= Column orthogonality
{title2=$\sum_{\chi\in\operatorname{Irr}(G)}\chi(g)\overline{\chi(h)}$}
For a <finite group>, the sum of products of <irreducible character> values is zero when $g,h$ are not conjugate, and equals $|C_G(g)|$ when $g,h$ are conjugate. To prove this from <row orthogonality relations for a character table>, choose representatives $g_j$ of its <conjugacy classes> and form $U_{ij}=\sqrt{|g_j^G|/|G|}\,\chi_i(g_j)$. Row orthogonality says $UU^*=I$. There are equally many <irreducible characters> and <conjugacy classes>, so this is a square <unitary matrix> and $U^*U=I$ as well. Its column equations give the result, using $|G|/|g_j^G|=|C_G(g_j)|$.
Back to article page
