Let $G$ be a compact Lie group, and let $dg$ denote the normalized Haar probability measure on $G$. Let $(\rho,V)$ and $(\sigma,W)$ be irreducible finite-dimensional unitary complex representations of $G$. Let $\chi_\rho:G\to\mathbb C$ and $\chi_\sigma:G\to\mathbb C$ be the characters defined by