CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6.1-sol)

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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.

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Omitting the exterior root-weight shifts gives the wrong dot weight

Statement refuted

False claim. The weight of the degree-one Kostant generator for ${s l}_{2}$ can be computed without the exterior root-weight contribution: taking only the extremal vector gives the weight $w λ = - m ω$ , and attaching the $ρ$ -shift to the top weight gives the weight $λ - α = (m - 2) ω$ . In particular the exterior factor $ε_{α}$ may be dropped from the generator, or the whole $ρ$ -shift counted only once off the exterior part.

Facts & Assumptions

Given: The Axiom of Choice, inherited from the cited Kostant example; $g = {s l}_{2} (C)$ , $λ = m ω$ with $m \in Z_{\geq 0}$ , $ρ = ω = α / 2$ , $s ρ = - ρ$ , the cochain $ε_{α} \otimes v_{s λ}$ of weight $- α + s λ$ , and the alternative weights $- m ω$ and $(m - 2) ω$ .

[L1]

In rank one, $s α = - α$ , $s ρ = - ρ$ , and the dot action is $s \cdot λ = s (λ + ρ) - ρ = - λ - 2 ρ = - (m + 2) ω$ (Root reflections and the Weyl group action, The Weyl vector rho for a chosen positive system, The special linear Lie algebra sl_2, The root sl_2 triple).

[L2]

The cochain $ε_{α} \otimes v$ has weight $- α + w t (v)$ ; the space $V = L (m ω)$ has the weight $s λ = - m ω$ with multiplicity one (Chevalley–Eilenberg cochains, Weight and weight space, Kostant cohomology for sl2).

Counterexample

technique · compare the three candidate weights in rank one

1.1L1L2

The correct weight is $- α - m ω = - 2 ω - m ω = - (m + 2) ω$ , and by [L1] this equals $s \cdot λ$ ; the exterior factor contributes $s ρ - ρ = - 2 ρ = - α$ and the vector factor contributes $s λ = - m ω$ , so the two shifts are distinct and both are needed.

2.1L1L2step 1.1

Dropping the exterior factor leaves the weight $s λ = - m ω$ , which differs from the correct weight by $2 ω$ for every $m \geq 0$ ; since $ω \neq 0$ , the difference never vanishes. Replacing the vector factor by the top weight and moving the whole shift to the exterior part would give $λ - α = m ω - 2 ω = (m - 2) ω$ , which differs from $- (m + 2) ω$ by $2 m ω$ , vanishing exactly when $m = 0$ , so it differs for every $m \geq 1$ .

3.1L1step 2.1∎

The true degree-one cohomology weight is therefore $- (m + 2) ω$ , and neither of the two listed modifications produces it uniformly in $m$ : the exterior root-weight shift $s ρ - ρ = - α$ must be added to the vector shift $s λ = - m ω$ , not dropped and not counted twice.

Depends on

Used by

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Dependency tree · two levels

39 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

Direct sl2 cochain-weight check against the sign conventions of the Chevalley–Eilenberg complex; this ai-generated row is not a dependency target for any item (standard reference, not scraped)