Polynomial conditions and homology of FI-modules

TL;DR

This paper explores polynomial conditions and homological invariants of FI-modules, improving stability ranges for symmetric groups and congruence subgroups through homological methods.

Contribution

It characterizes polynomial conditions via homological invariants and enhances stability ranges for symmetric and congruence groups.

Findings

Improved twisted homological stability ranges for symmetric groups.

Enhanced representation stability ranges for congruence subgroups.

Clarified relationships between polynomial conditions and homological invariants.

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.