We review recent (~20 years) advances in representation theory of symmetric groups with the emphasis on categorification techniques.
Branching rules describe restrictions of irreducible representations of a symmetric group to a smaller symmetric group. Studying branching rules leads one to a categorification idea: restriction and induction functors yield a categorical action of certain Lie algebras on representation categories of symmetric groups. The corresponding action of the Weyl group lifts to a derived equivalence between the blocks of the symmetric groups (Chuang-Rouquier). Connections with Khovanov-Lauda-Rouquier algebras tie the categorical actions to quantum groups.
In the very end, we present a joint result with Anton Evseev, which describes every block of a symmetric group up to derived equivalence as a certain Turner double algebra. This description was conjectured by Will Turner. Turner doubles are Schur-algebra-like 'local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups.
Most of the ideas presented in the talk hold much more generally, but we stick mostly to symmetric groups to make presentation non-technical.