Orthonormality of irreducible characters

Let $𝜒 𝜇 : 𝐺 \to ℂ$ be irreducible characters for each $𝜇 \in ˆ 𝐺$ . Then ${𝜒 𝜇}$ form an Orthonormal basis of the Centre of the group ring $𝑍 (ℂ [𝐺])$ (i.e. class functions into $ℂ$ ) under a certain inner product.¹ #m/thm/rep In particular,

(𝜒 𝛼 | 𝜒 𝛽) = 1 | 𝐺 | \sum 𝑔 \in 𝐺 ―――― 𝜒 𝛼 (𝑔) 𝜒 𝛽 (𝑔) = 𝛿 𝛼 𝛽

Proof of orthonormality and completeness

That each $𝜒 𝜇$ is central follows from Properties, since

𝜒 𝜇 (𝑦 𝑥 𝑦 - 1) = t r Γ 𝜇 (𝑦 𝑥 𝑦 - 1) = t r Γ 𝜇 (𝑥 𝑦 - 1 𝑦) = t r Γ 𝜇 (𝑥) = 𝜒 𝜇 (𝑥)

Orthonormality follows easily from Orthonormality of irreps:

(𝜒 𝛼 | 𝜒 𝛽) = 𝑑 𝛼 \sum 𝑗 = 1 𝑑 𝛽 \sum 𝑘 = 1 (Γ 𝛼 𝑗 𝑗 | Γ 𝛽 𝑘 𝑘) = 𝑑 𝛼 \sum 𝑗 = 1 𝑑 𝛽 \sum 𝑘 = 1 1 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑗 𝑘 = 𝛿 𝛼 𝛽

Completeness follows from that of irreps too, by first noting

1 | 𝐺 | \sum 𝑦 \in 𝐺 Γ 𝜇 𝑖 𝑗 (𝑦 𝑥 𝑦 - 1) = 1 | 𝐺 | \sum 𝑦 \in 𝐺 \sum 𝑘, 𝑙 Γ 𝜇 𝑖 𝑘 (𝑦) Γ 𝜇 𝑘 𝑙 (𝑥) ―――― Γ 𝜇 𝑗 𝑙 (𝑦) = \sum 𝑘, 𝑙 (Γ 𝜇 𝑗 𝑙 | Γ 𝜇 𝑖 𝑘) Γ 𝜇 𝑘 𝑙 (𝑥) = \sum 𝑘, 𝑙 1 𝑑 𝜇 𝛿 𝑗 𝑖 𝛿 𝑙 𝑘 Γ 𝜇 𝑘 𝑙 (𝑥) = 1 𝑑 𝜇 𝛿 𝑖 𝑗 𝜒 𝜇 (𝑥)

and therefore for any $𝑓 \in 𝑍 (ℂ [𝐺])$ , from completness of irreps $𝑓 = \sum 𝜇; 𝑖, 𝑗 𝑐 𝜇 𝑖 𝑗 Γ 𝜇 𝑖 𝑗$ for some $𝑐 𝜇 𝑖 𝑗$ , thus

𝑓 (𝑥) = 1 | 𝐺 | \sum 𝑦 \in 𝐺 𝑓 (𝑦 𝑥 𝑦 - 1) = \sum 𝜇; 𝑖, 𝑗 𝑐 𝜇 𝑖 𝑗 1 | 𝐺 | \sum 𝑦 \in 𝐺 Γ 𝜇 𝑖 𝑗 (𝑦 𝑥 𝑦 - 1) = \sum 𝜇; 𝑖, 𝑗 𝑐 𝜇 𝑖 𝑗 1 𝑑 𝜇 𝛿 𝑖 𝑗 𝜒 𝜇 (𝑦)

thus $𝑓 \in s p a n {𝜒 𝜇}$ .

Alternate proof of completeness via Schur's lemma and matrix algebra isomorphism

Let $𝑓 \in 𝑍 (ℂ [𝐺])$ . Then by the Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations, each $̂ 𝑓 𝛼$ commutes in its $𝑑 𝛼 \times 𝑑 𝛼$ matrix algebra, which includes with the concrete reälization of $Γ 𝛼$

̂ 𝑓 𝛼 Γ 𝛼 (𝑔) = Γ 𝛼 (𝑔) ̂ 𝑓 𝛼

and therefore by Schur's lemma $̂ 𝑓 𝛼 = 𝑐 𝛼 𝐈 𝛼$ so

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

as required.

Since $𝜒 𝛼$ are class functions, the orthonormality may be rewritten for the character table.

Corollaries

The number of conjugacy classes equals the number of non-equivalent irreps of a group
The decomposition of a character into irreducible characters is always possible reveals the composition of the characterised representation.
Character irreducibility criterion

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