  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.
