  We clarify the links between the graded Specht construction of modules over
cyclotomic Hecke algebras and the RSK construction for quiver Hecke algebras of
type A, that was recently imported from the setting of representations of
p-adic groups.
  For that goal we develop a theory of crystal derivative operators on quiver
Hecke algebra modules, that categorifies the Berenstein-Zelevinsky strings
framework on quantum groups, and generalizes a graded variant of the classical
Bernstein-Zelevinsky derivatives from the p-adic setting.
  Graded cyclotomic decomposition numbers are shown to be a special subfamily
of the wider concept of RSK decomposition numbers.
