DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-28

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.

A completely reducible representation as a finite direct sum of irreducible subrepresentations

Definition

Let $ρ : G \to GL (V)$ be a finite-dimensional representation of $G$ over a field $k$ . The representation $ρ$ is completely reducible if there are irreducible subrepresentations $V_{1}, \dots, V_{r} \leq V$ such that

$V = V_{1} \oplus \dots \oplus V_{r} .$

The empty direct sum is allowed, so the zero representation is completely reducible.

Under the dictionary of Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules, this is exactly the representation-side form of a semisimple left $k [G]$ -module (Semisimple modules as direct sums of simple modules).

A completely reducible representation as a finite direct sum of irreducible subrepresentations

Definition

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