In representation theory of finite groups, there is a number of intriguing `global-local' conjectures that relate representations of any finite group to those of its `local' subgroups. While some of these conjectures are about counting representations with certain properties, the Broué abelian defect group conjecture is more structural, as it predicts a derived equivalence between a block of a finite group with an abelian defect group and its Brauer correspondent.
Generalizing the Broué conjecture to blocks of arbitrary defect in the case of symmetric groups, Turner constructed certain `double' algebras. He conjectured that every block of a symmetric group is derived equivalent to an appropriate double and, more precisely, that every RoCK block is Morita equivalent to a double. The talk will discuss a recent proof of Turner's conjecture. I will focus on developing the theory of Turner doubles from first principles and will explain how the doubles may be explicitly described as sublattices in generalized Schur algebras, which are in Schur-Weyl duality with certain wreath products. The talk is based on joint work with Alexander Kleshchev.