On the symmetric group action on rigid disks on a strip

Abstract

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${}_{\ast }$-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 $2$.

arXiv comment33 pages, 8 figures