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

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 $\alpha ⊵\lambda$.

Young's rule givesthe Kostka number counting semistandard tableaux of shape $\alpha$ and content $\lambda$.

For $n\ge 4$, the only partitions dominating $(n-2,1,1)$ areThe entries of content $(n-2,1,1)$ consist of $n-2$ copies of $1$ and one copy each of $2$ and $3$. There is one semistandard tableau of shape $(n)$; two of shape $(n-1,1)$, according as the lower cell contains $2$ or $3$; one of shape $(n-2,2)$; and one of shape $(n-2,1,1)$. Thereforeand every other multiplicity is zero. For $n=3$, the coincident middle shapes combine to give $m_{(3)}=1$, $m_{(2,1)}=2$, and $m_{(1,1,1)}=1$.