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.

Elementary inversions detect the two initial prefixes

Statement

Under the finite-relative-root hypotheses of Finite relative longest element without spherical ambient rank, the article generators $v_{a, I}, v_{b, I}$ exist. For $c \in {a, b}$ , $N (v_{c, I}) = Φ_{I + c}^{+} ∖ Φ_{I}^{+}$ , this set contains $α_{c}$ , and $v_{c, I} ⪯ q$ , where $u ⪯ x$ means $ℓ (x) = ℓ (u) + ℓ (u^{- 1} x)$ . In particular the two generators are distinct.

Facts & Assumptions

[F1]

The definitions and results used in this argument are supplied, under their stated hypotheses, by Finite relative longest element without spherical ambient rank; Standard expressions and their lengths.

Proof

Given: The stated Coxeter and groupoid data and the explicit independent inputs cited below.

Goal: Establish the assertion in the Statement.

1.1F1given

For $c \in {a, b}$ , the inclusion $Φ_{I + c} ∖ Φ_{I} \subseteq Φ_{I + a + b} ∖ Φ_{I}$ makes the relative root difference finite. By the positive-root/reflection correspondence in Dyer–Gimenez §§1.4 and 1.28, $Φ_{I + c}^{+} ∖ Φ_{I}^{+}$ is in bijection with $(W_{I + c} ∖ W_{I}) \cap Ref (W)$ . Apply Lemma 1.25 to $Π_{I} \subseteq Π_{I + c}$ : the component $[I, c]$ containing $c$ is spherical. The inverse of $ν (c, I) = w_{[I, c]} w_{[I, c] - c}$ is exactly the generator $v_{c, I}$ of Standard expressions and their lengths. Lemma 1.25(a) gives $N (v_{c, I}) = Φ_{I + c}^{+} ∖ Φ_{I}^{+}$ .

2.1step 1.1∎

The inversion set is contained in $N (q)$ supplied by Finite relative longest element without spherical ambient rank, so §1.5 gives $v_{c, I} ⪯ q$ . Its intersection with the simple roots is precisely ${α_{c}}$ : every other simple root in $I + c$ lies in $I$ . Hence $v_{a, I} \neq v_{b, I}$ .

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §§1.4–1.5, PDF/printed p. 4; §1.24 and Lemma 1.25(a), PDF/printed p. 10; §1.28, PDF/printed p. 11

The map α↦s_α is a bijection from positive roots to reflections, and the root subsystem of W_L consists of roots whose reflections lie in W_L. For J⊆H⊆Π with (W_H\W_J)∩Ref(W) finite, the union D of components of H meeting H\J is spherical and ν(H\J,J)=w_D w_(D∩J) satisfies Φ^+∩ν(H\J,J)^(-1)Φ^-=Φ_H^+\Φ_J^+. For x,y∈W, inclusion Φ^+∩xΦ^-⊆Φ^+∩yΦ^- is equivalent to ℓ(y)=ℓ(x)+ℓ(x^(-1)y).

Hypotheses: A Coxeter system with its standard based geometric root system; arbitrary rank is allowed. In Lemma 1.25, J⊆H⊆Π and (W_H\W_J)∩Ref(W) is finite. The application uses J=Π_I and H=Π_(I+c); neither W nor W_I is assumed finite. Here the item’s N(x) denotes the source’s root inversion set Φ_x, rather than its reflection inversion set N(x).

Reading limits: For this proposal, read the frozen extracted §§1.4–1.5, §1.24 and Lemma 1.25, and §1.28 through its reflection-subgroup and standard-parabolic root conventions; visually checked PDF pp. 4, 10 and 11. Checked the source PDF URL. Only the positive-root/reflection correspondence, relative-longest statement and weak-order equivalence are imported. No reconstruction of their background proofs, complete paper reading, or reading of Brink–Howlett (1999) is claimed.

Sources

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §§1.4–1.5, PDF/printed p. 4; §1.24 and Lemma 1.25(a), PDF/printed p. 10; §1.28, PDF/printed p. 11