ExampleConstruction: Literature-sourcedVerification: 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.

Universal groupoid interpretation of the D4 graph

PDF p. 28: §7.2.1; (7.1); PDF p. 29: §7.2.1; (7.1)

Example

Use the $D_{4}$ Coxeter system with central vertex $2$ , $J = {2}$ , and generators $s = s_{1}$ , $t = s_{2}$ , $u = s_{3}$ , $v = s_{4}$ from D4 diagram and generator conventions. Take the 32 labelled vertices and 48 undirected adjacency pairs of D4 chamber labels and complete adjacency graph, replacing each adjacency by an arrow in each direction. A path is a finite composable sequence of these arrows, including the empty path at a vertex.

The universal wall groupoid $\tilde{B H} (J)$ has exactly one morphism between each ordered pair of these vertices. It is presented by this directed graph with a relation identifying every two paths with the same endpoints. For an arrow from $A = (x, I)$ to an adjacent vertex $B = (y, K)$ , its label in $W$ is $x^{- 1} y = v_{a, I}$ , where $a$ is the wall index at $A$ ; the reverse arrow has label $y^{- 1} x = v_{b, K}$ for its wall index $b$ at $B$ . Every two-cycle is an identity, and the two labels are inverses, although they need not be equal.

Facts & Assumptions

[F1]

The listed 32 vertices are all relative chamber labels, and the 48 listed pairs are exactly their adjacencies. Vertex 7 is $(e, 2)$ , vertex 4 is $(s t, 1)$ , and they are adjacent. D4 chamber labels and complete adjacency graph

[F2]

The universal wall groupoid has a singleton hom-set from $(x, I)$ to $(y, K)$ , labelled $x^{- 1} y$ ; composition multiplies labels in order. For adjacent chambers the arrow label is the corresponding simple wall-crossing element. Universal wall groupoid on relative chambers

[F3]

Adjacent-crossing arrows generate the universal wall groupoid for a finite-rank Coxeter system. The universal wall groupoid is generated by adjacent crossings

[F4]

A graph and endpoint-preserving path relations present a category whose arrows are path classes and whose composition is concatenation. Path presentations, congruences and formal inverses

[F5]

In the D4 representation, $s_{i}$ sends $z$ to $z - 2 B (z, α_{i}) α_{i}$ , with $B (α_{1}, α_{2}) = - 1 / 2$ and $B (α_{i}, α_{i}) = 1$ . D4 diagram and generator conventions

Verification

Given: The D4 data and the directed adjacency graph in the Example.

Goal: Establish the path presentation, labels, and inverse assertions.

1.1F1F2F3F4

By [F1] the graph has exactly the objects and adjacent crossings under consideration. Map each directed edge to its unique wall-crossing arrow. By [F2], composites of paths with the same endpoints agree. Conversely, by [F3], every arrow is represented by a crossing path. Thus the induced functor from the path category modulo all same-endpoint path relations is bijective on each hom-set and on objects. By the quotient construction in [F4], it is an isomorphism.

2.1F2step 1.1

For adjacent $A = (x, I)$ and $B = (y, K)$ , [F2] gives the forward label $x^{- 1} y = v_{a, I}$ and, applied in the reverse direction, the label $y^{- 1} x = v_{b, K}$ . Their products in either order are $1$ . The two composites have equal endpoints and hence are identities by the singleton hom-set property. This proves the two-cycle assertion.

3.1F1F2F5step 1.1step 2.1∎

For the adjacency from vertex 7 to vertex 4, [F1] and [F2] give the labels $s t$ and $t s$ . The generators are involutions, so these labels are inverse. They are unequal: applying the reflection formulas in [F5] from right to left gives $s t (α_{1}) = α_{2}$ and $t s (α_{1}) = - α_{1} - α_{2}$ . Distinct images under the group representation imply distinct group elements. Together with steps 1.1 and 2.1 this proves the example.

Source notes

The article reference attributes the example. The verification uses the declared suppliers and the displayed local calculation.

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.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 28–29, §7.2.1 and (7.1)