Idempotent of the complex group ring

Irreducible character as function of an idempotent

Let be a primitive idempotent generating the minimal left ideäl carrying irrep . Then the character of is given by #m/thm/rep

where denotes the conjugacy class of and its centraliser group with equal to the size of the group divided by the size of the conjugacy class.

Proof

Using the inner product and convolution on

and since ,

Applying the Orbit-stabilizer theorem (see its proof), it follows that

as required.1

It follows that .

#state/tidy | #lang/en | #SemBr

Footnotes

  1. An alternative proof is given in 2023, Groups and representations, pp. 60–61, but I like mine better. ↩