Graded Specht Modules as Bernstein-Zelevinsky Derivatives of the RSK Model

We clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the Robinson-Schensted-Knuth (RSK) construction for quiver Hecke algebras of type, which was recently imported from the setting of representations of -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 -adic setting. Graded cyclotomic decomposition numbers are shown to be a special subfamily of the wider concept of RSK decomposition numbers.