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.

The root-permuting subgroup intersects the parabolic trivially

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 $N_{J} = {n \in W : n A_{J} = A_{J}}$ . The subgroup $N_{J}$ lies in $M$ .

$W_{J} \cap N_{J} = {1}$ .

Statement references

Root-permuting normaliser complement

Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order

Facts & Assumptions

[F1]

By Root-permuting normaliser complement, $N_{J} = {n \in W : n A_{J} = A_{J}}$ is a subgroup of $M = {Norm}_{W} (W_{J})$ , with the root and parabolic conventions in the statement.

[F2]

Every $n \in N_{J}$ is the unique minimum-length element of $n W_{J}$ , by Root-permuting elements are minimum coset representatives.

[F3]

By Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order, $ℓ (w)$ is the least number of simple generators in a word representing $w$ , and the empty word represents $1$ . Thus $ℓ$ is nonnegative, $ℓ (1) = 0$ , and $ℓ (w) = 0$ implies $w = 1$ , since the only word with zero letters is the empty word. This definition allows arbitrary rank.

Proof

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

1.1F2F3given

Let $n \in W_{J} \cap N_{J}$ . By [F2], $n$ is the unique minimum-length element of $n W_{J} = W_{J}$ . The identity belongs to this coset and has length zero by [F3]; nonnegativity therefore gives $ℓ (n) = 0$ .

2.1F1F3step 1.1∎

By [F3] and step 1.1, $n = 1$ . Conversely the identity preserves $A_{J}$ and belongs to $W_{J}$ , so by [F1] it lies in both subgroups. Their intersection is therefore exactly ${1}$ .

Depends on

Used by · 1

Coxeter normaliser semidirect product

Dependency tree · 2 levels

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