Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

Let Rₙ denote the KLR algebra of type A⁽¹⁾ₑ₋₁. Using the presentation of Specht modules given by Kleschev--Mathas--Ram, Loubert completely determined Rₙ(S^μ,S^λ) where μ is an arbitrary partition, λ is a hook and e≠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^λ. We use this result to identify a large family of partitions μ such that there exists at least one non-zero homomorphism from S^μ to S^λ, explicitly describe these maps and give their grading. Finally, we generalise James's result for the trivial module.