FI-modules and stability for representations of symmetric groups

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.

54 pages. v4: new title, paper completely reorganized; final version, to appear in Duke Math Journal

References in corpus (10)

• FI-modules over Noetherian rings
• Representation stability and finite linear groups
• Cohomological Aspects of Magnus Expansions
• Discriminants in the Grothendieck Ring
• Introduction to twisted commutative algebras
• Tensor envelopes of regular categories
• Representation stability in cohomology and asymptotics for families of varieties over finite fields
• Noetherian property of infinite EI categories
• Hilbert series for twisted commutative algebras
• On the cohomology of pure mapping class groups as FI-modules

Cited by in corpus (168)

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• An algebraic model for commutative HZ-algebras
• Equivariant Hilbert Series in non-Noetherian Polynomial Rings
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• Representations of categories of G-maps
• Noetherianity of some degree two twisted commutative algebras
• A spectral sequence for stratified spaces and configuration spaces of points
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• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
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• Codimension and Projective Dimension up to Symmetry
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• Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• A long exact sequence for homology of FI-modules
• Higher order representation stability and ordered configuration spaces of manifolds
• Homological Invariants of FI-modules and FI_G-modules
• Periodicity in the cohomology of symmetric groups via divided powers
• Symmetric group characters as symmetric functions
• On the ideal generated by all squarefree monomials of a given degree
• Topological noetherianity for cubic polynomials
• Regularity of FI-modules and local cohomology
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• Upper bounds of homological invariants of $F{I}_G$-modules
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• Hilbert series for twisted commutative algebras
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Stability in the high-dimensional cohomology of congruence subgroups
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