Abstract: In this dissertation we study the ordered configuration space of open unit-diameter disks in the infinite strip of width w. For a sufficiently nice manifold X, Church--Ellenberg--Farb proved that the k-th homology of the ordered configuration space of n points in X stabilizes in a representation theoretic sense as n increases. Miller--Wilson extended this first-order representation stability to more general X, and they proved that there is a second-order representation stability pattern among the unstable homology classes.

The homology of the ordered configuration space of open unit-diameter disks in the infinite strip of width w has an algebraic structure, that of a twisted algebra, and we find a finite presentation for homology as such. We use this presentation to show that the ordered configuration space of open unit-diameter disks in the infinite strip of width w exhibits notions of first- through w/2-th-order representation stability, recovering several classical results for configurations of points in the plane in the process. In the case of the infinite strip of width 2, we use first-order representation stability to decompose homology as a direct sum of induced symmetric group representations. Additionally, we show that second-order representation stability for configurations of points in a manifold can be much more complex than first-order representation stability.

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