DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-29

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The kernel of a complex character

Definition

Let $ρ : G \to GL (V)$ be a finite-dimensional complex representation of a finite group $G$ , with character $χ = χ_{V}$ (The character $χ_{V} (g) = tr (ρ_{V} (g))$ of a finite-dimensional complex representation). The kernel of the character is

$\ker χ : = {g \in G : χ (g) = χ (1)} .$

The value $χ (1)$ is the reference point of the definition, not a random number: The kernel of a complex character agrees with the kernel of any representation affording it proves that this set equals the kernel $\ker ρ$ of the representation, so the two readings of "kernel" never conflict on this page.

Depends on

The character $\chi_V(g)=\operatorname{tr}(\rho_V(g))$ of a finite-dimensional complex representation

Used by

Dependency tree · two levels

4 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

Peter Webb, A Course in Finite Group Representation Theory, Proposition 3.1.1 (standard reference, not scraped)
Shani Meynet and Robert Moscrop, McKay quivers and decomposition, Appendix A.3 (standard reference, not scraped)