Symmetric-group character co-degree vanishing criterion
= Symmetric-group character co-degree vanishing criterion
For $\chi\in\operatorname{Irr}(S_n)$, if the order of $g\in S_n$ does not divide $|S_n|/\chi(1)$, then $\chi(g)=0$. The <Hook-length formula> turns the co-degree into a product of hook lengths, while the <Murnaghan–Nakayama rule> detects a cycle whose required prime-power hook cannot be removed.