FI-modules and stability for representations of symmetric groups

TL;DR

This paper develops the theory of FI-modules to analyze symmetric group representations, proving polynomial character and dimension stability for various geometric and algebraic structures as n grows.

Contribution

Introduces FI-modules and applies them to establish representation stability and polynomial behavior of characters and dimensions in symmetric group actions.

Findings

Characters are polynomial in cycle counts for large n

Dimensions of representations are eventually polynomial in n

Representation stability is equivalent to finite generation of FI-modules

Abstract

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.