Let \(G\) be a finite group and work with finite-dimensional complex representations. State the character orthogonality relations and explain how characters detect irreducible constituents.
Solution
For class functions \(\chi,\psi:G\to\CC\), define \[ \langle\chi,\psi\rangle =\frac1{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}. \]
If \(\chi_1,\ldots,\chi_r\) are the irreducible characters of \(G\), then the row orthogonality relations are \[ \langle\chi_i,\chi_j\rangle=\delta_{ij}. \] Equivalently, irreducible characters form an orthonormal basis of the space of complex class functions.
The column orthogonality relation says that for \(g,h\in G\), \[ \sum_{i=1}^r \chi_i(g)\overline{\chi_i(h)} = \begin{cases} |C_G(g)|,&g\text{ and }h\text{ are conjugate},\\ 0,&\text{otherwise}. \end{cases} \]
Now let \(V\) be a representation with character \(\chi_V\). Complete reducibility gives \[ V\cong\bigoplus_i m_i V_i, \] where \(V_i\) has irreducible character \(\chi_i\). Taking characters, \[ \chi_V=\sum_i m_i\chi_i. \] By orthogonality, \[ m_i=\langle\chi_V,\chi_i\rangle. \] Thus an irreducible representation \(V_i\) occurs in \(V\) exactly when \[ \langle\chi_V,\chi_i\rangle>0, \] and the inner product gives its multiplicity.