DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)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.

An irreducible complex character

Definition

A complex character $χ$ (The character $χ_{V} (g) = tr (ρ_{V} (g))$ of a finite-dimensional complex representation) is irreducible when $χ = χ_{V}$ for an irreducible representation $V$ of $G$ over $C$ (Subrepresentations, direct sums of representations, and irreducibility).

The phrase is well defined for characters rather than for individual representations: if $V$ and $W$ are equivalent, then $χ_{V} = χ_{W}$ by the invariance clause of (The character $χ_{V} (g) = tr (ρ_{V} (g))$ of a finite-dimensional complex representation), and equivalence preserves irreducibility. Throughout the page, "the irreducible characters" means one representative $χ_{1}, \dots, χ_{r}$ from each equivalence class of irreducible complex representations of the finite group $G$ ; the number $r$ is finite and equals the number of conjugacy classes of $G$ by the published count (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , the number of irreducible representations of $G$ equals the number of conjugacy classes).

Depends on

Used by

The character table of a finite group Definition
The first orthogonality relation for irreducible complex characters Theorem

Dependency tree · two levels

5 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, Chapter 3 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Section 3.5 (standard reference, not scraped)