Regular representation

The (left) Regular representation for a group is both a group representation $Λ : 𝐺 \to G L (ℂ [𝐺])$ of a group $𝐺$ and a ∗-representation $Λ ℂ [𝐺] : ℂ [𝐺] \to G L (ℂ [𝐺])$ of its complex group ring carried by the group ring itself and defined using the group ring's convolution operation. For $𝑎, 𝑏 \in ℂ [𝐺]$ and $𝑔 \in 𝐺$

Λ ℂ [𝐺] (𝑎) 𝑏 = 𝑎 * 𝑏 Λ (𝑔) 𝑏 = 𝛿 𝑔 * 𝑏

and thus for $𝑔, ℎ \in 𝐺$ and $𝑎 \in ℂ [𝐺]$

Λ (𝑔) 𝛿 ℎ = 𝛿 𝑔 ℎ Λ (𝑔) 𝑎 (ℎ) = 𝑎 (𝑔 - 1 ℎ)

Proof these are representations

If we prove that $Λ ℂ [𝐺]$ is a ∗-representation it follows that $Λ$ is a unitary representation. Properties 1, 2, and 4 follow from properties of the ∗-algebra (distributivity, associativity, monoid identity), hence all that is left to prove is that $⟨ 𝑐 | 𝑎 * 𝑏 ⟩ = ⟨ 𝑎 † * 𝑐 | 𝑏 ⟩$ for any $𝑎, 𝑏, 𝑐 \in ℂ [𝐺]$ . Using $⟨ 𝑎 | 𝑏 ⟩$ as defined in the Zettel for Group ring

⟨ 𝑐 | 𝑎 * 𝑏 ⟩ = \sum 𝑥 \in 𝐺 ――― 𝑐 (𝑥) (𝑎 * 𝑏) (𝑥) = \sum 𝑥 \in 𝐺 \sum 𝑦 \in 𝐺 ――― 𝑐 (𝑥) 𝑎 (𝑥 𝑦 - 1) 𝑏 (𝑥) = \sum 𝑥 \in 𝐺 \sum 𝑦 \in 𝐺 ――――――― 𝑐 (𝑥) 𝑎 † (𝑦 𝑥 - 1) 𝑏 (𝑥) = \sum 𝑥 \in 𝐺 ―――――― (𝑎 † * 𝑐) (𝑥) 𝑏 (𝑥) = ⟨ 𝑎 † * 𝑐 | 𝑏 ⟩

as required.

The right regular representation $P$ is defined the same way using right multiplication.

Matrix

If group elements are identified with indices for a matrix then for each $𝑥 \in 𝐺$

Λ 𝑔 ℎ (𝑥) = {1 𝑥 ℎ = 𝑔 0 𝑥 ℎ \neq 𝑔

i.e. $⟨ 𝛿 𝑔 | Λ (𝑥) 𝛿 ℎ ⟩ = 𝛿 𝑔 (𝑥 ℎ)$ , so each $Λ (𝑥)$ is basically the group table.

Properties

The regular representation contains all irreducible representations
Its character is $| 𝐺 |$ times the indicator function $𝛿 1$ .

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