  Let $S_\lambda$ denote the Specht module defined by Dipper and James for the
Iwahori-Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$.
When $e=2$ we determine the decomposability of all Specht modules corresponding
to hook partitions $(a,1^b)$. We do so by utilising the Brundan-Kleshchev
isomorphism between $\mathscr{H}$ and a Khovanov-Lauda-Rouquier algebra and
working with the relevant KLR algebra, using the set-up of
Kleshchev-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
$S_\lambda$.
