Generalized representation stability and FI_{^}-modules

Abstract

In this note we consider the complex representation theory of F I d \mathrm {FI}_d , a natural generalization of the category F I \mathrm {FI} of finite sets and injections. We prove that finitely generated F I d \mathrm {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 F I \mathrm {FI} -modules to representation stability.