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Let $S_\lambda$ denote the Specht module defined by Dipper and James for theIwahori-Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$.When $e=2$ we determine the decomposability of all Specht modules correspondingto hook partitions $(a,1^b)$. We do so by utilising the Brundan-Kleshchevisomorphism between $\mathscr{H}$ and a Khovanov-Lauda-Rouquier algebra andworking with the relevant KLR algebra, using the set-up ofKleshchev-Mathas-Ram. When $n$ is even, we easily arrive at the conclusion that$S_\lambda$ is indecomposable. When $n$ is odd, we find an endomorphism of$S_\lambda$ and use it to obtain a generalised eigenspace decomposition of$S_\lambda$.

Decomposable Specht modules for the Iwahori-Hecke algebra $\mathscr{H}_{\mathbb{F},-1}(\mathfrak{S}_n)$