Prime-degree irreducible Kronecker product criterion for a symmetric group

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.