Tensor identity for an induced character

For a subgroup $H\le G$, a character $\chi$ of $G$ and a character $\varphi$ of $H$,It follows directly from the induced-character formula because $\chi (g)=\chi (x^{-1}gx)$.

Let $n$ be prime. If the Kronecker product ${\chi }^{\alpha }{\chi }^{\beta }$ of two irreducible characters of $S_n$ is irreducible, then one of $\alpha ,\beta$ is $(n)$ or $(1^n)$. Comparing the two self-products shows that only one may contain the standard character; the restriction branching rule then makes one partition rectangular, and primality makes that rectangle a single row or column.

For the standard inclusions $S_2,S_3\le S_4$,