FI$^m$-modules over Noetherian rings

TL;DR

This paper extends the representation theory of the category FI to its m-fold product FI^m, demonstrating that many properties, including representation stability, generalize to this broader setting over fields of characteristic zero.

Contribution

It introduces and studies FI^m-modules, generalizing FI-module properties, and proves their representation stability over characteristic zero fields.

Findings

Representation stability for finitely generated FI^m-modules

Generalization of FI properties to FI^m

Homological properties similar to FI

Abstract

In this paper we study representation theory of the category FI$^m$ introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI$^m$ in a natural way. In particular, we prove the representation stability property of finitely generated FI$^m$-modules over fields of characteristic 0.