The regular representation contains all irreducible representations

The regular representation $Λ : 𝐺 \to G L (ℂ [G])$ contains all irreps of $𝐺$ , where an irrep $Γ 𝜇$ appears with multiplicity $d i m Γ 𝜇 = 𝑑 𝜈$ . #m/thm/rep

Λ ≅ ⨁ 𝜇 \in ̂ 𝐺 𝑑 𝜇 Γ 𝜇

Proof

The Group character of this representation is

𝜒 Λ (𝑔) = \sum ℎ \in 𝐺 ⟨ 𝛿 ℎ | Λ (𝑔) 𝛿 ℎ ⟩ = \sum ℎ \in 𝐺 ⟨ 𝛿 ℎ | 𝛿 𝑔 ℎ ⟩ = {| 𝐺 | 𝑔 = 𝑒 0 𝑔 \neq 𝑒

and using the Orthonormality of irreducible characters we find that the multiplicity $𝑎 𝑘$ of each $Γ 𝑘$ is

𝑎 𝑘 = 1 | 𝐺 | \sum 𝑔 \in 𝐺 ―――― 𝜒 𝑘 (𝑔) 𝜒 Λ (𝑔) = 1 | 𝐺 | 𝜒 𝑘 (𝑒) | 𝐺 | = 𝑑 𝑘

as required.

As a corollary, the squares of the dimensions of all irreps sum to $| 𝐺 |$ , i.e.

| 𝐺 / \sim | \sum 𝑘 = 1 (𝑑 𝑛) 2 = | 𝐺 |

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