Induction and Decomposition Numbers for RoCK Blocks

Abstract

This work is concerned with RoCK blocks (also known as Rouquier blocks) of symmetric groups. A RoCK block, bρ,w, with abelian defect group is Morita equivalent to a certain block of a wreath product of symmetric group algebras (Chuang and Kessar). Turner specified an idempotent, e, and conjectured that, for arbitrary weight w, ebρ,we should be Morita equivalent to this block of the wreath product. In this work we provide evidence in support of this conjecture. We prove that the decomposition matrices of these two algebras are identical. As a corollary to the proof, we obtain some knowledge of the composition factors of induced and restricted simple modules.