FI-modules and stability for representations of symmetric groups

By Thomas Church, Jordan S. Ellenberg, Benson Farb. 2012

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5ba66f4e9c337fdf33447dd217fc47990b73957a

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Citations in the corpus

Higher-order representation stability and ordered configuration spaces of manifolds (2016, unknown): “Free FI–modules are highly constrained; all FI morphisms act by injective maps, and they are completely determined by their minimal generators; see Theorem 2.16, quoting [9, Theorem 4.1.5].”
Correction to the article A spectral sequence for stratified spaces and configuration spaces of points (2016, methodology): “= V (n)Sn for n ≫ 0; if {V (n)} were a sequence of homology groups, the Sn-invariants would satisfy homological stability in the usual sense. The theory was clariﬁed by the introduction of FI-modules [CEF15]. The key point is that the underlying sequence of Sn-representations of an FI-module in Q-ve”
Representation stability and finite linear groups (2014, methodology): “on The language of representation stability was introduced by Church–Farb [ChFa] to describe representation-theoretic patterns they observed in many parts of mathematics. In later work with Ellenberg [ChEFa], they introduced FI-modules, which gave a clearer framework for these patterns for the repre”
Homological stability for automorphism groups (2014, background): “ the homogeneous category we associate is the category FIof nite sets and injections, much studied recently by Church, Ellenberg, Farb, and others in the context of representation stability (see e.g. [18]). For the groupoid of nitely-generated R-modules and their isomorphisms, the construction yield”
FI- and OI-modules with varying coefficients (2017, methodology): “Our exposition is influenced by the approach in [3].”