Orthonormality of irreducible characters

Let $G$ be a finite group, and let $\mathrm{Cl}(G;\mathbb C)$ denote the $\mathbb C$ -vector space of complex-valued class functions on $G$ .

Define the standard inner product by

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

Orthonormality theorem. If $\chi$ and $\psi$ are irreducible complex characters of $G$ , then

\langle\chi,\psi\rangle=\delta_{\chi,\psi}.

Equivalently, summing over conjugacy classes $C$ ,

\langle f,g\rangle=\sum_C\frac{|C|}{|G|}f(C)\overline{g(C)}.

Consequences

Multiplicity formula. If $V$ is a complex representation with character $\chi_V$ and $\chi_i$ is irreducible, then the multiplicity $m_i$ of the corresponding irreducible representation in $V$ is $m_i \;=\; \langle \chi_V,\chi_i\rangle \in \mathbb Z_{\ge 0}.$ This uses Maschke's theorem / complete reducibility over $\mathbb C$ .

Orthonormal basis of class functions (over $\mathbb C$ ). The irreducible characters form an orthonormal basis of $\mathrm{Cl}(G;\mathbb C)$ . In particular, every class function $f$ has a unique expansion $f \;=\; \sum_i \langle f,\chi_i\rangle\, \chi_i.$ The spanning/basis part is tied to the number of irreducibles equals the number of conjugacy classes.

Character tables. Writing the character table with rows $\chi_i$ and columns indexed by conjugacy classes, orthonormality implies the rows are orthonormal with respect to the weights $|C|/|G|$ . (There is also a “column orthogonality” relation, equivalent to the same set of facts.)

Examples

Let $G=C_n=\langle a\rangle$ with $|G|=n$ , and fix $\zeta=e^{2\pi i/n}$ . The irreducible characters are 1-dimensional:

\chi_k(a^m)=\zeta^{km}\qquad (k=0,1,\dots,n-1).

Then

\langle \chi_k,\chi_\ell\rangle =\frac1n\sum_{m=0}^{n-1}\zeta^{km}\overline{\zeta^{\ell m}} =\frac1n\sum_{m=0}^{n-1}\zeta^{(k-\ell)m} =\begin{cases} 1,&k=\ell,\\ 0,&k\ne \ell, \end{cases}

since the sum is a geometric series.