Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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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.

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False statement: a completely reducible representation has a unique decomposition into irreducible subrepresentations

Statement

False claim. A completely reducible representation has a unique decomposition into irreducible subrepresentations.

Facts & Assumptions

Given: The two-dimensional trivial representation of C2.

[L1]

The two-dimensional trivial representation of C2 has two different decompositions into irreducible subrepresentations, while its isotypic component is still unique (The two-dimensional trivial representation of C2 has many irreducible splittings but one isotypic component).

[L2]

The uniqueness theorem concerns the isotypic decomposition, not the choice of individual irreducible summands (A decomposition into irreducible summands need not be unique even when the isotypic decomposition is).

Refutation

technique · direct
1.1

The example [L1] exhibits one completely reducible representation with two different decompositions into irreducible subrepresentations.

L1given
2.1

By [L2], this does not contradict isotypic uniqueness: the isotypic block is canonical, but the individual irreducible splitting is not. Therefore the displayed example refutes the claim.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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