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

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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

1.1

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.

givenalgebra

2.1

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

givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Pavel Etingof, Representations of Lie Groups, Exercise 8.15(iv) (standard reference, not scraped)