Polynomial conditions and homology of FI-modules

Abstract

We identify two recursively defined polynomial conditions for FI-modules in the literature. We characterize these conditions using homological invariants of FI-modules (namely the local degree and regularity, together with the stable degree) and clarify their relationship. For one of these conditions, we give improved twisted homological stability ranges for the symmetric groups. As another application, we improve the representation stability ranges for congruence subgroups with respect to the action of an appropriate linear group by a factor of 2 in its slope.

arXiv comment21 pages. To appear in Pacific Journal of Mathematics

References in corpus (8)

• FI-modules and stability for representations of symmetric groups
• FI-modules over Noetherian rings
• Representation stability and finite linear groups
• Homology of FI-modules
• Stability in the homology of congruence subgroups
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Regularity of FI-modules and local cohomology
• Central stability homology