Regularity and stable ranges of FI-modules

TL;DR

This paper refines bounds on the regularity and stable ranges of FI-modules, providing sharper results and explicit stable ranges for various algebraic and topological structures, advancing the understanding of their stabilization properties.

Contribution

It offers improved bounds for FI-modules' regularity and stable ranges, with sharpness in several cases, and applies these to specific algebraic and topological contexts.

Findings

Bounds for FI-modules' regularity are sharpened.

Explicit stable ranges are derived for diagonal coinvariant algebras.

Improved stable ranges are established for configuration spaces and congruence subgroups.

Abstract

We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We apply these to get explicit stable ranges for diagonal coinvariant algebras, and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.