Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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A nonintegral reflection does not produce a singular power

Statement refuted

For every complex λ, the formal power f⟨λ+ρ,α∨⟩vλ is a singular vector in the Verma module.

Counterexample

Given: The Verma-module convention Verma modules.

Proof technique: direct.

1.1

In sl2, take λ=−12, so ⟨λ+ρ,α∨⟩=12. PBW gives vectors fnvλ only for integers n≥0; f1/2vλ is not a vector of M(λ).

given
2.1

Thus the asserted formal power does not even define a candidate singular vector. The integrality condition is necessary before the rank-one singular-vector calculation can begin.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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