Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 2 / d / i

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