Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

Let Rₙ denote the KLR algebra of type A^ (1) ₄-₁. Using the presentation of Specht modules given by Kleschev--Mathas--Ram, Loubert completely determined ₑ䂸 (S^, S^) where is an arbitrary partition, is a hook and e2. 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^. We use this result to identify a large family of partitions such that there exists at least one non-zero homomorphism from S^ to S^, explicitly describe these maps and give their grading. Finally, we generalise James's result for the trivial module.