# nLab Schur orthogonality relation

Contents

group theory

### Cohomology and Extensions

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Given a finite group $G$ of cardinality ${\vert G \vert}$, then the complex valued class functions $\chi \colon G \to \mathbb{C}$ have an inner product – the Schur inner product – given by

$\langle \alpha, \beta\rangle \coloneqq \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} \alpha(g) \overline{\beta(g)} \,.$

The Schur orthogonality relation is the statement that the irreducible group characters $\{\chi_i\}_i$ form an orthonormal basis of the space of class functions under this inner product:

$\langle \chi_i, \chi_j \rangle = \left\{ \array{ 1 & if \; i = j \\ 0 & otherwise } \right.$

## References

Created on November 18, 2014 at 01:07:55. See the history of this page for a list of all contributions to it.