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.

Component roots intersect the complementary parabolic exactly

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 $Φ_{K} \cap Φ_{I} = Φ_{J}$ . In particular $Φ_{K} ∖ Φ_{J} \subseteq D$ . No rank-finiteness assumption is needed.

Facts & Assumptions

Given: the diagram and reflection formula from Coxeter-system, roots, length and weak-order conventions, the root orbits from Standard parabolic subgroups and parabolic roots, and the component $K = [I, a]$ from Connected component notation [I,a]. Put $L = I ∖ K$ . [F1] There is no edge between $K$ and $L$ ; hence $B (α_{k}, α_{l}) = 0$ and $s_{k} s_{l} = s_{l} s_{k}$ for $k \in K, l \in L$ . [F2] The subspaces $V_{K}$ and $V_{L}$ have zero intersection, and every root is nonzero because it is the invertible image of a basis vector.

Proof

Goal: Establish the assertion in the Statement.

Proof technique: direct.

1.1F1given

The two disjoint parts of $I$ are $J$ and $L$ . By [F1] every finite word in $I$ can be reordered to a word in $J$ followed by a word in $L$ . Generators in $L$ act identically on $V_{K}$ , and generators in $J$ act identically on $V_{L}$ , by the reflection formula. Consequently the orbit under $W_{I}$ of a simple root indexed by $J$ is exactly its orbit under $W_{J}$ ; the orbit of a simple root indexed by $L$ stays in $V_{L}$ . Thus $Φ_{I} = Φ_{J} \cup Φ_{L}$ , with $Φ_{J} \subseteq V_{K}$ and $Φ_{L} \subseteq V_{L}$ .

2.1F2step 1.1given∎

Every root in $Φ_{K}$ lies in $V_{K}$ , because its generators preserve $V_{K}$ . By [F2] and step 1.1, a root belonging also to $Φ_{I}$ must lie in $Φ_{J}$ . Conversely $J \subseteq K \cap I$ gives $Φ_{J} \subseteq Φ_{K} \cap Φ_{I}$ directly from the orbit definitions. This proves equality. Finally $Φ_{K} \subseteq Φ_{I + a}$ , so a root in $Φ_{K} ∖ Φ_{J}$ is outside $Φ_{I}$ by the equality and hence lies in $D$ .

Depends on

Used by · 1

Finite root difference forces finite connected parabolic

Dependency tree · 2 levels

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