Let $\ell \in \llbracket 2, n \rrbracket$ and let $\gamma \in \mathfrak{S}_\ell$ be a cycle of length $\ell$. Show that $\chi_\gamma(X) = X^\ell - 1$.
One may reduce to the case $\gamma = (12\cdots\ell)$ and consider the matrix
$$\Gamma_\ell = \left( \begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & 1 \\ 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & \vdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & 0 \\ 0 & \cdots & \cdots & 0 & 1 & 0 \end{array} \right) \in \mathcal{M}_\ell(\mathbb{C})$$