On the symmetric group action on rigid disks in a strip

Abstract

In this paper we decompose the rational homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 as a direct sum of induced \(S_{n}\)-representations. In (Alpert in Generalized representation stability for disks in a strip and no-k-equal spaces, arXiv:2006.01240, 2020), Alpert proves that the \(k^{\text {th}}\)-integral homology of the ordered configuration space of n open unit-diameter disks in 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 \(H_{k}\big (\text {Conf}(n,2)\big )\) are free abelian, and Alpert computes a basis for \(H_{k}\big (\text {Conf}(n,2)\big )\) as an abelian group. In this paper, we study the rational homology groups as \(S_{n}\)-representations. We find a new basis for \(H_{1}\big (\text {Conf}(n,2);\mathbb {Q}\big ),\) and use this, along with results of Ramos (Proc Am Math Soc 145(11):4647–4660, 2017), to give an explicit description of \(H_{k}\big (\text {Conf}(n,2);\mathbb {Q}\big )\) as a direct sum of induced \(S_{n}\)-representations arising from free FI\(_{d}\)-modules. We use this decomposition to calculate the rank of the rational homology of the unordered configuration space of n open unit-diameter disks in the infinite strip of width 2.