Fayers’ conjecture and the socles of cyclotomic weyl modules

Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by p-restricted partitions. We prove an analogue of this result in the very general setting of “Schur pairs”. As an application we show that the socle of a Weyl module of a cyclotomic q-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type A. Finally, we prove a cyclotomic analogue of the Carter-Lusztig theorem.