Higher order representation stability and ordered configuration spaces of manifolds

Using the language of twisted skew-commutative algebras, we define secondary representation stability, a stability pattern in the unstable homology of spaces that are representation stable in the sense of Church, Ellenberg, and Farb [CEF15]. We show that the rational homology of configuration spaces of ordered particles in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing particles near the boundary, secondary representation stability involves stabilizing by introducing pairs of orbiting particles. This result can be thought of as a representation-theoretic analogue of secondary homological stability in the sense of Galatius, Kupers, and Randal-Williams [GKRW]. In the course of the proof we establish some additional results: we give a new characterization of the integral homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds. In an appendix, we use results on FI-homology to give explicit stable ranges for the integral cohomology of configuration spaces of closed manifolds.