关于对无限带宽为2的刚性圆盘上对称群作用
On the symmetric group action on rigid disks on a strip
AI总结：
本文研究了无限带宽为2的带上的刚性圆盘配置空间的对称群作用，通过同调分解和基计算，给出了有理同调作为诱导表示的显式描述。
摘要：
在本文中，我们将$p$个开单位直径圆盘在无限带宽为2的带上的有序配置空间的有理同调分解为诱导的$S_{n}$-表示的直接和。Alpert证明了通过研究称为'高插入映射'的某些同调操作，$n$个开单位直径圆盘在无限带宽为2的带上的有序配置空间的第$k$积分同调是一个FI$_{k+1}$-模块。积分同调群$H_{k}(\text{cell}(n,2))$是自由交换群，Alpert计算了$H_{k}(\text{cell}(n,2))$作为交换群的一个基。在本文中，我们研究有理同调群作为$S_{n}$-表示。我们发现$H_{k}(\text{cell}(n,2);\mathbb{Q})$的新基，并利用这一结果以及Ramos的结果，给出了$H_{k}(\text{cell}(n,2);\mathbb{Q})$作为来自自由FI$_{*}$-模诱导的$S_{n}$-表示的直接和的显式描述。我们利用这一分解来计算无限带宽为2的带上的$p$个开单位直径圆盘的无序配置空间的有理同调的维数。
英文摘要：
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 $2$.
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