Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-09-23 (gpt-6-sol)
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.

The zero-weight singular central character

Example

For a supplied positive-root system, the weight −ρ is fixed by the dot action of every Weyl-group element: for every w∈W,

w⋅(−ρ)=−ρ.

Its highest-weight central character is singular in the sense that its label has full dot-action stabilizer W.

Facts & Assumptions

Given: A complex semisimple Lie algebra with a supplied positive-root system, Weyl group W, and Weyl vector ρ.

Verification

technique · direct
1.1givenalgebra

By definition of the dot action, w⋅(−ρ)=w((−ρ)+ρ)−ρ=w(0)−ρ=−ρ for every w∈W.

2.1step 1.1∎

The cyclic highest-weight module M(−ρ) has a central character χ−ρ because The Harish-Chandra projection computes the highest-weight scalar makes every central element act by a scalar. Step 1.1 shows that the label −ρ has the full Weyl group as its dot-action stabilizer, which is the asserted singularity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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