Spectrum orbit bound for a faithful symmetric-group representation
= Spectrum orbit bound for a faithful symmetric-group representation
For a $p$-cycle $g\in S_p$, each $g^k$ with $1\leq k<p$ 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 $p-1$.