Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 19G (Representation Theory)

The irreducible characters are an orthonormal basis of the space of complex class functions for the inner productThus the row orthogonality relation and completeness statement areIn particular the number of irreducible characters equals the number of conjugacy classes. In class-representative notation, the orthogonality relation is ${\sum }_j\overline{{\chi }_i(g_j)}{\chi }_k(g_j)/|Z(g_j)|={\delta }_{ik}$.

Apply completeness to the indicator $f_j$ of the conjugacy class of $g_j$. Its expansion coefficient is $⟨{\chi }_i,f_j⟩=\overline{{\chi }_i(g_j)}/|Z(g_j)|$, because that class has size $|G|/|Z(g_j)|$. Evaluating at $g_k$ givesThe complex conjugation is part of the printed formula; dropping it would not give the general column orthogonality relation.

The matrix $A$ is square by completeness. Column orthogonality says $A^{\ast }A=\mathrm{diag}(|Z(g_j)|)$. Taking determinants givesIn particular its determinant is nonzero.

Complex conjugation permutes the irreducible characters, because the conjugate of an irreducible representation is irreducible. This involution fixes the real-valued characters and interchanges the other characters in pairs. If there are $r$ such pairs, then conjugating $A$ permutes its rows by $r$ transpositions. ThereforeIt follows that the determinant is real when $r$ is even and purely imaginary when $r$ is odd. It never vanishes, by the preceding part. Real-valued characters include both real and quaternionic representation types; the criterion concerns values of the characters, not those two representation types separately.