论文标题
在刚性磁盘上的对称组动作上
On the symmetric group action on rigid disks on a strip
论文作者
论文摘要
在本文中，我们将$ p $打开的单位直径磁盘的有序配置空间的合理同源性分解为无限宽度$ 2 $的合理同源性，作为诱导$ s_ {n} $表示的直接总和。 Alpert证明了$ k^{\ text {th}} $ - $ n $ open单位直径磁盘的订购配置空间的积分同源性$ 2 $是一个fi $ _ {k+1} $ - 通过研究某些操作，称为“高in-inserology”，称为“高in-nigh-nigh-nigh-insology”。 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.在本文中，我们将理性同源性小组研究为$ s_ {n} $ - 表示。我们找到了$ h_ {k}（\ text {celt}（n，2）; \ mathbb {q}），$的新基础，并将其与ramos的结果一起使用$ h_ {k {k {k}（\ text {celt} {celt}（n，2）; \ mathbb {q} $ s y night and and Direction s y and and and and and and and and and and and and and and and and and and and and and and and and and and and and。免费fi $ _ {*} $ - 模块。我们使用这种分解来计算$ p $开放单位直径磁盘的无序配置空间的理性同源性的维度，该磁盘是$ 2 $的无限条带。
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 $2$.
