Generalized Representation Stability and FI_d-modules

TL;DR

This paper extends the theory of FI-modules to FI_d, showing that finitely generated FI_d-modules display representation stability properties similar to those in the classical FI setting, broadening the understanding of algebraic stability phenomena.

Contribution

It generalizes the Church-Farb representation stability theorem from FI-modules to FI_d-modules, establishing similar stability behaviors in a more complex category.

Findings

Finitely generated FI_d-modules exhibit representation stability.

Generalization of Church-Farb stability theorem to FI_d.

Provides new insights into complex representation theory.

Abstract

In this note we consider the complex representation theory of FI_d, a natural generalization of the category FI of finite sets and injections. We prove that finitely generated FI_d-modules exhibit behaviors in the spirit of Church-Farb representation stability theory, generalizing a theorem of Church, Ellenberg, and Farb which connects finite generation of FI-modules to representation stability.