Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

Let $R_n$ denote the KLR algebra of type $A_{e-1}^{(1)}$. Using the presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely determined ${\mathrm{hom}}_{R_n}(S^{\mu },S^{\lambda })$ where $\mu$ is an arbitrary partition, $\lambda$ is a hook and $e\mathrm{\ne }2$. 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.