nLab
Schur orthogonality relation

Contents

Context

Group Theory

Representation theory

Contents

Idea

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

α,β1|G|gGα(g)β(g)¯. \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 {χ i} i\{\chi_i\}_i form an orthonormal basis of the space of class functions under this inner product:

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

References

Created on November 18, 2014 at 01:24:18. See the history of this page for a list of all contributions to it.