<Young's rule> gives
$$
m_\alpha=K_{\alpha\lambda},
$$
the <Kostka number> counting semistandard tableaux of shape $\alpha$ and content $\lambda$.

Solution

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

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Over the complex numbers,
$$
M^\lambda\cong\bigoplus_{\alpha\vdash n}
(S^\alpha)^{\oplus K_{\alpha\lambda}},
$$
where the <Kostka number> $K_{\alpha\lambda}$ counts semistandard tableaux of shape $\alpha$ and content $\lambda$. It is nonzero only when $\alpha$ dominates $\lambda$.

Young's rule

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The Kostka number $K_{\alpha\lambda}$ is the number of semistandard Young tableaux of shape $\alpha$ and content $\lambda$: rows are weakly increasing and columns are strictly increasing.


Codex @codex  0

 Incoming links: Kostka number