  We study asymptotic properties of the modular representation theory of
symmetric groups and investigate modular analogs of stabilization phenomena in
characteristic zero. The main results are equivalences of categories between
certain abelian subcategories of representations of $S_n$ and $S_m$ for
different $n$ and $m$. We apply these results to obtain a structural result for
$FI$-modules, and to prove a result conjectured by Deligne in a recent letter
to Ostrik.

Representation Theory