Character orthogonality

Let $G$ be a finite group. A (complex) character $\chi:G\to\mathbb C$ is a class function: it is constant on each conjugacy class.

Inner product on class functions. On the space $\mathrm{Cl}(G)$ of complex class functions, define

\langle f, g\rangle \;=\; \frac{1}{|G|}\sum_{x\in G} f(x)\,\overline{g(x)}.

This is a Hermitian inner product (compare inner product and orthogonality).

Theorem (orthogonality of irreducible characters). Let $\chi_1,\dots,\chi_r$ be the distinct irreducible characters of $G$ over $\mathbb C$ . Then:

Row orthogonality (orthonormality): $\langle \chi_i, \chi_j\rangle = \delta_{ij}.$ In particular, for any character $\chi$ , $\langle \chi,\chi\rangle \in \mathbb Z_{\ge 0},$ and $\chi$ is irreducible iff $\langle \chi,\chi\rangle=1$ (see also character orthonormality).

Completeness: The set $\{\chi_1,\dots,\chi_r\}$ is an orthonormal basis of $\mathrm{Cl}(G)$ . Hence $r=\dim_\mathbb C \mathrm{Cl}(G)$ , which equals the number of conjugacy classes (cf. number of irreducibles equals number of conjugacy classes).

Column orthogonality (one common form): for $g,h\in G$ , $\sum_{i=1}^r \chi_i(g)\,\overline{\chi_i(h)} = \begin{cases} 0, & \text{if } g \text{ and } h \text{ are not conjugate},\\[4pt] |C_G(g)|, & \text{if } g \text{ and } h \text{ are conjugate}, \end{cases}$ where $C_G(g)$ is the centralizer of $g$ . In particular, $\sum_{i=1}^r |\chi_i(g)|^2 = |C_G(g)|.$

These identities are proved using complete reducibility (via Maschke's theorem), the decomposition of tensor products, and Schur's lemma.

Examples

Cyclic group $C_n$ : discrete Fourier orthogonality. Let $C_n=\langle t\rangle$ . Its irreducible characters are $\chi_j(t^m)=\zeta_n^{jm}$ for $j=0,\dots,n-1$ . Then $\langle \chi_j,\chi_\ell\rangle = \frac{1}{n}\sum_{m=0}^{n-1} \zeta_n^{jm}\overline{\zeta_n^{\ell m}} = \frac{1}{n}\sum_{m=0}^{n-1} \zeta_n^{(j-\ell)m} = \delta_{j\ell}.$

$S_3$ : checking row orthogonality from the character table. $S_3$ has three conjugacy classes: $()$ , transpositions, and 3-cycles. Let $\chi_{\mathrm{triv}},\chi_{\mathrm{sgn}},\chi_{\mathrm{std}}$ be the irreducible characters (degrees $1,1,2$ ). Using the class sizes $1,3,2$ and values $\chi_{\mathrm{triv}}=(1,1,1),\quad \chi_{\mathrm{sgn}}=(1,-1,1),\quad \chi_{\mathrm{std}}=(2,0,-1),$ one computes for example $\langle \chi_{\mathrm{std}},\chi_{\mathrm{sgn}}\rangle =\frac{1}{6}\Big(1\cdot 2\cdot 1 + 3\cdot 0\cdot(-1) + 2\cdot (-1)\cdot 1\Big)=0,$ and $\langle \chi_{\mathrm{std}},\chi_{\mathrm{std}}\rangle =\frac{1}{6}\Big(1\cdot |2|^2 + 3\cdot |0|^2 + 2\cdot |-1|^2\Big)=1,$ confirming orthogonality and irreducibility.

Column orthogonality in $S_3$ : centralizer sizes. In $S_3$ , a transposition $\tau$ has centralizer size $|C_{S_3}(\tau)|=2$ . Column orthogonality predicts $|\chi_{\mathrm{triv}}(\tau)|^2 + |\chi_{\mathrm{sgn}}(\tau)|^2 + |\chi_{\mathrm{std}}(\tau)|^2 = |1|^2 + |-1|^2 + |0|^2 = 2,$ matching $|C_{S_3}(\tau)|$ .