Regular representation ↩

Metadata 
aliases: []
shorthands: {}
created: 2021-10-18 18:21:22
modified: 2022-01-10 04:13:04

The space

is a finite group, then is a finite dimensional Hermitian space over the field of complex numbers (the components of the vectors in it are indexed by Group elements). The scalar product is defined as

Notes

Regular representation definition

We define the following orthonormal basis vectors in , with :

Now is such that .

The regular representation contains every irreducible representation

The regular representation contains every irreducible representation and exactly as many times as what the dimension of the irreducible representation is.

Intuitively

The matrices just permutate the coordinates of the vectors in the way the abstract group does.

Algorithm for finding the regular representation of a group

Examples