Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
How statement and proof provenance work

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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The sl2 Shapovalov norm product

Example

For the normalized form on the sl2 Verma module,

Sλ(fnvλ,fnvλ)=n!∏j=0n−1(λ−j)(n≥0),

where the empty product is 1.

Facts & Assumptions

Given: Contravariance from Existence and uniqueness of the Shapovalov form and the e-action in The sl2 Verma action in the PBW basis.

Verification

technique · direct
1.1

For n=0 the formula is the normalization. For n>0, contravariance gives S(fnv,fnv)=S(fn−1v,efnv)=n(λ−n+1)S(fn−1v,fn−1v).

givenalgebra
2.1

Iterating that recurrence from 0 to n gives exactly the displayed factorial product, including its empty-product boundary.

givenalgebra∎

Depends on

Used by

Dependency tree · two levels

5 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