Representation stability, secondary stability, and polynomial functors
arXiv:1910.05574
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.
arXiv comment62 pages, 9 figures. Major revisions
References in corpus (10)
- FI-modules and stability for representations of symmetric groups
- Representation theory and homological stability
- Stabilization for mapping class groups of 3-manifolds
- Representation stability and finite linear groups
- Homology of FI-modules
- Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map
- Central stability for the homology of congruence subgroups and the second homology of Torelli groups
- Stable homology of surface diffeomorphism groups made discrete
- Improved homological stability for configuration spaces after inverting 2
- On the edge of the stable range