Solution
= 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 $\alpha\mathrel{\unrhd}\lambda$.