ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

A two-dimensional A2 Verma weight space

Example

In type $A_{2}$ , choose Chevalley generators and $f_{12} = [f_{2}, f_{1}]$ , with PBW order $f_{1}, f_{12}, f_{2}$ . At weight $λ - α_{1} - α_{2}$ the basis is $f_{1} f_{2} v_{λ}, f_{12} v_{λ}$ . Writing $λ_{i} = λ (h_{i})$ , its Shapovalov matrix is

$(\begin{array}{cc} λ_{2} (λ_{1} + 1) & - λ_{2} \\ - λ_{2} & λ_{1} + λ_{2} \end{array}),$

and its determinant is $λ_{1} λ_{2} (λ_{1} + λ_{2} + 1)$ .

Facts & Assumptions

Given: The induced module Verma modules, the prescribed PBW basis from The PBW model of a Verma module, and its normalized contravariant form Existence and uniqueness of the Shapovalov form.

Verification

technique · direct

1.1givenalgebra

PBW gives exactly the two displayed monomials. The Chevalley relations give $e_{1} f_{1} f_{2} v_{λ} = (λ_{1} + 1) f_{2} v_{λ}$ , $e_{1} f_{12} v_{λ} = - f_{2} v_{λ}$ , $e_{12} f_{1} f_{2} v_{λ} = - λ_{2} v_{λ}$ , and $e_{12} f_{12} v_{λ} = (λ_{1} + λ_{2}) v_{λ}$ . Contravariance, together with $S_{λ} (f_{2} v_{λ}, f_{2} v_{λ}) = λ_{2}$ , gives the four displayed matrix entries.

2.1givenalgebra∎

Taking the $2 \times 2$ determinant gives $λ_{2} (λ_{1} + 1) (λ_{1} + λ_{2}) - λ_{2}^{2} = λ_{1} λ_{2} (λ_{1} + λ_{2} + 1)$ .

Depends on

Used by

Nothing in the library uses this result yet.

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, Exercise 8.15(iv) (standard reference, not scraped)