Standard-character multiplicity in a Specht self-product
= Standard-character multiplicity in a Specht self-product
For $\lambda\vdash n$ with $n\geq2$,
$$
\langle\chi^\lambda\chi^\lambda,\chi^{(n-1,1)}\rangle=|\lambda^-|-1.
$$
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.