Representation stability for disks in a strip

Abstract

In this paper, we consider the ordered configuration space of [Formula: see text] open unit-diameter disks in the infinite strip of width [Formula: see text]. 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 [Formula: see text] exhibits a notion of first-order representation stability similar to Church–Ellenberg–Farb and Miller–Wilson’s first-order representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large [Formula: see text] this disk configuration space exhibits notions of second- (and higher) order representation stability.