 Spectrum orbit bound for a faithful symmetric-group representation

For a $p$-cycle $g\in S_p$, each $g^k$ with $1\le k<p$ is conjugate to $g$. In a faithful complex representation, $\rho (g)$ has a nontrivial $p$th-root eigenvalue $\lambda$, and similarity of $\rho (g)$ and $\rho (g{)}^k$ puts all $p-1$ values ${\lambda }^k$ in its spectrum. Hence the representation has dimension at least $p-1$.