On the tails of FI-modules

TL;DR

This paper investigates the asymptotic properties of integer-valued FI-modules, introducing tail invariants and establishing an equivalence of categories with finitely supported modules in a new category called FJ.

Contribution

It defines tail invariants for FI-modules and proves an equivalence of categories between FI-tails and finitely supported FJ-modules, advancing understanding of their end-behavior.

Findings

High degrees of FI-modules characterized by tail invariants

Equivalence of categories between FI-tails and FJ-finitely supported modules

Introduction of the FJ category with Lie bracket-based morphisms

Abstract

We study the end-behavior of integer-valued FI-modules. Our first result describes the high degrees of an FI-module in terms of newly defined tail invariants. Our main result provides an equivalence of categories between FI-tails and finitely supported modules for a new category that we call FJ. Objects of FJ are natural numbers, and morphisms are infinite series with summands drawn from certain modules of Lie brackets.