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

FI-modules over Noetherian rings

“…This paper has three main results; all three are proved in §2 below. When k is a field of characteristic 0, Theorem A was proved earlier in [Sn,Theorem 2.3] and [CEF,Theorem 2.60], and Theorem B was proved in [CEF,Theorem 2.67].…”

“…When char k = 0, we proved in [CEF,Theorem 2.67] that not only the dimensions but also the characters of V n are eventually polynomial. In the situation of Theorem B, it is reasonable to expect that when k is a field of positive characteristic the Brauer characters of V n similarly have polynomial behavior.…”

Stable Decompositions of Certain Representations of the Finite General Linear Groups

“…The proof of Proposition 17(i) (given in [5, Proposition 4.1 and Proposition 4.3]) is an adaptation of their arguments for consistent sequences of representations of the symmetric groups to our situation of consistent sequences for representations of the finite general linear groups. Let us also mention that [1] also proved eventual polynomiality of characters for finitely generated FI-modules over a field of characteristic zero. We do not know if there are analogous results for VI-modules or VIC-modules.…”

“…Our motivation to consider the question of multiplicity stability for the sequence in the above theorem comes from representation stability theory, in which a similar phenomenon for representations of symmetric groups is studied in [1] and [2]; see [3] for a survey. In particular, an analog of Theorem 2 for symmetric groups was proved by Hemmer [7,Theorem 2.4] using Pieri's rule.…”

“…In [5, Theorem 1.6], the authors proved a representation stability theorem for finitely generated VI-modules over k. This result can also be deduced from Theorem 4: there is a forgetful functor from the category VIC to the category VI, and the pullback of any finitely generated VI-module is a finitely generated VIC-module; see [8 (iii) We should mention that the categories VI and VIC are analogues of the category FI of finite sets and injective maps. FI-modules were introduced and studied by Church, Ellenberg and Farb in [1]. One of the main results of their paper is a representation stability theorem for finitely generated FI-modules over a field of characteristic zero.…”