Convolution of matrix representations

Let $Γ 𝛼 𝑖 𝑗, Γ 𝛽 𝑘 𝑙 \in ℂ [𝐺]$ be the entries of concrete realizations of unitary irreps. Then the group ring convolution is given by¹ #m/thm/rep

Γ 𝛼 𝑖 𝑗 * Γ 𝛽 𝑘 𝑙 = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 Γ 𝛼 𝑖 𝑙

Proof

Using orthogonality of irreps and the fact that $Γ 𝛼 (𝑔 - 1) = Γ 𝛼 (𝑔) †$

(Γ 𝛼 𝑖 𝑗 * Γ 𝛽 𝑘 𝑙) (𝑥) = \sum 𝑔 \in 𝐺 Γ 𝛼 𝑖 𝑗 (𝑥 𝑔 - 1) Γ 𝛽 𝑘 𝑙 (𝑔) = 𝑑 𝛼 \sum 𝑝 = 1 \sum 𝑔 \in 𝐺 Γ 𝛼 𝑖 𝑝 (𝑥) Γ 𝛼 𝑝 𝑗 (𝑔 - 1) Γ 𝛽 𝑘 𝑙 (𝑔) = 𝑑 𝛼 \sum 𝑝 = 1 \sum 𝑔 \in 𝐺 Γ 𝛼 𝑖 𝑝 (𝑥) ―――― Γ 𝛼 𝑗 𝑝 (𝑔) Γ 𝛽 𝑘 𝑙 (𝑔) = 𝑑 𝛼 \sum 𝑝 = 1 Γ 𝛼 𝑖 𝑝 (𝑥) ⟨ Γ 𝛼 𝑗 𝑝 | Γ 𝛽 𝑘 𝑙 ⟩ = 𝑑 𝛼 \sum 𝑝 = 1 | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 𝛿 𝑝 𝑙 Γ 𝛼 𝑖 𝑝 (𝑥) = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 Γ 𝛼 𝑖 𝑙 (𝑥)

as required.

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

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