TheoremStatement: Literature-sourcedProof: 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.

Brink–Howlett presentation by atomic and quadratic relations

PDF p. 19: Theorem 4.19

Statement

For any Coxeter diagram and $J \subseteq Γ_{0}$ , the groupoid $B H$ is generated by the wall-crossing morphisms in Corollary 4.13(1), with defining relations all atomic braid relations and $v_{a, I} v_{a^{'}, K} = e$ , where $a^{'} = ι_{[I, a]} (a)$ and $K = I + a - a^{'}$ .

Facts & Assumptions

[F1]

The arbitrary-rank Coxeter system, associates, Brink–Howlett groupoid, left-to-right label composition, component $D = [I, a]$ , finite longest elements, opposition permutation and elementary labels $v_{a, I} = w_{D ∖ {a}} w_{D}$ are as in Standard expressions and their lengths. Only the crossing component $W_{D}$ is required to be finite.

[F2]

The elementary morphisms generate $B H (J)$ by Brink–Howlett generators and invariance of standard length.

[F3]

Atomic braid relations are those of Atomic braid relations. Path categories and their endpoint-preserving congruence quotients are defined in Path presentations, congruences and formal inverses.

[F4]

The quotient by all atomic braids and both ordered elementary inverse-pair quadratics is a groupoid with faithful label evaluation to $B H (J)$ by The atomic-quadratic path quotient is faithful. Its Proof, steps 1.1 and 2.1, establishes for every elementary edge that, with $b = ι_{D} (a)$ and $K = (I \cup {a}) ∖ {b}$ , the reverse edge exists, $[K, b] = D$ , $ι_{D} (b) = a$ and $v_{b, K} = v_{a, I}^{- 1}$ . That argument applies the inverse calculation inside the finite Coxeter subsystem on $D$ and then identifies its labels in the ambient group; it imposes no ambient rank bound or finiteness assumption on $W_{I}$ .

Proof

Given: An arbitrary Coxeter diagram and $J \subseteq Γ_{0}$ , with the notation and elementary morphisms of [F1] and inputs [F2]–[F4].

Goal: Establish the presentation in the Statement.

Proof technique: direct.

1.1F1F4

For each elementary arrow $v_{a, I}$ , put $D = [I, a]$ , $a^{'} = ι_{D} (a)$ and $K = (I \cup {a}) ∖ {a^{'}}$ . By [F4], the reverse elementary arrow is $v_{a^{'}, K} = v_{a, I}^{- 1}$ , with $[K, a^{'}] = D$ and $ι_{D} (a^{'}) = a$ . Hence the displayed quadratic evaluates to the identity at $I$ . The same displayed family applied to the reverse edge gives $v_{a^{'}, K} v_{a, I} = e$ at $K$ , so this family contains both ordered inverse-pair relations. Conversely each relation in that ordered family is an instance of the displayed family. Thus the quadratic families agree for arbitrary ambient rank.

2.1F1F3F4step 1.1

Use [F3] to form the path-category quotient on the elementary-edge quiver by the atomic braids and the displayed quadratics. Step 1.1 identifies its relations with exactly those of [F4]. Therefore this quotient is a groupoid, and label evaluation is a faithful functor to $B H (J)$ , identical on objects.

3.1F2F3step 2.1∎

By [F2], every Brink–Howlett morphism is the label of a finite composable elementary path. Thus evaluation is surjective on every hom-set. Its faithfulness makes it injective on every hom-set as well. Since it is the identity on objects, its inverse sends each morphism to its unique path class; this respects identities and composition because evaluation does and is injective. Evaluation is consequently an isomorphism of groupoids. By the definition of path presentation in [F3], the atomic braids and displayed quadratics are therefore defining relations, proving the Statement.

Source notes

The original Theorem 4.19 is cited for attribution only. Presentation completeness is supplied by the current local faithfulness lemma and generation corollary. The inverse-pair application uses the explicit finite-subsystem argument in the faithfulness lemma, rather than an application of a finite-rank adjacent-chamber proposition to arbitrary rank. This repair supplies no library publication, independent review or audit stamp.

Depends on

Used by · 1

The ordinary Coxeter presentation is the empty-J case

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.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 19; Theorem 4.19