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

MSC:
55N25 Homology with local coefficients, equivariant cohomology
05E10 Combinatorial aspects of representation theory
20J06 Cohomology of groups
20G10 Cohomology theory for linear algebraic groups
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