TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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.

The BGG criterion for homomorphisms between Verma modules

Statement

Assume the Axiom of Choice.

For weights $λ, μ$ , a nonzero homomorphism $M (μ) \to M (λ)$ exists if and only if $μ ↑ λ$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), positive-root embeddings Verma embedding for an arbitrary positive root, the definition The strong linkage order on weights, and Choice-qualified necessity A Verma embedding implies strong linkage.

Proof

technique · direct

1.1givenconstruct

If $μ ↑ λ$ , each edge in a witnessing chain gives a positive-root embedding; composing them gives a nonzero homomorphism $M (μ) \to M (λ) .$

2.1givenalgebra∎

Conversely, write the image of the source highest vector as $a v_{λ}$ in the PBW model. For $a \neq 0$ , a kernel vector $u v_{μ}$ would give $u a = 0$ in $U (n^{-})$ , impossible because its PBW-degree associated graded algebra is the domain $S (n^{-})$ . Thus the map is an embedding, and the necessity lemma gives $μ ↑ λ$ .

Depends on

Used by

Dependency tree · two levels

14 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

Pavel Etingof, Representations of Lie Groups, Theorem 15.11 (standard reference, not scraped)