Induction branching rule for a symmetric group (${\mathrm{Ind}}_{{\Sigma }_{n-1}}^{{\Sigma }_n}{S}^{\lambda }={⨁}_{\mu \supset \lambda ,|\mu |=|\lambda |+1}{S}^{\mu }$)

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.