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

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Cited by in corpus (169)

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• Introduction to twisted commutative algebras
• FI-modules and the cohomology of modular representations of symmetric groups
• An algebraic model for commutative HZ-algebras
• Equivariant Hilbert Series in non-Noetherian Polynomial Rings
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• Noetherianity of some degree two twisted commutative algebras
• FI- and OI-modules with varying coefficients
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• Noetherianity of some degree two twisted skew-commutative algebras
• Codimension and Projective Dimension up to Symmetry
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• Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
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• Upper bounds of homological invariants of $F{I}_G$-modules
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• Homological degrees of representations of categories with shift functors
• Twisted homological stability for configuration spaces
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• Ideals of bounded rank symmetric tensors are generated in bounded degree
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• A generating function approach to new representation stability phenomena in orbit configuration spaces
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• On monoids up to symmetry
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