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

Abstract

Abstract 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 $A$, which 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.