Group ring

∗-representations of the complex group ring

Let be a finite group, and be a Unitary representation, and be the complex Group ring. Then induces a ∗-representation of the group ring rep where

which satisfies the following properties for

Conversely, any representation of the group ring with these properties corresponds to a Unitary representation,¹ defined by

Proof

Let . Then

satisfying property 1; and

satisfying property 2; and

satisfying property 3; and

satisfying property 4.

For the converse, let be a -representation obeying properties 1–4. We define . It follows that

as required above, but is a unitary representation? From the property 2 it follows that , so is indeed a representation of . From property 3 it follows that , so is unitary as required.

The Regular group representation is a ∗-representation of the group ring carried by the group ring itself.

Properties

• Invariant subspaces of ∗-representations and unitary representations coïncide. Thus is an irrep iff is irreducible.

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Footnotes

1. 1996, Representations of finite and compact groups, §II.3, p 26 ↩