The (co)homology of vertical configuration spaces is representation stable as S_k≀S_n-representations, with explicit stable ranges and an improved homological stability range for the unordered spaces.
Homology of FI-modules
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abstract
We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.
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Representation stability in the (co)homology of vertical configuration spaces
The (co)homology of vertical configuration spaces is representation stable as S_k≀S_n-representations, with explicit stable ranges and an improved homological stability range for the unordered spaces.