An inductive machinery for representations of categories with shift functors

TL;DR

This paper introduces an inductive method to establish key properties of representations of categories with shift functors, simplifying proofs of Noetherianity, regularity, and growth in representation stability contexts.

Contribution

It develops a systematic inductive machinery that leverages shift functors to prove properties of finitely generated representations in categories like FI_G and OI_G.

Findings

Proves finitely generated representations are Noetherian under certain conditions.

Establishes bounds on Castelnuovo-Mumford regularity.

Shows polynomial growth of dimensions in specific categories.

Abstract

We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories $\FI_G$ and $\OI_G$ which appear in representation stability theory.