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

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 $S_\lambda$.