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.

Outside parabolic roots form an invariant set

Statement

Let $(W, S)$ be a Coxeter system indexed by a possibly infinite set $Γ_{0}$ , acting geometrically on the real vector space $V$ with simple-root basis $(α_{i})_{i \in Γ_{0}}$ . For $L \subseteq Γ_{0}$ , write $W_{L} = ⟨ s_{i} : i \in L ⟩$ and $Φ_{L} = {u α_{i} : u \in W_{L}, i \in L}$ . Let $I \subseteq Γ_{0}$ and $a \in Γ_{0} ∖ I$ , and define $I + a = I \cup {a}$ and $D = Φ_{I + a} ∖ Φ_{I}$ . Then $α_{a} \in D$ and $w D = D$ for every $w \in W_{I}$ .

Statement references

Coxeter-system, roots, length and weak-order conventions

Standard parabolic subgroups and parabolic roots

Facts & Assumptions

Given: the Coxeter system, geometric action, parabolic subgroups and root sets in the Statement. Goal: prove $α_{a} \in D$ and $w D = D$ for all $w \in W_{I}$ .

[F1]

The vectors $(α_{i})_{i \in Γ_{0}}$ form a basis of $V$ , every vector has finite support, and $s_{i} (v) = v - 2 B (v, α_{i}) α_{i}$ , by Coxeter-system, roots, length and weak-order conventions. Define $V_{I}$ to be the subspace consisting of all finite real linear combinations of $α_{i}$ with $i \in I$ .

[F2]

For each $L \subseteq Γ_{0}$ , the definitions $W_{L} = ⟨ s_{i} : i \in L ⟩$ and $Φ_{L} = {u α_{i} : u \in W_{L}, i \in L}$ are supplied by Standard parabolic subgroups and parabolic roots. In particular, elements of $W_{L}$ are finite words in these involutive generators.

Proof

technique · direct

1.1F1F2given

For $i \in I$ and $v \in V_{I}$ , the reflection formula gives $s_{i} (v) \in V_{I}$ , since both $v$ and $α_{i}$ belong to $V_{I}$ . Thus every finite $W_{I}$ -word preserves $V_{I}$ , so $Φ_{I} \subseteq V_{I}$ . Since $a \in I + a$ , the identity word shows $α_{a} \in Φ_{I + a}$ . Basis independence and $a \notin I$ give $α_{a} \notin V_{I}$ , hence $α_{a} \notin Φ_{I}$ . Therefore $α_{a} \in D$ .

2.1F2given∎

Since $I \subseteq I + a$ , every $W_{I}$ -word is a $W_{I + a}$ -word, giving $W_{I} \leq W_{I + a}$ . For $L = I$ or $L = I + a$ , $w \in W_{I}$ and $u α_{i} \in Φ_{L}$ , one has $w (u α_{i}) = (w u) α_{i} \in Φ_{L}$ . The same argument for $w^{- 1}$ yields $w Φ_{L} = Φ_{L}$ . If $β \in D$ , then $w β \in Φ_{I + a}$ ; membership of $w β$ in $Φ_{I}$ would imply $β \in Φ_{I}$ by applying $w^{- 1}$ , a contradiction. Thus $w D \subseteq D$ , and applying this inclusion to $w^{- 1}$ gives $w D = D$ .

Depends on

Used by · 2

Dependency tree · 2 levels

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