= Induction branching rule for a symmetric group
{title2=$\operatorname{Ind}_{\Sigma_{n-1}}^{\Sigma_n}S^\lambda=\bigoplus_{\mu\supset\lambda,\ |\mu|=|\lambda|+1}S^\mu$}
In <characteristic zero>, inducing a <Specht module> from $\Sigma_{n-1}$ to $\Sigma_n$ gives one copy of each <Specht module> obtained by adding one box in a valid Young-diagram position. By <Frobenius reciprocity>, its multiplicity in the induced <module> equals the multiplicity of $S^\lambda$ in the restriction of that larger <Specht module>. The <restriction branching rule for a symmetric group> says this multiplicity is one exactly when removing one corner gives $\lambda$, and zero otherwise. Semisimplicity in <characteristic zero> turns these multiplicities into the displayed direct-sum decomposition.
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