Graded Specht modules as Bernstein-Zelevinsky derivatives of the RSK model

TL;DR

This paper connects graded Specht modules and RSK models for quiver Hecke algebras, introducing crystal derivative operators that extend Bernstein-Zelevinsky derivatives and relate to quantum group structures.

Contribution

It develops a theory of crystal derivative operators on quiver Hecke modules, linking Specht modules with RSK models and generalizing Bernstein-Zelevinsky derivatives.

Findings

Crystal derivative operators categorify Berenstein-Zelevinsky strings.

Graded cyclotomic decomposition numbers are a subset of RSK decomposition numbers.

Established links between Specht modules and RSK models in the quiver Hecke algebra setting.

Abstract

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.