DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring

Definition

Fix $n \geq 1$ and a commutative ring $R$ . A function $F : M_{n} (R) \to R$ is column-multilinear when, with all other columns fixed, it is additive and compatible with scalar multiplication in each selected column.

It is alternating when $F (A) = 0$ whenever two columns of $A$ are equal. It is antisymmetric when interchanging two columns negates its value. It is normalized when $F (I_{n}) = 1$ . These conditions are stated over the ring itself and do not assume that $2$ is invertible.

Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring

Definition

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