On the symmetric group action on rigid disks on a strip

TL;DR

This paper decomposes the rational homology of configuration spaces of disks on a strip into induced symmetric group representations, providing explicit descriptions and dimension calculations for these homology groups.

Contribution

It introduces a new basis for the rational homology and explicitly describes it as a sum of induced $S_{n}$-representations, advancing understanding of symmetric group actions on configuration spaces.

Findings

Explicit decomposition of rational homology as induced $S_{n}$-representations.

New basis for the rational homology groups.

Calculated dimensions of the homology of unordered configurations.

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$_{*}$-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 $2$.