Let G be a finite group. A (complex) character χ:G→C is a class function: it is constant on each conjugacy class.
Inner product on class functions. On the space Cl(G) of complex class functions, define
⟨f,g⟩=∣G∣1x∈G∑f(x)g(x).
This is a Hermitian inner product (compare inner product and orthogonality).
Theorem (orthogonality of irreducible characters). Let χ1,…,χr be the distinct irreducible characters of G over C. Then:
- Row orthogonality (orthonormality):
⟨χi,χj⟩=δij. In particular, for any character χ, ⟨χ,χ⟩∈Z≥0, and χ is irreducible iff ⟨χ,χ⟩=1 (see also character orthonormality).
- Completeness: The set {χ1,…,χr} is an orthonormal basis of Cl(G). Hence r=dimCCl(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∈G,
i=1∑rχi(g)χi(h)={0,∣CG(g)∣,if g and h are not conjugate,if g and h are conjugate, where CG(g) is the centralizer of g. In particular, i=1∑r∣χi(g)∣2=∣CG(g)∣.
These identities are proved using complete reducibility (via Maschke's theorem), the decomposition of tensor products, and Schur's lemma.