# nLab Schur orthogonality relation

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

group 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 complex irreducible group characters $\{\chi^{(a)}\}_a$ form an orthonormal basis of the space of class functions under this inner product:

(1)$\langle \chi^{(a)}, \chi^{(b)} \rangle = \left\{ \array{ 1 & if \; a = b \\ 0 & otherwise } \right.$

In fact, this holds also at the level of complex irreducible representations $\rho^{(a)}$ (“grand Schur orthogonality”), regarded for each $g \in G$ as unitary matrices $\rho^{(a)}(g) \in U(\chi^{(a)}(e))$ with matrix entries $\big(\rho^{(a)}(g)_{i j}\big)_{i j}$:

(2)$\underset{g \in G}{\sum} \rho^{(a)}(g)_{i_1 j_1} \cdot \overline{\rho^{(b)}(g)_{i_2 j_2}} \;=\; \delta^{a b} \delta_{i_1 i_2} \delta_{j_1 j_2} \cdot \frac{ \left\vert G \right\vert }{ dim(\rho^{(a)}) } \,.$

(e.g. tom Dieck 09 (2.2.7))

Conversely, the sum over all (isomorphism classes) of irreducible characters $\chi_i$ yields:

$\underset{\chi^{(a)}}{\sum} \chi^{(a)}(g) \cdot \overline{ \chi^{(a)}(h) } \;=\; \left\{ \array{ \left\vert Central_G(g) \right\vert & \text{if} \; h,g \;\text{are conjugate} \\ 0 & \text{otherwise} } \right.$

## References

Textbook accounts:

Lecture notes: