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

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:

1. 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 ).

2. 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 ).

3. 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

1. 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

   $$\langle \chi_j,\chi_\ell\rangle = \frac{1}{n}\sum_{m=0}^{n-1} \zeta_n^{jm}\overline{\zeta_n^{\ell m}} = \frac{1}{n}\sum_{m=0}^{n-1} \zeta_n^{(j-\ell)m} = \delta_{j\ell}.$$

2. $S_3$: 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.

3. 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 $|C_{S_3}(\tau)|$.