False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: every value of an irreducible complex character is real

Statement

The statement "every value of an irreducible complex character is real" is false: the nontrivial one-dimensional characters of $A_{4}$ take the primitive cube root of unity $ω = e^{2 π i / 3}$ , which is not real, on the $3$ -cycles.

Facts & Assumptions

Given: The group $A_{4}$ , a primitive cube root of unity $ω$ , and the irreducible character $χ$ of the character table with $χ ((123)) = ω$ .

[F1]

The character table of $A_{4}$ contains the irreducible character $χ$ with $χ ((123)) = ω$ and $χ ((132)) = ω^{2}$ (The character table of $A_{4}$ ).

[A1]

A primitive cube root of unity $ω$ satisfies $ω \notin R$ : its imaginary part is nonzero.

Refutation

technique · construct

1.1

By [F1], the irreducible character $χ$ of $A_{4}$ satisfies $χ ((123)) = ω$ .

F1given

2.1

By [A1], $ω \notin R$ , so this value of the irreducible character $χ$ is not real.

A1step 1.1

3.1

Hence the claimed statement, that every value of an irreducible complex character is real, is refuted by the character $χ$ at $(123)$ .

step 2.1discharge-construct: counterexample∎

Depends on

The character table of $A_4$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

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