Real-valued character ($\chi (g)\in \mathbb{R}(g\in G)$)

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.