Permutation matrix/Cycle/Characteristic polynomial/Fact

characteristic polynomial of a permutation matrix ${\displaystyle {}M_{\rho }}$ for a cycle ${\displaystyle {}\rho \in S_{n}}$ of order ${\displaystyle {}k}$ is

${\displaystyle {}\chi _{M}=(X-1)^{n-k}(X^{k}-1)\,.}$