Orthogonality Relations

The orthogonality relations are fundamental identities satisfied by matrix coefficients of irreducible representations, providing a complete understanding of their linear independence and decomposition.

TheoremSchur Orthogonality Relations (First Form)

Let $V$ and $W$ be finite-dimensional irreducible representations of a finite group $G$ over $\mathbb{C}$ . Choose bases and let $\rho_V(g)_{ij}$ and $\rho_W(g)_{kl}$ denote matrix entries. Then: $\frac{1}{|G|} \sum_{g \in G} \overline{\rho_V(g)_{ij}} \rho_W(g)_{kl} = \begin{cases} \frac{1}{\dim V} \delta_{il} \delta_{jk} & \text{if } V \cong W \\ 0 & \text{if } V \not\cong W \end{cases}$

where $\delta$ is the Kronecker delta.

This theorem states that matrix coefficients of distinct irreducible representations are orthogonal with respect to the inner product: $\langle f, h \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{f(g)} h(g)$ on the space of complex-valued functions on $G$ .

ExampleConsequences

Linear independence: Matrix coefficients of non-isomorphic irreducible representations are linearly independent.

Dimension formula: If $V_1, \ldots, V_k$ are all irreducible representations of $G$ (up to isomorphism), then: $|G| = \sum_{i=1}^k (\dim V_i)^2$

For $S_3$ : The irreducibles have dimensions $1, 1, 2$ , giving $6 = 1^2 + 1^2 + 2^2$ .

TheoremSchur Orthogonality Relations (Second Form)

Let $\chi_V$ and $\chi_W$ be the characters of irreducible representations $V$ and $W$ of a finite group $G$ . Then: $\langle \chi_V, \chi_W \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi_V(g)} \chi_W(g) = \begin{cases} 1 & \text{if } V \cong W \\ 0 & \text{if } V \not\cong W \end{cases}$

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

ProofSketch

The key is to construct a specific linear map and apply Schur's Lemma. For irreducible $V, W$ and a linear map $T: V \to W$ , define: $S = \sum_{g \in G} \rho_W(g) \circ T \circ \rho_V(g^{-1})$

This is a $G$ -equivariant map. By Schur's Lemma:

If $V \not\cong W$ , then $S = 0$
If $V \cong W$ , then $S = \lambda \cdot \text{id}_V$ for some $\lambda$

Computing the trace gives the orthogonality relations.

■

Remark

The orthogonality relations are the foundation of character theory. They provide:

A criterion for irreducibility: $\langle \chi, \chi \rangle = 1$ if and only if $\chi$ is irreducible
Decomposition formulas: for any representation with character $\chi$ , the multiplicity of irreducible $V_i$ is $\langle \chi, \chi_i \rangle$
Explicit Fourier analysis on finite groups, generalizing classical Fourier series

These relations extend to compact groups via Haar measure, becoming the Peter-Weyl theorem, which underlies much of modern harmonic analysis.