RemarkRemark: 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.

Opposite-category convention for Brink–Howlett

PDF p. 15: Remark 4.1

Statement

The article uses the opposite of the groupoid in Brink–Howlett [5]: composition $Hom (I, K) \times Hom (K, L) ⟶ Hom (I, L), (x, y) ⟼ x y$ matches left-to-right path composition.

External input

Use the independent explicit restatement in Dyer–Gimenez, §1.9 (PDF p. 5). For simple-root subsets $A, B$ , their full Brink–Howlett groupoid $B$ has arrows $w : A \to B$ exactly when $w A = B$ , tagged as $(B, w, A)$ . Its composition is $(C, v, B) (B, w, A) = (C, v w, A) .$ Section 1.9 attributes this definition to Brink–Howlett [9]. Restrict $B$ to the objects $Π_{I} = {α_{i} : i \in I}$ with $I$ an associate of the fixed $J$ ; retain every arrow between these objects. The reading limit for the original 1999 source is recorded below and in the source metadata.

Proof

Given: The article groupoid, the independent external convention above, and the definition of the opposite category.

1.1given

By Brink–Howlett groupoid, a labelled arrow $x : I \to K$ in the article's groupoid is exactly an element satisfying $x Π_{K} = Π_{I}$ . Thus it is exactly the reversed arrow of $(Π_{I}, x, Π_{K})$ in $B$ . The identification $I \leftrightarrow Π_{I}$ is a bijection on the chosen object sets because the simple roots form a basis. This identifies the hom-sets with those of $B^{o p}$ as defined in Opposite category Cᵒᵖ.

2.1givenstep 1.1

Let $x : I \to K$ and $y : K \to L$ be article arrows. Their corresponding arrows in $B$ go $Π_{K} \overset{x}{\to} Π_{I}$ and $Π_{L} \overset{y}{\to} Π_{K}$ . The external composition formula gives $(Π_{I}, x, Π_{K}) (Π_{K}, y, Π_{L}) = (Π_{I}, x y, Π_{L})$ . Reversing this composite gives an arrow $I \to L$ with label $x y$ . Identities have label $1$ in both groupoids. Hence the hom-set identification preserves identities and composition and identifies the article's groupoid with this opposite category.

3.1givenstep 2.1∎

The notation of Set-sized groupoids, functors and composition conversion writes the composite of $x : I \to K$ followed by $y : K \to L$ from left to right. Step 2.1 proves that its label is $x y$ , exactly the displayed rule. No commutativity or finiteness hypothesis is needed.

Source notes

The 1999 Brink–Howlett original was not accessible through the public publisher page and was not read here. Dyer–Gimenez §1.9 is the independently read explicit source for its arrow and composition conventions. The comparison is a local calculation from that stated definition; it does not claim an independent reconstruction of any theorem of the 1999 paper. Original article PDF p. 15 was inspected to verify the formulation being compared. No source defect is asserted: opposite categories and differing composition conventions are compatible.

Depends on

Used by · 2

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 · PDF/printed p. 5, §§1.8–1.9; arbitrary-rank convention p. 4

The full Brink–Howlett groupoid has objects the subsets of the simple-root set Π and arrows J to K given by triples (K,w,J) with K=w(J); composition is (L,v,K)(K,w,J)=(L,vw,J). Section 1.9 explicitly attributes this groupoid to Brink and Howlett [9].

Hypotheses: A Coxeter system with its based root system, in particular the standard geometric root system. Finite rank is not assumed. Restricting to the associates of a fixed subset gives the groupoid used in this comparison.

Reading limits: Read PDF pp. 1–5 as extracted text and visually inspected p. 5, including §§1.8–1.9 formulas. Only the groupoid definition and composition convention are imported. No later result or proof is imported. Brink–Howlett (1999), DOI 10.1007/s002220050312, was not read: its public publisher PDF request redirected to a subscription preview, and the author homepage offered only that publisher link or an emailed offprint. The historical comparison uses this independent explicit restatement, not a claimed reading of the 1999 paper.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · §4.1, PDF p. 15, definition and Remark 4.1
Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · PDF/printed p. 5, §§1.8–1.9; arbitrary-rank convention p. 4