LemmaStatement: AI-generatedProof: AI-generated

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.

Each normaliser coset has exactly one root-permuting representative

Statement

Let $(W, S)$ be an arbitrary-rank Coxeter system in its standard real root representation, with simple roots $Π = {α_{i} : i \in Γ_{0}}$ and length $ℓ$ . Fix $J \subseteq Γ_{0}$ , put $A_{J} = {α_{j} : j \in J}$ and $W_{J} = ⟨ s_{j} : j \in J ⟩$ , and write $M = {Norm}_{W} (W_{J})$ and $N_{J} = {n \in W : n A_{J} = A_{J}}$ . These are the conventions of Root-permuting normaliser complement; $N_{J}$ is a subgroup of $M$ .

Each coset $g W_{J}$ with $g \in M$ contains exactly one element of $N_{J}$ .

Facts & Assumptions

[F1]

The declared suppliers provide the subgroup and previously established assertions:

Proof

Given: The data in the statement. Goal: Prove its conclusion.

Root-permuting normaliser complement

Minimum representatives of normaliser cosets preserve the simple roots

Root-permuting elements are minimum coset representatives

1.1F1

The minimum-normaliser-root-basis supplier in [F1] gives an element of $N_{J}$ in every indicated coset. This proves existence of a root-permuting representative.

2.1F1step 1.1∎

The root-permutation-minimum supplier in [F1] says that every element of $N_{J}$ in this coset must be its unique minimum-length element. Thus any two root-permuting representatives in that coset are equal. Together with step 1.1, this proves existence and uniqueness.

Depends on

Used by · 1

Coxeter normaliser semidirect product

Dependency tree · 2 levels

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