Restriction branching rule for a symmetric group

Over the complex numbers, restriction from $S_n$ to $S_{n-1}$ is multiplicity-free:Equivalently, one removes one Removable node of a Young diagram in every possible distinct way.

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.

For $\lambda ⊢n$ with $n\ge 2$,Indeed, adjoining the trivial character to the standard character gives the point-permutation character, and Frobenius reciprocity turns its multiplicity in the self-product into the norm of the multiplicity-free restriction.