On the symmetric group action on rigid disks in a strip

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