The homology groups of ordered Hurwitz spaces, seen as representations of symmetric groups, have stable multiplicities in a range that grows linearly with homological degree.
Representation stability, secondary stability, and polynomial functors
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abstract
We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum.
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Representation stability for ordered Hurwitz spaces
The homology groups of ordered Hurwitz spaces, seen as representations of symmetric groups, have stable multiplicities in a range that grows linearly with homological degree.