Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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Equal central character does not give every Verma embedding direction

Statement refuted

Equal central character of λ and μ implies both M(μ)→M(λ) and M(λ)→M(μ) are nonzero.

Counterexample

Given: Dot-orbit central characters Central characters are dot-Weyl orbits and the BGG criterion The BGG criterion for homomorphisms between Verma modules.

Proof technique: direct.

1.1

In a regular dominant integral A2 dot orbit take μ=s1⋅λ. The weights have equal central character because they are dot conjugate.

given
2.1

The BGG criterion gives M(μ)↪M(λ), but not the reverse map: λ↑̸μ, since the only directed reflection lowers the dominant weight. Thus central-character equality does not determine both directions.

step 1.1algebra∎

Depends on

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