is an FI$_{k+1}$-module by studying certain operations on homology called "high-insertion maps." The integral homology groups $H_{k}(\text{cell}(n,2))$ are free abelian, and Alpert computed a basis for $H_{k}(\text{cell}(n,2))$ as an abelian group. In this paper, we study the rational homology groups as $S_{n}$-representations. We find a new basis for $H_{k}(\text{cell}(n,2);\mathbb{Q}),$ and use this, along with results of Ramos, to give an explicit description of $H_{k}(\text{cell}(n,2);\mathbb{Q})$ as a direct sum of induced $S_{n}$-representations arising from free FI$_{*}$-modules. We use this decomposition to calculate the dimension of the rational homology of the unordered configuration space of $p$ open unit-diameter disks on the infinite strip of width $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_00718 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the symmetric group action on rigid disks on a strip Wawrykow, Nicholas Algebraic Topology Combinatorics Representation Theory 55R80 In this paper we decompose the rational homology of the ordered configuration space of $p$ open unit-diameter disks on the infinite strip of width $2$ as a direct sum of induced $S_{n}$-representations. Alpert proved that the $k^{\text{th}}$-integral homology of the ordered configuration space of $n$ open unit-diameter disks on the infinite strip of width $2$ is an FI$_{k+1}$-module by studying certain operations on homology called "high-insertion maps." The integral homology groups $H_{k}(\text{cell}(n,2))$ are free abelian, and Alpert computed a basis for $H_{k}(\text{cell}(n,2))$ as an abelian group. In this paper, we study the rational homology groups as $S_{n}$-representations. We find a new basis for $H_{k}(\text{cell}(n,2);\mathbb{Q}),$ and use this, along with results of Ramos, to give an explicit description of $H_{k}(\text{cell}(n,2);\mathbb{Q})$ as a direct sum of induced $S_{n}$-representations arising from free FI$_{*}$-modules. We use this decomposition to calculate the dimension of the rational homology of the unordered configuration space of $p$ open unit-diameter disks on the infinite strip of width $2$. |
| title | On the symmetric group action on rigid disks on a strip |
| topic | Algebraic Topology Combinatorics Representation Theory 55R80 |
| url | https://arxiv.org/abs/2201.00718 |