---
tag: AY0B20QC04
kind: statement
role: theorem
name: Schur orthogonality relations
msc: [20C15]
uses: [G5S1M9, WYGRXCG2RN, FX3NEFYFFN, RMWMYBY8TC, 0DEBXAJ5SA, 02YYV36BFM, 0AR68JTMDT, 0FTH2E9TAS]
about: [RMWMYBY8TC, FX3NEFYFFN]
sources:
  - url: https://en.wikipedia.org/wiki/Schur_orthogonality_relations
same_as:
  wikidata: Q7433034
  wikipedia: Schur orthogonality relations
---

## Statement

Let $G$ be a finite [[group|G5S1M9]].
For functions $\alpha, \beta \colon G \to \C$ write
$$\ip{\alpha}{\beta} = \frac{1}{\abs{G}} \sum_{g \in G} \alpha(g) \overline{\beta(g)},$$
where the bar is the [[complex conjugate|0FTH2E9TAS]].

1. Let $\rho$ and $\sigma$ be [[irreducible|FX3NEFYFFN]] complex representations of $G$ by [[unitary matrices|0AR68JTMDT]], of degrees $d_\rho$ and $d_\sigma$. If $\rho$ and $\sigma$ are not isomorphic, then $\ip{\rho_{ij}}{\sigma_{kl}} = 0$ for all indices. For all indices $i, j, k, l$ one has $\ip{\rho_{ij}}{\rho_{kl}} = \delta_{ik}\delta_{jl} / d_\rho$.
2. Let $\chi$ and $\psi$ be irreducible complex [[characters|RMWMYBY8TC]] of $G$. Then $\ip{\chi}{\psi} = 1$ if $\chi = \psi$ and $\ip{\chi}{\psi} = 0$ otherwise.
3. Let $g, h \in G$, and let $C_G(g)$ be the [[centraliser|02YYV36BFM]] of $g$. Then $\sum_{\chi} \chi(g) \overline{\chi(h)}$, summed over all irreducible complex characters $\chi$ of $G$, equals $\abs{C_G(g)}$ if $g$ and $h$ are [[conjugate|0DEBXAJ5SA]], and $0$ otherwise.

## Remarks

The proof of the first relation applies Schur's lemma to the averaged map $\sum_{g} \rho(g) A \sigma(g)^{-1}$, which commutes with the action for every matrix $A$.
Taking traces in the first relation gives the second.
The second relation shows that the irreducible characters are orthonormal in the space of class functions; they span this space, and the third relation follows from that.
For example, for $G = \Z/2\Z$ the two irreducible characters are $(1, 1)$ and $(1, -1)$, and $\ip{\chi}{\psi} = \tfrac{1}{2}(1 - 1) = 0$.
A consequence: a complex representation with character $\chi$ is irreducible if and only if $\ip{\chi}{\chi} = 1$.
Every complex representation of a finite group is isomorphic to a unitary one, so the first relation applies to all irreducible representations after a change of basis.

## Explanations

### L1 human

The irreducible characters of a finite group behave like perpendicular unit vectors.
If one averages the product of two different irreducible characters over the group, the result is zero.
This makes it easy to decompose any representation: the number of times an irreducible piece occurs is an inner product, like a coordinate along an axis.

### L2 human

The relations are the main computational tool of [[character tables|P16AFZT3K0]].
Row orthogonality (relation 2) gives the multiplicity $\ip{\chi_V}{\chi}$ of an irreducible $\chi$ in a representation $V$.
Column orthogonality (relation 3) with $g = h = 1$ gives $\sum_\chi \chi(1)^2 = \abs{G}$.
Together they show that the number of irreducible characters equals the number of conjugacy classes, and they let one fill in missing entries of a character table.
They are the finite group case of the Peter–Weyl theory for compact groups, where the average becomes an integral against Haar measure.