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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
##### Keywords:
FI-modules; representations; symmetric groups
Full Text:
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