Representation theory and homological stability

Church, Thomas; Farb, Benson

doi:10.1016/j.aim.2013.06.016

Cited by 167 publications

(206 citation statements)

References 62 publications

(69 reference statements)

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“…(This uses the nontrivial result that the mod-p reduction SL n O K → SL n F is actually surjective.) In keeping with the philosophy of [CF,§8] one might ask whether the groups H m (Γ n (p); Z) obey an appropriate notion of "representation stability" with respect to the action of the family {SL n F}.…”

Section: Congruence Fi-groups (Proof Of Theorem D)mentioning

confidence: 99%

FI-modules over Noetherian rings

et al. 2014

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FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S nrepresentations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely-generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.

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Section: Congruence Fi-groups (Proof Of Theorem D)mentioning

confidence: 99%

FI-modules over Noetherian rings

et al. 2014

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“…Previously known results were: Nakaoka [60] (Σ n with constant coefficients), Betley [6] (Σ n with more restrictive polynomial coefficients) and Hausmann [48] (A n with constant coefficients), Arnold [1] (β n with constant coefficients) and Church-Farb [19] (β n more restrictive polynomial coefficients), Segal [74] (β S n with constant coefficients), Frenkel and Callegaro [34,10] (complete computation for β n with constant coefficients), and Hatcher-Wahl [47] (G Σ n and G β S n with constant coefficients). Automorphism groups of free groups.…”

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confidence: 99%

Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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Abstract. Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability.As a byproduct of our work, we construct the braided analogue of the category F I of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.A family of groupsis said to satisfy homological stability if the induced mapsare isomorphisms in a range 0 ≤ i ≤ f (n) increasing with n. In this paper, we prove that homological stability always holds if there is a monoidal category C satisfying a certain hypothesis and a pair of objects A and X in C, such that G n is the group of automorphisms of A ⊕ X ⊕n in C. We show that stability holds not just for constant coefficients, but also for both polynomial and abelian coefficients, without any further assumption on C. The polynomial coefficient systems considered here are functors F : C → Z -Mod satisfying a finite degree condition. They are generalisations of polynomial functors in the sense of functor homology, classically considered in homological stability, and include new examples such as the Burau representation of braid groups. Abelian coefficients are given by functors F : C → ZG ab ∞ -Mod for G ab ∞ the abelianisation of the limit group G ∞ , satisfying the same finite degree condition. These include coefficients such as the sign representation, or determinant-twisted polynomial functors. Such coefficient systems are newer to the subject. One consequence of stability with abelian coefficients is that stability with polynomial coefficients also holds (under the same conditions) for the commutator subgroups G n ≤ G n .Our theorem applies to all the classical examples and gives new stability results with twisted coefficients in particular for symmetric groups, alternating groups, unitary groups, braid groups, mapping class groups of non-orientable surfaces, automorphisms and symmetric automorphisms of free groups, and it proves stability for these groups...

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“…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.…”

Section: Theorem 2 Fix a Non-negative Integer M And Letmentioning

confidence: 99%