A Survey of Representation Stability Theory
Anastasia Khomenko, Dhaniram Kesari

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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Advanced Operator Algebra Research
