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.

Connected propagation spans the component

Statement

Use the arbitrary-rank geometric representation of Coxeter-system, roots, length and weak-order conventions and the orbit definition $Φ_{L} = {w α_{i} : w \in W_{L}, i \in L}$ from Standard parabolic subgroups and parabolic roots. Let $I \subseteq Γ_{0}$ , $a \in Γ_{0} ∖ I$ , $K = [I, a]$ , $J = K ∖ {a}$ , $V_{L} = {span}_{R} {α_{i} : i \in L}$ and $D = Φ_{I + a} ∖ Φ_{I}$ , with connected-component notation from Connected component notation [I,a]. Vectors have finite support, even if the vertex set is infinite. Then $V_{K} \subseteq {span}_{R} D$ . This assertion does not assume that $K$ or $I$ is finite.

Facts & Assumptions

Given: the notation and reflection formula of Coxeter-system, roots, length and weak-order conventions, the root orbits of Standard parabolic subgroups and parabolic roots, and $K$ as the vertices joined to $a$ by finite paths in Connected component notation [I,a]. [F1] The set $D$ contains $α_{a}$ and is $W_{I}$ -invariant by Outside parabolic roots form an invariant set. [F2] Across a diagram edge $i - j$ , $B (α_{i}, α_{j}) \neq 0$ : it is $- \cos (π / m_{i j}) < 0$ for finite $m_{i j} \geq 3$ and $- 1$ for infinite $m_{i j}$ .

Proof

Goal: Establish the assertion in the Statement.

Proof technique: direct.

1.1F1given

Let $U = {span}_{R} D$ . By [F1], $U$ is $W_{I}$ -invariant and contains $α_{a}$ . Fix $v \in K$ and choose a path of least length $a = i_{0}, i_{1}, \dots, i_{r} = v$ . Such a path has no repeated vertex, so every $i_{t}$ with $t > 0$ belongs to $I$ .

2.1F1F2step 1.1algebra∎

Induct along this finite path. If $α_{i_{t - 1}} \in U$ , then $s_{i_{t}} α_{i_{t - 1}} \in U$ by invariance. Subtracting gives $- 2 B (α_{i_{t - 1}}, α_{i_{t}}) α_{i_{t}} \in U$ . The coefficient is nonzero by [F2], so $α_{i_{t}} \in U$ . Thus every basis vector indexed by $K$ lies in $U$ , proving $V_{K} \subseteq U$ .

Depends on

Used by · 1

Finite outside roots give a finite complementary parabolic

Dependency tree · 2 levels

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