DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

The kernel of a complex character agrees with the kernel of any representation affording it Theorem

Dependency tree · two levels

3 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)