Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Integral reflection linkage is an equivalence relation

Statement

Fix a finite-dimensional complex semisimple Lie algebra g, a Cartan subalgebra h, and a positive Borel b=h⊕n+. Write Q+=∑iZ≥0αi, μ≤λ when λ−μ∈Q+, and w⋅λ=w(λ+ρ)−ρ.

The set Φλ is preserved by its root reflections. If μ∈Wλ⋅λ, then Φμ=Φλ and Wμ=Wλ. The equivalence classes generated by moves η↦sα⋅η with α∈Φη are exactly Wλ⋅λ. Every strong-linkage chain stays in one such class.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra g, a Cartan subalgebra h, and a positive Borel b=h⊕n+. Write Q+=∑iZ≥0αi, μ≤λ when λ−μ∈Q+, and w⋅λ=w(λ+ρ)−ρ. For a weight λ, define Φλ={α∈Φ:⟨λ+ρ,α∨⟩∈Z},Wλ=⟨sα:α∈Φλ⟩⊂W. Reflections and coroots are those of def-root-reflections-and-the-weyl-group-action, with the shift from def-weyl-vector-rho-for-a-chosen-positive-system. The integral-reflection linkage class through λ is Wλ⋅λ. The word integral includes zero and negative integral pairings. If Φλ is empty the generated group is {1}. This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed. (The integral Weyl group of a weight)

[F2]

For the shifted (dot) action sα⋅η:=sα(η+ρ)−ρ, write μ↑λ if there are weights λ=η0≻η1≻⋯≻ηr=μ and positive roots αj such that ηj=sαj⋅ηj−1and⟨ηj−1+ρ,αj∨⟩∈Z>0 for every j. The empty chain is allowed, so λ↑λ. (The strong linkage order on weights)

Proof

1.1

Put a=λ+ρ and let α∈Φλ. For any root β, reflection invariance of the root-coroot pairing gives ⟨sαa,β∨⟩=⟨a,sαβ∨⟩=⟨a,β∨⟩−⟨α,β∨⟩⟨a,α∨⟩. The subtracted term is an integer by crystallographic integrality. Thus the left pairing is integral if and only if the first pairing on the right is integral.

F1algebra
2.1

Since sαβ is a root with coroot sαβ∨, the same identity says β∈Φλ if and only if sαβ∈Φλ. It also says Φsα⋅λ=Φλ. Iteration along any generating word proves the claimed equality of root sets and therefore of the generated groups at every point in the orbit.

F1algebrastep 1.1
3.1

Every allowed sequence of moves now uses generators of the original Wλ, so ends in its orbit. Conversely every word in those generators is an allowed sequence, since the root set stays unchanged at each intermediate weight. Each move reverses itself, and concatenation and the empty word give symmetry, transitivity and reflexivity. Strong-linkage moves use positive integral pairings, a subset of these allowed moves. A zero pairing fixes the weight.

F2algebrastep 2.1∎

Depends on

Used by

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