A classification of injective FI$^m$-modules

TL;DR

This paper extends a shift theorem for FI$^m$-modules and classifies finitely generated injective modules over a characteristic zero field, advancing the understanding of their structure.

Contribution

It generalizes a key shift theorem and provides a classification of finitely generated injective FI$^m$-modules, a significant step in representation theory.

Findings

Generalized the shift theorem for FI$^m$-modules

Classified finitely generated injective FI$^m$-modules over characteristic zero fields

Enhanced understanding of the structure of FI$^m$-modules

Abstract

In this paper we generalize a shift theorem, which plays a key role in studying representations of FI$^m$, the product category of the category of finite sets and injections, and classify finitely generated injective FI$^m$-modules over a field of characteristic 0.