[Paper Review] FI-modules: a new approach to stability for S_n-representations

Thomas M. Church, Jordan S. Ellenberg+1 more|arXiv (Cornell University)|Apr 20, 2012

Advanced Combinatorial Mathematics8 citations

TL;DR

This paper introduces FI-modules as a framework to study stability in symmetric group representations, showing that in fixed degree, the character of Sn-representations is eventually given by a polynomial in cycle-counting functions—implying dimensions are eventually polynomial in n. The theory unifies and extends representation stability across diverse algebraic and topological settings.

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 ordered n-tuples on an arbitrary 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 × n matrices • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 and more. The symmetric group Sn 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. FI-modules are a refinement of Church–Farb’s theory of representation stability for Snrepresentations. In this framework, a complicated sequence of Sn-representations becomes a

Motivation & Objective

• To develop a systematic framework for understanding stability in sequences of symmetric group representations (S_n-representations).
• To address the lack of knowledge about characters and dimensions in key algebraic and topological representations of S_n.
• To generalize Church–Farb’s theory of representation stability by introducing FI-modules as a categorical refinement.
• To unify diverse settings—configuration spaces, moduli spaces, cohomology of Torelli groups—under a common algebraic structure.
• To prove that in fixed degree, the character of the representation is eventually given by a universal polynomial in cycle-counting functions.

Proposed method

• Introduce FI-modules as functors from the category of finite sets and injections to the category of vector spaces.
• Use the FI-module structure to encode sequences of S_n-representations with compatible actions.
• Apply homological algebra techniques to analyze the structure of FI-modules, especially their stability properties.
• Leverage the fact that FI-modules of finite type have characters that stabilize in fixed degree to deduce polynomial behavior.
• Establish that the character of a fixed-degree component is a polynomial in cycle-counting functions independent of n for large n.
• Apply the framework to specific settings like configuration spaces, diagonal coinvariant algebras, and moduli spaces of curves.

Experimental results

Research questions

• RQ1 Can a unified framework explain the stability of S_n-representations across diverse algebraic and topological contexts?
• RQ2 To what extent can the characters of S_n-representations in fixed degree be described by universal polynomials in cycle-counting functions?
• RQ3 Does the dimension of the representation in fixed degree become a polynomial in n for sufficiently large n?
• RQ4 How does the FI-module structure refine and generalize Church–Farb’s theory of representation stability?
• RQ5 Can FI-modules be used to derive new results about cohomology rings and tautological classes in moduli spaces?

Key findings

• For each fixed degree, the character of the S_n-representation is eventually given by a polynomial in the cycle-counting functions, independent of n.
• The dimension of the representation in fixed degree is eventually a polynomial in n, resolving long-standing questions about asymptotic behavior.
• The theory applies uniformly to configuration spaces of manifolds, diagonal coinvariant algebras, moduli spaces of curves, and Torelli group cohomology.
• FI-modules provide a categorical framework that refines and extends the theory of representation stability beyond the scope of previous approaches.
• The cohomology of the genus n Torelli group, generated by H^1, exhibits stable character behavior described by universal polynomials.
• The diagonal coinvariant algebra on r sets of n variables admits a stable character description in fixed degree via FI-module structure.