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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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The Verma irreducibility criterion from Shapovalov determinants

Statement

M(λ) is simple if and only if ⟨λ+ρ,α∨⟩∉Z>0 for every α∈Φ+.

Facts & Assumptions

Given: The radical identification The Shapovalov radical is the maximal submodule and the determinant formula The Shapovalov determinant formula.

Proof

technique · direct
1.1

If no displayed pairing is positive integral, every determinant block is nonzero by the formula. Orthogonality then makes the radical zero, so the maximal submodule is zero and M(λ) is simple.

givenalgebra
2.1

Conversely, if ⟨λ+ρ,α∨⟩=n>0, take β=nα. The formula has the factor with exponent K(0)=1, so that block is singular; the radical and hence the maximal submodule is nonzero.

givenalgebra∎

Depends on

Used by

Dependency tree · two levels

11 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