DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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The central character of an irreducible complex character

Definition

Let $G$ be a finite group, let $χ$ be an irreducible complex character of $G$ , and let $C$ be a conjugacy class of $G$ with class sum $\hat{C}$ (The class sum $\hat{C}$ of a conjugacy class $C$ ).

The central character of $χ$ is the function on class sums defined by

$ω_{χ} (\hat{C}) : = \frac{∣ C ∣ χ (g)}{χ (1)} (g \in C) .$

This is well defined because $χ$ is a class function (For a complex character, $χ (1) = \dim V$ , $χ$ is a class function, and $∣ χ (g) ∣ \leq χ (1)$ with equality exactly at scalars), so the value does not depend on the chosen $g \in C$ , and $χ (1) > 0$ for an irreducible character.

Depends on

Used by

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anupam Singh, Representation Theory of Finite Groups, Chapter 15 (standard reference, not scraped)
Peter Webb, A Course in Finite Group Representation Theory, Section 3.5 (standard reference, not scraped)