Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 4 / b / i / Solution

Choose a copy of $S^{\alpha }$ inside $M^{\lambda }$ and a nonzero polytabloid in it. Its image under a suitable Column antisymmetrizer of a Young tableau is nonzero. By the fact allowed in the question, this can happen only if $\alpha$ dominates $\lambda$. Hence $m_{\alpha }>0$ implies $\alpha ⊵\lambda$.