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Homomorphisms into Specht modules labelled by hooks in quantum characteristic two (2402.08942v2)

Published 14 Feb 2024 in math.RT

Abstract: 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.

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