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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. 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