Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

Let $R_n$ denote the KLR algebra of type $A^{(1)}_{e-1}$. Using the presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely determined $\hom_{R_n}(S^\mu,S^\lambda)$ where $\mu$ is an arbitrary partition, $\lambda$ is a hook and $e\neq2$. In this paper, we investigate the same problem when $e=2$. First we give a complete description of the action of the generators on the basis elements of $S^\lambda$. We use this result to identify a large family of partitions $\mu$ such that there exists at least one non-zero homomorphism from $S^\mu$ to $S^\lambda$, explicitly describe these maps and give their grading. Finally, we generalise James's result for the trivial module.