Real-valued character
= Real-valued character
{title2=$\chi(g)\in\mathbb R\quad(g\in G)$}
= Real character
{synonym}
A <character> is real-valued when all its values are <real numbers>, equivalently $\chi=\overline\chi$. By the <character inversion identity>, this is equivalent to $\chi(g)=\chi(g^{-1})$ for every $g$. This condition alone does not assert that the underlying <complex representation> can be realized by real <matrices>.