∗-representations of the complex group ring

Let $𝐺$ be a finite group, and $Γ : 𝐺 \to G L (𝑉)$ be a Unitary representation, and $ℂ [𝐺]$ be the complex Group ring. Then $Γ$ induces a ∗-representation of the group ring $Γ ℂ [𝐺] : ℂ [𝐺] \to G L (𝑉)$ #m/thm/rep where

Γ ℂ [𝐺] (𝑎) = \sum 𝑔 \in 𝐺 𝑎 (𝑔) Γ (𝑔)

which satisfies the following properties for $𝑎, 𝑏 \in ℂ [𝐺]$

$Γ ℂ [𝐺] (𝑎 + 𝑏) = Γ ℂ [𝐺] (𝑎) + Γ ℂ [𝐺] (𝑏)$
$Γ ℂ [𝐺] (𝑎 * 𝑏) = Γ ℂ [𝐺] (𝑎) Γ ℂ [𝐺] (𝑏)$
$Γ ℂ [𝐺] (𝑎 †) = Γ ℂ [𝐺] (𝑎) †$
$Γ ℂ [𝐺] (𝛿 𝑒) = 𝐈$

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

Γ (𝑔) = Γ ℂ [𝐺] (𝛿 𝑔)

Proof

Let $Γ ℂ [𝐺] (𝑎) = \sum 𝑔 \in 𝐺 𝑎 (𝑔) Γ (𝑔)$ . Then

Γ ℂ [𝐺] (𝑎 + 𝑏) = \sum 𝑔 \in 𝐺 𝑎 (𝑔) Γ (𝑔) + \sum ℎ \in 𝐺 𝑏 (ℎ) Γ (ℎ) = Γ ℂ [𝐺] (𝑎) + Γ ℂ [𝐺] (𝑏)

satisfying property 1; and

Γ ℂ [𝐺] (𝑎 * 𝑏) = \sum 𝑥 \in 𝐺 \sum 𝑦 \in 𝐺 𝑎 (𝑥 𝑦 - 1) 𝑏 (𝑦) 𝑈 (𝑥) = \sum 𝑔 \in 𝐺 \sum ℎ \in 𝐺 𝑎 (𝑔) 𝑏 (ℎ) 𝑈 (𝑔 ℎ) = (\sum 𝑔 \in 𝐺 𝑎 (𝑔) 𝑈 (𝑔)) (\sum ℎ \in 𝐺 𝑏 (ℎ) 𝑈 (ℎ)) = Γ ℂ [𝐺] (𝑎) Γ ℂ [𝐺] (𝑏)

satisfying property 2; and

Γ ℂ [𝐺] (𝑎 †) = \sum 𝑔 \in 𝐺 ―――― 𝑎 (𝑔 - 1) Γ (𝑔) = \sum ℎ \in 𝐺 ――― 𝑎 (ℎ) Γ (ℎ) † = Γ ℂ [𝐺] (𝑎) †

satisfying property 3; and

Γ ℂ [𝐺] (𝛿 𝑒) = \sum 𝑔 \in 𝐺 𝛿 𝑒 (𝑔) Γ (𝑔) = Γ (𝑒) = 𝐈

satisfying property 4.

For the converse, let $Γ ℂ [𝐺] : ℂ [𝐺] \to G L (𝑉)$ be a $*$ -representation obeying properties 1–4. We define $Γ (𝑔) = Γ ℂ [𝐺] (𝛿 𝑔)$ . It follows that

\sum 𝑔 \in 𝐺 𝑎 (𝑔) Γ (𝑔) = \sum 𝑔 \in 𝐺 𝑎 (𝑔) Γ ℂ [𝐺] (𝛿 𝑔) = Γ ℂ [𝐺] (\sum 𝑔 \in 𝐺 𝑎 (𝑔) 𝛿 𝑔) = Γ ℂ [𝐺] (𝑎)

as required above, but is $Γ$ a unitary representation? From the property 2 it follows that $Γ (𝑔 ℎ) = Γ ℂ [𝐺] (𝛿 𝑔 * 𝛿 ℎ) = Γ ℂ [𝐺] (𝛿 𝑔) Γ ℂ [𝐺] (𝛿 ℎ) = Γ (𝑔) Γ (ℎ)$ , so $Γ$ is indeed a representation of $𝐺$ . From property 3 it follows that $Γ (𝑔) † = Γ ℂ [𝐺] (𝛿 𝑔) † = Γ ℂ [𝐺] (𝛿 † 𝑔) = Γ (𝑔 - 1)$ , so $Γ$ 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 $Γ$ is an irrep iff $Γ ℂ [𝐺]$ is irreducible.

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

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

∗-representations of the complex group ring

Properties

Footnotes