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.

Two elementary prefixes force a finite rank-two relative root set

Statement

If distinct elementary arrows $v_{a, I}, v_{b, I}$ both left-divide an article arrow $w : I \to K$ with Coxeter-length additivity, then $Φ_{I + a + b} ∖ Φ_{I}$ is finite. The morphism $q = v_{a, b, I}$ exists, is a length-additive prefix of $w$ , and has the two standard expressions defining its atomic braid relation.

Statement references

Standard expressions and their lengths

Two-generator minimal common left multiple

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

Standard parabolic subgroups and parabolic roots

Atomic braid relations

Facts & Assumptions

[F1]

Use the arbitrary-rank Coxeter system, geometric action, roots $Φ^{\pm}$ , and prefix weak order of Coxeter-system, roots, length and weak-order conventions, the parabolic roots of Standard parabolic subgroups and parabolic roots, and the elementary arrows and standard expressions of Standard expressions and their lengths. Write $I + c = I \cup {c}$ , $I + a + b = I \cup {a, b}$ , $Π_{L} = {α_{i} : i \in L}$ , and $Π = Π_{Γ_{0}}$ . In this proof define ${Inv}_{L} (x) = {α \in Φ^{+} : x^{- 1} α \in Φ^{-}}$ . Thus ${Inv}_{L} (x)$ is the source's $Φ_{x}$ ; the subscript $L$ means left inversions.

[F2]

The root-sign decomposition, inversion count, length-additive prefix criterion, elementary inversion formula, longest-element involutions, and relative-root inclusion are the independent inputs recorded in External input and source reference 0.

[F3]

The reversal of article arrows to root-action arrows, with unchanged labels, is supplied by Transfer to the root-action Brink–Howlett groupoid.

[F4]

Under finiteness of $Φ_{I + a + b} ∖ Φ_{I}$ , Two-generator minimal common left multiple supplies $v_{a, b, I}$ , its two standard expressions and its universal length-additive prefix property. Their equality is the atomic braid relation of Atomic braid relations.

External input

Use Dyer–Gimenez, sections 1.4–1.5 and 1.7, Lemma 1.25(a), and the inclusion in the proof of Lemma 3.12(a), with the hypotheses and notation translation in source reference 0. In particular, $∣ {Inv}_{L} (x) ∣ = ℓ (x)$ and a length-additive prefix $y$ of $x$ satisfies ${Inv}_{L} (y) \subseteq {Inv}_{L} (x)$ . For a root-action arrow $u : Π_{I} \to Π_{K}$ and $M \subseteq Π \cap {Inv}_{L} (u^{- 1})$ , that proof gives $Φ_{Π_{I} \cup M}^{+} ∖ Φ_{I}^{+} \subseteq {Inv}_{L} (u^{- 1})$ . Here a root subsystem indexed by simple roots means the parabolic subsystem on their indices. Only these premises are imported; the consequence below is proved locally.

Proof

Given: Distinct $a, b \notin I$ , elementary arrows $v_{a, I}, v_{b, I}$ that are length-additive left prefixes of $w : I \to K$ , and the conventions and premises [F1]–[F4].

Goal: Establish the Statement.

Proof technique: direct.

1.1givenF1F2

For an elementary arrow $v_{c, I}$ put $D = [I, c]$ . The source's label is $ν (α_{c}, Π_{I}) = w_{D} w_{D ∖ {c}}$ . The elementary definition and the longest-element involutions give $v_{c, I} = ν (α_{c}, Π_{I})^{- 1}$ . Lemma 1.25(a), in the left-inversion convention, therefore gives ${Inv}_{L} (v_{c, I}) = Φ_{I + c}^{+} ∖ Φ_{I}^{+}$ . Each parabolic root is supported on its index set: the simple reflections in that set preserve its span by the geometric reflection formula. Consequently ${Inv}_{L} (v_{c, I}) \cap Π = {α_{c}}$ . Applying the prefix criterion to the two given prefixes yields $α_{a}, α_{b} \in {Inv}_{L} (w)$ , equivalently $w^{- 1} α_{a}, w^{- 1} α_{b} \in Φ^{-}$ .

2.1step 1.1F1F2F3

By reversal, $u = w^{- 1}$ is a root-action arrow $Π_{I} \to Π_{K}$ , so $u Π_{I} = Π_{K}$ . Set $M = {α_{a}, α_{b}}$ . Step 1.1 gives $M \subseteq Π \cap {Inv}_{L} (u^{- 1})$ , since $u^{- 1} = w$ . The inclusion proved in Lemma 3.12(a) now gives $Φ_{I + a + b}^{+} ∖ Φ_{I}^{+} \subseteq {Inv}_{L} (u^{- 1}) = {Inv}_{L} (w)$ . Its right-hand side has $ℓ (w)$ elements. The full relative root set is the union of this finite positive difference and its negative, because roots have exactly one sign and parabolic root sets are closed under negation. Hence $Φ_{I + a + b} ∖ Φ_{I}$ is finite, with no finite-rank or spherical assumption on $I$ .

3.1step 2.1F1F4∎

Apply the two-generator lemma with the finiteness just proved. It supplies $q = v_{a, b, I} : I \to H$ , its two standard expressions, and the universal prefix property, which applies to $w$ . Thus $w = q z$ with $z = q^{- 1} w : H \to K$ and $ℓ (w) = ℓ (q) + ℓ (z)$ . By the atomic-braid definition, the equality of these two expressions for $q$ is its atomic braid relation. This proves the Statement with the required endpoints and length additivity.

Depends on

Used by · 1

Standard expressions are connected by atomic braid moves

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 · Sections 1.3–1.5 (especially (1.4.1), (1.4.4) and the first equivalence in 1.5); sections 1.7 and 1.9; Lemma 1.25, opening formula and (a); Lemma 3.12(a) and its proof.

Roots split disjointly into Phi^+ and Phi^-=-Phi^+. Write Inv_L(x)={alpha in Phi^+: x^{-1}alpha in Phi^-}, the source's Phi_x. Then |Inv_L(x)|=ell(x), and u is a length-additive prefix of x iff Inv_L(u) is a subset of Inv_L(x). Spherical longest elements are involutions. For D=[I,c] spherical, nu(alpha_c,Pi_I)=w_D w_{D minus c}, and Inv_L(nu(alpha_c,Pi_I)^{-1})=Phi_{I+c}^+ minus Phi_I^+. For a root-action morphism u:Pi_I to Pi_K and M a subset of Pi intersect Inv_L(u^{-1}), the proof of Lemma 3.12(a) gives Phi_{Pi_I union M}^+ minus Phi_I^+ as a finite subset of Inv_L(u^{-1}).

Hypotheses: A Coxeter system of arbitrary rank with its standard based root system. In the single-root formula only D=[I,c] is spherical. The root-action morphism satisfies u(Pi_I)=Pi_K; M consists of simple roots made negative by u. No finiteness or spherical hypothesis is imposed on I, K or the ambient system.

Reading limits: Read the HTML at sections 1.3–1.5, 1.7, 1.9, Lemma 1.25 and the statement and proof of Lemma 3.12(a). The displayed inversion conventions, hypotheses and applications were checked. No PDF inspection, reconstruction of the imported results, whole-paper reading, or reading of Brink–Howlett (1999) is claimed in this repair. No presentation or completeness-of-relations assertion is used.

Sources

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · Sections 1.3–1.5 (especially (1.4.1), (1.4.4) and the first equivalence in 1.5); sections 1.7 and 1.9; Lemma 1.25, opening formula and (a); Lemma 3.12(a) and its proof.