Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations

Consider unitary irreps $Γ 𝛼 : 𝐺 \to G L (ℂ 𝑑 𝛼)$ for $𝛼 \in ˆ 𝐺$ . Then there exists a unitary isomorphism from the Group ring to the direct sum of matrix algebras

ℂ [𝐺] \to ⨁ 𝛼 \in ˆ 𝐺 M 𝑑 𝛼, 𝑑 𝛼 (ℂ)

defined by $𝑓 \mapsto ̂ 𝑓$ with¹ #m/thm/rep

̂ 𝑓 𝛼 𝑗 𝑘 = \sum 𝑔 \in 𝐺 ―――― Γ 𝛼 𝑗 𝑘 (𝑔) 𝑓 (𝑔) = ⟨ Γ 𝛼 𝑗 𝑘 | 𝑓 ⟩

and likewise

𝑓 = 1 | 𝐺 | \sum 𝛼; 𝑗 𝑘 𝑑 𝛼 ̂ 𝑓 𝛼 𝑗 𝑘 Γ 𝛼 𝑗 𝑘

which is unitary from $⟨ \cdot | \cdot ⟩$ to the following inner product

⟨ ̂ 𝑓 | ̂ ℎ ⟩ = 1 | 𝐺 | \sum 𝛼 𝑑 𝛼 t r [(ˆ 𝑓 𝛼) † ̂ ℎ 𝛼] = 1 | 𝐺 | \sum 𝛼; 𝑖, 𝑗 𝑑 𝛼 ――― ̂ 𝑓 𝛼 𝑖 𝑗 ̂ ℎ 𝛼 𝑖 𝑗

and a homomorphism in the sense that

̂ (𝑓 * ℎ) 𝛼 𝑖 𝑗 = 𝑑 𝛼 \sum 𝑘 = 1 ̂ 𝑓 𝛼 𝑖 𝑘 ̂ ℎ 𝛼 𝑘 𝑗

Proof

To verify the given inverse, note that by orthogonality of irreps, ${\sqrt 𝑑 𝛼 Γ 𝛼 𝑗 𝑘}$ form an orthonormal basis with respect to the inner product $(\cdot | \cdot)$

𝑓 = 1 | 𝐺 | \sum 𝛼; 𝑗, 𝑘 𝑑 𝛼 ̂ 𝑓 𝛼 𝑗 𝑘 Γ 𝛼 𝑗 𝑘 = \sum 𝛼; 𝑗, 𝑘 \sqrt 𝑑 𝛼 Γ 𝛼 𝑗 𝑘 (\sqrt 𝑑 𝛼 Γ 𝛼 𝑗 𝑘 | 𝑓) = 𝑓 ̂ 𝑓 𝛼 𝑖 𝑗 = ⟨ Γ 𝛼 𝑖 𝑗 | 𝑓 ⟩ = (Γ 𝛼 𝑖 𝑗 | \sum 𝛽; 𝑘, 𝑙 𝑑 𝛼 ̂ 𝑓 𝛼 𝑘 𝑙 Γ 𝛽 𝑘 𝑙) = \sum 𝛽; 𝑘, 𝑙 ̂ 𝑓 𝛼 𝑘 𝑙 (\sqrt 𝑑 𝛼 Γ 𝛼 𝑖 𝑗 | \sqrt 𝑑 𝛼 Γ 𝛽 𝑘 𝑙) = ̂ 𝑓 𝛼 𝑗 𝑘

and hence it is a linear bijection. Since

⟨ 𝑓 | ℎ ⟩ = ⟨ 1 | 𝐺 | \sum 𝛼; 𝑖, 𝑗 𝑑 𝛼 ̂ 𝑓 𝛼 𝑖 𝑗 Γ 𝛼 𝑖 𝑗 | 1 | 𝐺 | \sum 𝛽; 𝑘 𝑙 𝑑 𝛽 ̂ ℎ 𝛽 𝑘 𝑙 Γ 𝛽 𝑘, 𝑙 ⟩ = 1 | 𝐺 | 2 \sum 𝛼; 𝑖, 𝑗 \sum 𝛽; 𝑘, 𝑙 𝑑 𝛼 𝑑 𝛽 ――― ̂ 𝑓 𝛼 𝑖 𝑗 ̂ ℎ 𝛽 𝑘 𝑙 ⟨ Γ 𝛼 𝑖 𝑗 | Γ 𝛽 𝑘 𝑙 ⟩ = 1 | 𝐺 | \sum 𝛼; 𝑖, 𝑗 \sum 𝛽; 𝑘, 𝑙 𝑑 𝛽 ――― ̂ 𝑓 𝛼 𝑖 𝑗 ̂ ℎ 𝛽 𝑘 𝑙 𝛿 𝛼 𝛽 𝛿 𝑖 𝑘 𝛿 𝑗 𝑙 = 1 | 𝐺 | \sum 𝛼; 𝑖, 𝑗 𝑑 𝛼 ――― ̂ 𝑓 𝛼 𝑖 𝑗 ̂ ℎ 𝛼 𝑖 𝑗 = ⟨ ̂ 𝑓 | ̂ ℎ ⟩

it is unitary. From Convolution of matrix representations, it follows that

̂ (Γ 𝛼 𝑖 𝑗 * Γ 𝛽 𝑘 𝑙) 𝛾 𝑝 𝑞 = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 ̂ (Γ 𝛼 𝑖 𝑙) 𝛾 𝑝 𝑞 = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 ⟨ Γ 𝛾 𝑝 𝑞 | Γ 𝛼 𝑖 𝑙 ⟩ = | 𝐺 | 2 (𝑑 𝛼) 2 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 𝛿 𝛾 𝛼 𝛿 𝑝 𝑖 𝛿 𝑙 𝑞 = 𝑑 𝛾 \sum 𝑚 = 1 | 𝐺 | (𝑑 𝛾) 2 𝛿 𝛾 𝛼 𝛿 𝑝 𝑖 𝛿 𝑚 𝑗 𝛿 𝛾 𝛽 𝛿 𝑚 𝑘 𝛿 𝑞 𝑙 = 𝑑 𝛾 \sum 𝑚 = 1 ⟨ Γ 𝛾 𝑝 𝑚 | Γ 𝛼 𝑖 𝑗 ⟩ ⟨ Γ 𝛾 𝑚 𝑞 | Γ 𝛽 𝑘 𝑙 ⟩ = 𝑑 𝛾 \sum 𝑚 = 1 ̂ (Γ 𝛼 𝑖 𝑗) 𝛾 𝑝 𝑚 ̂ (Γ 𝛽 𝑘 𝑙) 𝛾 𝑚 𝑞

hence $̂ \cdot$ preserves the algebra operations.

The isomorphism is denoted in such a way to evoke the Fourier transform due to similar properties. This may be viewed as a special case of the Wedderburn–Artin theorem.

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

1996, Representations of finite and compact groups, §III.1, pp. 38–39 ↩

Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations

Footnotes