FI-modules and stability for representations of symmetric groups. (English) Zbl 1339.55004

Church, Thomas; Ellenberg, Jordan S.; Farb, Benson

Duke Math. J. 164, No. 9, 1833-1910 (2015).

Let \(\mathrm{FI}\) be the category whose objects are finite sets and whose morphisms are injections. An \(\mathrm{FI}\)-module over a commutative ring \(k\) is a functor \(V: \mathrm{FI} \rightarrow \mathcal{M}od_{k}\), where \(\mathcal{M}od_{k}\) denotes the category of \(k\)-modules. Since \(\mathrm{End}_{\mathrm{FI}}\big(\{1,2,\dots,n\}\big)=\mathfrak S_n\), the symmetric group, then each \(k\)-module \(V(\{1,2,\dots,n\})\) is a \(k[\mathfrak S_n]\)-module.
In this paper the authors give some results about \(\mathfrak S_n\)-representations, show the relation between finite generation of \(\mathrm{FI}\)-modules and representation stability for a sequence of \(\mathfrak S_n\)-representations (see [T. Church and B. Farb, Adv. Math. 245, 250–314 (2013; Zbl 1300.20051)]), and apply their theory to obtain results about cohomology of configuration spaces, cohomology of moduli spaces, coinvariant algebras and rank varieties, etc.

Reviewer: Saïd Zarati (Tunis)

MSC:

55N25	Homology with local coefficients, equivariant cohomology
05E10	Combinatorial aspects of representation theory
20J06	Cohomology of groups
20C30	Representations of finite symmetric groups
18B99	Special categories
55R80	Discriminantal varieties and configuration spaces in algebraic topology

Keywords:

FI-modules; representations; symmetric groups

Citations:

Zbl 1300.20051

References:

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