TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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The BGG criterion for homomorphisms between Verma modules

Statement

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

Facts & Assumptions

Given: Positive-root embeddings Verma embedding for an arbitrary positive root, the definition The strong linkage order on weights, and necessity A Verma embedding implies strong linkage.

Proof

technique · direct

1.1

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

givenconstruct

2.1

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 $μ ↑ λ$ .

givenalgebra∎

Depends on

Used by

Equal central character does not give every Verma embedding direction Counterexample
The A2 regular integral-dominant Verma embedding poset Example
The sl2 Verma embedding chain Example

Dependency tree · two levels

8 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)