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

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

Table of Contents

Ideal of the complex group ring

Projection operators

Properties

Graph View

Backlinks

Table of Contents

Ideal of the complex group ring

Projection operators

Properties

Footnotes

Graph View

Backlinks