A Survey of Representation Stability Theory

TL;DR

This survey reviews the development of representation stability theory, highlighting three main approaches and their applications to symmetric groups and other groups, summarizing research progress up to 2015.

Contribution

It provides a comprehensive overview of the three approaches to representation stability theory and their recent advancements.

Findings

Summarizes three approaches: FI-modules, Schur-Weyl duality, and modules over infinite-dimensional spaces.

Highlights the stability of symmetric group representations and generalizations to other groups.

Synthesizes research efforts from 2015 on representation stability theory.

Abstract

In this survey article we summarize the current state of research in representation stability theory. We look at three different, yet related, approaches, using (1) the category of FI-modules, (2) Schur-Weyl duality, and (3) finitely-generated modules over certain infinite dimensional vector spaces. The main example is the stability of representations of the symmetric group, though there have also been some notable generalizations of representation stability to other groups. This work summarizes the research that both authors engaged in over the course of the summer of 2015.