Ideal of the complex group ring

A left ideal $𝐿$ of the group ring $ℂ [𝐺]$ is a subspace of the group ring that is invariant under left-convolution, i.e. $𝑏 * 𝑎 \in 𝐿$ for all $𝑎 \in 𝐿$ and $𝑏 \in ℂ [𝐺]$ . In other words, $𝐿$ is an invariant subspace of the Regular group representation $Λ$ and $Λ ℂ [𝐺]$ ¹. If $𝐿$ is irreducible it is called a minimal left-ideal.

Since The regular representation contains all irreducible representations, each irrep $Γ 𝜇$ of $𝐺$ is carried by $𝑑 𝜇$ left-ideäls $𝐿 𝜇 𝛼$ with $1 \leq 𝛼 \leq 𝑑 𝜇$ , which collectively form a (non-minimal) left-ideäl $𝐿 𝜇$ transforming under $Γ 𝜇$ .

Projection operators

If $𝑃 𝜇 𝛼$ is a projection operator onto $𝐿 𝜇 𝛼$ , i.e.

$𝑃 𝜇 𝛼 ℂ [𝐺] = 𝐿 𝜇 𝛼$
$𝑃 𝜇 𝛼 ↾ 𝐿 𝜇 𝛼 = 𝐈$
$𝑃 𝜈 𝛽 𝑃 𝜇 𝛼 = 𝛿 𝜇 𝜈 𝛿 𝛼 𝛽 𝑃 𝜇 𝛼$

it follows

$Λ ℂ [𝐺] (𝑞) 𝑃 𝜇 𝛼 = 𝑃 𝜇 𝛼 Λ ℂ [𝐺] (𝑞)$ for all $𝑞 \in ℂ [𝐺]$

Proof

Let $𝑟 \in ℂ [𝐺]$ , with its unique decomposition into minimal left-ideäls $𝑟 = \sum 𝜇; 𝛼 𝑟 𝜇 𝛼$ . Then

𝑞 * 𝑃 𝜇 𝛼 𝑟 = 𝑞 * 𝑃 𝜇 𝛼 \sum 𝜇; 𝛼 𝑟 𝜇 𝛼 = 𝑞 * 𝑟 𝜇 𝛼

and

𝑃 𝜇 𝛼 (𝑞 * 𝑟) = 𝑃 𝜇 𝛼 \sum 𝜇; 𝛼 𝑞 * 𝑟 𝜇 𝛼 ⏟ \in 𝐿 𝜇 𝛼 = 𝑞 * 𝑟 𝜇 𝛼

so $Λ ℂ [𝐺] (𝑞) 𝑃 𝜇 𝛼 = 𝑃 𝜇 𝛼 Λ ℂ [𝐺] (𝑞)$ for all $𝑞 \in ℂ [𝐺]$ .

The projection operator $𝑃 𝜇 = \sum 𝑑 𝜇 𝛼 = 1 𝑃 𝜇 𝛼$ onto $𝐿 𝜇$ is given by right multiplication by an Idempotent of the complex group ring $𝑒 𝜇$ , i.e. $𝑃 𝜇 = P ℂ [𝐺] (𝑒 𝜇)$ where $P$ is the right Regular group representation.

Properties

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

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

Ideal of the complex group ring

Projection operators

Properties

Footnotes