TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge 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.

Finite-dimensional complex representations of a finite group are determined up to isomorphism by their characters

Statement

Let $G$ be a finite group and let $V$ and $W$ be finite-dimensional complex representations of $G$ . Then

$V ≅ W ⟺ χ_{V} = χ_{W} .$

Facts & Assumptions

Given: A finite group $G$ and finite-dimensional complex representations $V$ and $W$ of $G$ .

[F1]

Every finite-dimensional representation of a finite group over a field of characteristic not dividing $∣ G ∣$ is completely reducible (If $char k ∤ ∣ G ∣$ , every finite-dimensional representation of $G$ is completely reducible).

[F2]

For irreducible $V_{i}$ , the multiplicity of $V_{i}$ in a completely reduced representation $U$ is $m_{i} (U) = ⟨ χ_{U}, χ_{i} ⟩$ (The multiplicity of an irreducible summand is a character inner product).

[A1]

Equivalent representations have equal characters.

Proof

technique · direct

1.1F1F2given

By [F1] there are decompositions $V ≅ ⨁_{i} m_{i} V_{i}$ and $W ≅ ⨁_{i} n_{i} V_{i}$ over the irreducible representations $V_{i}$ . By [F2] the multiplicities are $m_{i} = ⟨ χ_{V}, χ_{i} ⟩$ and $n_{i} = ⟨ χ_{W}, χ_{i} ⟩$ .

1.2A1given

Conversely, if $V ≅ W$ via an invertible intertwiner, then [A1] gives $χ_{V} = χ_{W}$ . This proves the reverse implication.

2.1step 1.1algebra

Assume $χ_{V} = χ_{W}$ . Then every inner product in step 1.1 agrees, so $m_{i} = n_{i}$ for every $i$ ; the two direct sums are built from the same irreducibles with the same multiplicities, hence $V ≅ W$ . This proves the forward implication.

3.1step 2.1step 1.2∎

Steps 2.1 and 1.2 prove both implications, hence the equivalence.

Depends on

Used by

Outer induction is not the Kronecker product Counterexample
Monomial representations, monomial characters, and M-groups Definition
An invariant irreducible normal representation yields projective inertia operators Lemma
The character field is the stabilizer fixed field Lemma
Sign twist corresponds to the omega involution Proposition
Schur index as minimal realization multiplicity Theorem
The outer Littlewood–Richardson rule Theorem

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

Peter Webb, A Course in Finite Group Representation Theory, Corollary 3.3.3 (standard reference, not scraped)