The regular representation contains every irreducible representation ↩

aliases: []
shorthands: {}
created: 2021-11-05 19:26:25
modified: 2022-01-10 04:13:04

Based on this, the regular representation of a group can be just written like this:

Where is the direct sum, are the irreducible representations of , is the dimension of the corresponding irreducible representation, is the number of irreducible representations, is the character of and is the character of the irreducible representation indexed by .

Proof

We just consider the character of the regular representation . We can do it in two ways:

1. Directly from the definition: .
2. The other way: we know from Consequences of the grand orthogonality theorem#C6 that , to its character can also be written in this form: . From this, we can determine the multiplicity coefficients by the projection: and by using the definition of the character shown in 1.
   We see that the multiplicity and the dimension is the same.