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The restriction form of the restriction branching rule for a symmetric group iswhere ${\lambda }^{-}$ contains the distinct partitions obtained by deleting one Removable node of a Young diagram. In particular, the restriction is multiplicity-free.

Restrict the alternating expressionThe supplied restriction formula for a Young permutation character, with $k=1$, says that each term restricts by subtracting one from each possible component. After collecting the alternating sums, this givesIf row $i$ has no removable node, part i straightens ${\psi }^{\lambda -{ϵ}_i}$ against the adjacent term with the opposite sign, or makes it zero when two shifted entries coincide. The surviving terms are exactly ${\psi }^{\mu }$ for $\mu \in {\lambda }^{-}$. Since $\lambda$ and each surviving $\mu$ are partitions, ${\psi }^{\lambda }={\chi }^{\lambda }$ and ${\psi }^{\mu }={\chi }^{\mu }$. We obtainComplex representations of a finite group are semisimple, so equality of characters proves the asserted module decomposition.

The point-permutation character isBy the tensor identity for an induced character and Frobenius reciprocity,The restriction branching rule for a symmetric group is multiplicity-free with one constituent for each member of ${\lambda }^{-}$, so the right side is $|{\lambda }^{-}|$. Also $⟨{\chi }^{\lambda }{\chi }^{\lambda },{\chi }^{(n)}⟩=1$ because every symmetric-group character is real and irreducible. Subtracting the trivial constituent proves the standard-character multiplicity in a Specht self-product formula