Representation Stability for Disks in a Strip

Abstract

We consider the ordered configuration space of n open unit-diameter disks in the infinite strip of width w. In the spirit of Arnol'd and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that the ordered configuration space of open unit-diameter disks in the infinite strip of width w exhibits a notion of first-order representation stability similar to Church-Ellenberg-Farb and Miller-Wilson's firstorder representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large w this disk configuration space exhibits notions of second-(and higher) order representation stability.