Decomposable Specht modules for the Iwahori–Hecke algebra HF,−1(Sn)HF,−1(Sn)

Let SλSλ denote the Specht module defined by Dipper and James for the Iwahori–Hecke algebra HnHn of the symmetric group SnSn. When e=2e=2 we determine the decomposability of all Specht modules corresponding to hook partitions (a,1b)(a,1b). We do so by utilising the Brundan–Kleshchev isomorphism between HH 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 SλSλ is indecomposable. When n   is odd, we find an endomorphism of SλSλ and use it to obtain a generalised eigenspace decomposition of SλSλ.