Character Orthogonality Theorem

The orthogonality relations for characters are fundamental identities that completely determine the structure of the representation theory of finite groups.

TheoremCharacter Orthogonality (Row Orthogonality)

Let $\chi$ and $\psi$ be characters of finite-dimensional representations of a finite group $G$ over $\mathbb{C}$ . Then: $\langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi(g)} \psi(g)$

For irreducible characters $\chi_i$ and $\chi_j$ : $\langle \chi_i, \chi_j \rangle = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$

The irreducible characters form an orthonormal basis for the space of class functions on $G$ .

This theorem has profound consequences: it provides a complete criterion for irreducibility, a formula for multiplicities, and a method to decompose arbitrary representations.

TheoremCharacter Orthogonality (Column Orthogonality)

Let $C_1, \ldots, C_k$ be the conjugacy classes of $G$ , and let $\chi_1, \ldots, \chi_k$ be the irreducible characters. For conjugacy classes $C_i$ and $C_j$ with representatives $g_i, g_j$ : $\frac{1}{|G|} \sum_{\ell=1}^k |C_\ell| \chi_i(g_\ell) \overline{\chi_j(g_\ell)} = \delta_{ij}$

Equivalently, treating the character table as a matrix $X = [\chi_i(g_j)]$ and letting $D = \text{diag}(|C_1|, \ldots, |C_k|)$ : $X^* D X = |G| \cdot I$

ExampleApplications

Irreducibility criterion: A representation with character $\chi$ is irreducible if and only if $\langle \chi, \chi \rangle = 1$ .

Multiplicity formula: For a representation with character $\chi$ , the multiplicity of irreducible $V_i$ in its decomposition is: $m_i = \langle \chi, \chi_i \rangle$

Example: For $S_3$ , the permutation representation on $\mathbb{C}^3$ has character $(3, 1, 0)$ . Computing:

$\langle \chi, \chi_{\text{triv}} \rangle = \frac{1}{6}(3 \cdot 1 + 3 \cdot 1 \cdot 1 + 0) = 1$
$\langle \chi, \chi_{\text{std}} \rangle = \frac{1}{6}(3 \cdot 2 + 0 + 0) = 1$

Thus the permutation representation decomposes as $\mathbb{C}^3 \cong V_{\text{triv}} \oplus V_{\text{std}}$ .

TheoremNumber of Irreducible Representations

The number of irreducible representations of $G$ (up to isomorphism) equals the number of conjugacy classes of $G$ .

Proof sketch: The irreducible characters are an orthonormal basis for the space of class functions, which has dimension equal to the number of conjugacy classes.

Remark

These orthogonality relations are the discrete analogue of Fourier analysis. Characters are "Fourier modes" on the group, and the orthogonality relations are analogous to the orthogonality of $e^{inx}$ in Fourier series.

For compact Lie groups, these relations generalize to the Peter-Weyl theorem, which establishes that matrix coefficients of irreducible representations form a complete orthonormal system in $L^2(G)$ , the foundation of non-commutative harmonic analysis.