Irreducible orthonormal basis

Let $Γ : 𝐺 \to G L (𝑉)$ be a unitary representation of a finite group with decomposition

Γ ≅ ⨁ 𝜈 \in ̂ 𝐺 𝑎 𝜈 Γ 𝜈

and let ${𝑒 𝜈 𝛼 𝛽} 𝑎 𝜈 𝑑 𝜈 𝛼, 𝛽 = 1 \subseteq 𝑉$ be an orthonormal basis transforming under $𝐺$ in¹ a unitary irrep $Γ 𝜈$ for each $𝜈$ , i.e. for all $𝑔 \in 𝐺$

Γ (𝑔) 𝑒 𝜈 𝛼 𝛽' = 𝑑 𝜈 \sum 𝛽 = 1 𝑒 𝜈 𝛼 𝛽 Γ 𝜈 𝛽 𝛽' (𝑔)

then $⟨ 𝑒 𝜈 𝛼 𝛽 | 𝑒 𝜇 𝛼' 𝛽' ⟩ = 𝛿 𝜈 𝜇 𝛿 𝛼 𝛼' 𝛿 𝛽 𝛽'$ .² #m/thm/rep

Proof

Applying orthogonality or irreps on the third line:

⟨ 𝜓 𝜈 𝛼 | 𝜓 𝜇 𝛽 ⟩ = 1 | 𝐺 | \sum 𝑔 \in 𝐺 ⟨ Γ (𝑔) 𝜓 𝜈 𝛼 | Γ (𝑔) 𝜓 𝜇 𝛽 ⟩ = 1 | 𝐺 | \sum 𝑔 \in 𝐺 ⟨ 𝑑 𝜈 \sum 𝛾 = 1 𝜓 𝜈 𝛾 Γ 𝜈 𝛾 𝛼 (𝑔) | 𝑑 𝜇 \sum 𝛾' = 1 𝜓 𝜇 𝛾' Γ 𝜇 𝛾' 𝛽 (𝑔) ⟩ = 𝑑 𝜈 \sum 𝛾 = 1 𝑑 𝜇 \sum 𝛾' = 1 1 | 𝐺 | \sum 𝑔 \in 𝐺 ―――― Γ 𝜈 𝛾 𝛼 (𝑔) Γ 𝜇 𝛾' 𝛽 (𝑔) ⟨ 𝜓 𝜈 𝛾 | 𝜓 𝜇 𝛾' ⟩ = 𝑑 𝜈 \sum 𝛾 = 1 𝑑 𝜇 \sum 𝛾' = 1 1 𝑑 𝜈 𝛿 𝜈 𝜇 𝛿 𝛾 𝛾' 𝛿 𝛼 𝛽 ⟨ 𝜓 𝜈 𝛾 | 𝜓 𝜇 𝛾' ⟩ = 1 𝑑 𝜈 𝑑 𝜈 \sum 𝛾 = 1 𝛿 𝜈 𝜇 𝛿 𝛼 𝛽 ⟨ 𝜓 𝜈 𝛾 | 𝜓 𝜈 𝛾 ⟩ = 𝛿 𝜈 𝜇 𝛿 𝛼 𝛽

as required.

$𝑒 𝜈 𝛼 𝛽$ are thus called irreducible basis vectors transforming under irrep $Γ 𝜈$ . Every $𝜓 \in 𝑉$ may then be expressed as such, with

𝜓 = \sum 𝜈; 𝛼, 𝛽 𝑐 𝜈 𝛼 𝛽 𝑒 𝜈 𝛼 𝛽

and the application of $Γ$ gives

Γ (𝑔) 𝜓 = \sum 𝜈; 𝛼, 𝛽, 𝛽' 𝑐 𝜈 𝛼 𝛽 𝑒 𝜈 𝛼 𝛽' Γ 𝜈 𝛽' 𝛽 (𝑔)

which motivates the Generalized projection operator of a representation.

Explanation

Irreducible basis functions $𝑒 𝜈 𝛽$ have special symmetry properties under $𝐺$ , and the above theorem basically states these functions are orthogonal to each other.

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

the invariant subspace of $Γ$ corresponding to ↩
2023, Groups and representations, p. 44 (§4.1 lemma 8) ↩

Irreducible orthonormal basis

Explanation

Footnotes