Using the language of twisted skew-commutative algebras, we define
secondaryrepresentation stability, a stability pattern in the
unstable homology of spaces that
are representation stable in the sense of Church, Ellenberg and Farb (2015). We
show that the rational homology of configuration spaces of ordered points in
noncompact manifolds satisfies secondary representation stability. While
representation stability for the homology of configuration spaces involves stabilizing
by introducing a point “near infinity”, secondary representation stability
involves stabilizing by introducing a pair of orbiting points — an operation that
relates homology groups in different homological degrees. This result can be
thought of as a representation-theoretic analogue of
secondary homologicalstability in the sense of Galatius, Kupers and Randal-Williams (2018). In
the course of the proof we establish some additional results: we give a new
characterization of the 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.
