Homomorphisms from an arbitrary Specht module to one corresponding to a hook

TL;DR

This paper investigates homomorphisms between Specht modules, providing a comprehensive description of these maps when targeting hook-shaped modules, and extends previous results by James.

Contribution

It introduces a general method to determine all homomorphisms from arbitrary Specht modules to hook Specht modules, utilizing Kleshchev-Mathas-Ram presentations.

Findings

Complete characterization of homomorphisms to hook Specht modules

Explicit description of KLR generator actions on hook modules

Generalization of James' earlier results

Abstract

We use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James.