Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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The strong linkage principle for Verma modules

Statement

If [M(λ):L(μ)]≠0, then μ↑λ.

Facts & Assumptions

Given: Strong linkage The strong linkage order on weights, finite multiplicities Verma composition multiplicities are finite, the first filtration term The first Jantzen filtration term is the maximal Verma submodule, and the Jantzen sum formula The Jantzen sum formula for a Verma module.

Proof

technique · induction
1.1

Induct on the height of λ−μ. Height zero gives L(λ) and the empty linkage chain.

givenbase
1.2

For positive height, the factor is in the maximal submodule M1(λ). The Jantzen sum and finite multiplicities place it in some M(sα⋅λ) with ⟨λ+ρ,α∨⟩∈Z>0.

givenih
2.1

The new difference has smaller height, so induction gives μ↑sα⋅λ; appending the indicated positive-integral reflection gives μ↑λ.

step 1.2ihdischarge-induction∎

Depends on

Used by

Dependency tree · two levels

14 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