A Survey of 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.

15 pages, One figure

References in corpus (11)

• Representation stability and finite linear groups
• Homology of FI-modules
• Introduction to twisted commutative algebras
• FI-modules and the cohomology of modular representations of symmetric groups
• Noetherianity of some degree two twisted commutative algebras
• Representation Stability
• Proof of Stembridge's conjecture on stability of Kronecker coefficients
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• Homological degrees of representations of categories with shift functors
• FI_W-modules and constraints on classical Weyl group characters
• The partition algebra and the Kronecker coefficients