Irreps collectively distinguish group elements

Let $Γ 𝛼 𝑗 𝑘 \in ℂ [𝐺]$ be entries of matrix representations of each irrep $𝛼 \in ˆ 𝐺$ . Then the function subspace spanned by all such entries distinguishes all group elements in $𝐺$ , where for any $𝑥 \in 𝐺$ there exists a linear combination

𝑓 𝑥 = \sum 𝛾 \in ˆ 𝐺 𝑑 𝛾 \sum 𝑗 = 1 𝑑 𝛾 \sum 𝑘 = 1 𝐶 𝛾 𝑗 𝑘 Γ 𝛾 𝑗 𝑘

such that $𝑓 𝑥 𝑦 (𝑥) = 1$ and $𝑓 𝑥 𝑦 (𝑦) = 0$ for $𝑦 \neq 𝑥$ . #m/thm/rep

Proof

This follows from the existence of the Regular group representation $Λ : 𝐺 \to ℂ [𝐺]$ . For we can define

𝑓 𝑥 (𝑦) = ⟨ 𝛿 𝑥 | Λ (𝑦) 𝛿 𝑒 ⟩

using the unnormalised inner product on $ℂ [𝐺]$ , which has the required property, and since $Λ$ is a representation it is unitarily equivalent to a direct sum of irreps, i.e.

Λ = ⨁ 𝜇 𝑇 Γ 𝜇 𝑇 † Λ 𝑗 𝑘 (𝑦) = \sum 𝜇, 𝑝, 𝑞 𝑇 𝜇 𝑗 𝑝 Γ 𝛼 𝜇 𝑝 𝑞 (𝑦) ――― 𝑇 𝜇 𝑘 𝑞

and so treating $𝑥$ and $𝑒$ as indices

𝑓 (𝑦) = Λ 𝑥 𝑒 (𝑦) = \sum 𝜇, 𝑝, 𝑞 𝑇 𝜇 𝑥 𝑝 Γ 𝛼 𝜇 𝑝 𝑞 (𝑦) ――― 𝑇 𝜇 𝑒 𝑞

as required.

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