PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-30

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Class sums act on an irreducible representation by central-character scalars

Statement

Let $G$ be a finite group, let $C$ be a conjugacy class of $G$ , let $\hat{C} \in C [G]$ be its class sum, and let $V$ be an irreducible complex representation of $G$ with character $χ$ . Then $\hat{C}$ acts on $V$ as the scalar $ω_{χ} (\hat{C})$ :

ρ_{V} (\hat{C}) = ω_{χ} (\hat{C}) {id}_{V} .

Facts & Assumptions

Given: A finite group $G$ , a conjugacy class $C$ of $G$ , its class sum $\hat{C}$ , and an irreducible complex representation $V$ of $G$ with character $χ$ .

[F1]

The class sums form a basis of the center of $C [G]$ , so each class sum lies in the center (For a finite group, the class sums form a basis of $Z (k [G])$ ).

[F2]

The central character is $ω_{χ} (\hat{C}) = ∣ C ∣ χ (g) / χ (1)$ for $g \in C$ (The central character of an irreducible complex character).

[F3]

On an irreducible complex representation, every $G$ -endomorphism is scalar (Over an algebraically closed field, every endomorphism of an irreducible representation is scalar).

Proof

technique · direct

1.1

Because $\hat{C}$ is central by [F1], the operator $ρ_{V} (\hat{C})$ commutes with $ρ_{V} (g)$ for every $g \in G$ , so it is a $G$ -endomorphism of the irreducible representation $V$ . By [F3], there is a scalar $λ_{C}$ with $ρ_{V} (\hat{C}) = λ_{C} {id}_{V}$ .

F1F3given

2.1

Taking traces gives $λ_{C} χ (1) = tr ρ_{V} (\hat{C}) = \sum_{x \in C} tr ρ_{V} (x) = \sum_{x \in C} χ (x) = ∣ C ∣ χ (g)$ for any $g \in C$ , because $χ$ is constant on $C$ . Hence $λ_{C} = ∣ C ∣ χ (g) / χ (1) = ω_{χ} (\hat{C})$ by [F2].

F2step 1.1algebra

3.1

Substituting the scalar from step 2.1 into step 1.1 yields $ρ_{V} (\hat{C}) = ω_{χ} (\hat{C}) {id}_{V}$ .

F2step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

Peter Webb, A Course in Finite Group Representation Theory, Proposition 3.5.3 (standard reference, not scraped)
Anupam Singh, Representation Theory of Finite Groups, Chapter 15 (standard reference, not scraped)