  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.

Algebraic Topology;Algebraic Geometry;Representation Theory