Group ring

Ideal of the complex group ring

A left ideal of the group ring is a subspace of the group ring that is invariant under left-convolution, i.e. for all and . In other words, is an invariant subspace of the Regular group representation and ¹. If is irreducible it is called a minimal left-ideal.

Since The regular representation contains all irreducible representations, each irrep of is carried by left-ideäls with , which collectively form a (non-minimal) left-ideäl transforming under .

Projection operators

If is a projection operator onto , i.e.

it follows

4. for all

Proof

Let , with its unique decomposition into minimal left-ideäls . Then

and

so for all .

The projection operator onto is given by right multiplication by an Idempotent of the complex group ring , i.e. where is the right Regular group representation.

Properties

• Equivalence of irreps on left ideals criterion
• Trivial irrep carrying ideal of the group ring

---

tidy | en | sembr

Footnotes

1. since Invariant subspaces of ∗-representations and unitary representations coïncide ↩