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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FALSE: every complex class function with self-inner-product $1$ is a character

Statement

The statement "every complex class function with self-inner-product $1$ is a character" is false: for a nontrivial finite group $G$ with trivial character $1$ , the class function $- 1$ has self-inner-product $1$ but is not a character.

Facts & Assumptions

Given: A finite cyclic group $G$ of order $n \geq 2$ , its trivial character $1$ , and the class function $- 1$ defined by $(- 1) (g) = - 1$ for every $g$ .

[F1]

The cyclic group of order $n \geq 2$ has the trivial irreducible character $1$ with $1 (g) = 1$ for every $g$ (The character table of a finite cyclic group over $C$ ).

[F2]

A complex character is irreducible exactly when its self-inner-product is $1$ (A complex character is irreducible if and only if its self-inner-product is $1$ ).

[A1]

The inner product satisfies $⟨ c φ, c ψ ⟩ = ∣ c ∣^{2} ⟨ φ, ψ ⟩$ for a scalar $c$ , so $⟨ - 1, - 1 ⟩ = ⟨ 1, 1 ⟩$ .

[A2]

Every character $χ$ of a representation satisfies $χ (1) = \dim V \geq 0$ .

Refutation

technique · construct

1.1

The constant function $- 1$ is a class function, because it is constant on $G$ , hence constant on every conjugacy class.

given

1.2

By [F1] and [F2], the trivial character satisfies $⟨ 1, 1 ⟩ = 1$ ; by [A1], $⟨ - 1, - 1 ⟩ = ⟨ 1, 1 ⟩ = 1$ .

F1F2A1given

1.3

If $- 1$ were a character of some representation, then by [A2] its value at $1$ would be nonnegative, but $(- 1) (1) = - 1 < 0$ . Hence $- 1$ is not a character.

A2given

2.1

Steps 1.1 through 1.3 exhibit a class function with self-inner-product $1$ that is not a character, so the claimed statement is refuted.

step 1.1step 1.2step 1.3discharge-construct: counterexample∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)

FALSE: every complex class function with self-inner-product 11 is a character