Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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.

Dot-conjugate type-A2 weights have the same central character

Example

In type A2, let λ=0 and let μ=s1⋅λ=−α1. Then λ and μ have the same central character. Concretely,

λ+ρ=(1,0,−1),μ+ρ=s1(1,0,−1)=(0,1,−1),

and the basic symmetric invariants take the same values on those two triples.

Facts & Assumptions

Given: Type A2 in the realization {(x1,x2,x3)∈C3:x1+x2+x3=0}, the simple reflection s1, and the weight λ=0.

Verification

technique · direct
1.1

In the A2 realization, ρ=(1,0,−1) and s1 swaps the first two coordinates. Hence μ=s1⋅0=s1(ρ)−ρ=−α1, and μ+ρ=s1(ρ)=(0,1,−1).

givenalgebra
2.1

The degree-two and degree-three symmetric invariants from the A2 Cartan take the same values on ρ and s1(ρ) because those points are in the same ordinary Weyl orbit. Therefore Central characters are dot-Weyl orbits gives χμ=χλ.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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