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D4 Deligne and ribbon graphs without quadratic relations

PDF p. 30: §7.2.2; PDF p. 31: §7.2.2

Example

Take the $D_{4}$ Coxeter system with central vertex $2$ , leaves $1, 3, 4$ , and $J = {2}$ . Write $s = s_{1}$ , $t = s_{2}$ , $u = s_{3}$ and $v = s_{4}$ . A chamber label $(x, i)$ means $(x, {i})$ . Let $D^{+}$ be the category of directed chamber paths modulo identifications of geodesics with common endpoints, and let $D$ be its groupoid completion. A geodesic is a path of minimum length among paths with those endpoints. The reduced ribbon groupoid $R$ has the associates of $J$ as objects and is generated by elementary ribbon arrows and their inverses.

The generating graph for $D$ is the 32-vertex chamber adjacency graph of the preceding labelled-chamber example, with each edge replaced by one arrow in each direction. Its relations are generated by identifying the two geodesic arcs around each codimension-two face: these faces are squares or hexagons. In particular, both pairs of paths displayed in (7.2), with the second atomic relation corrected to $s \cdot t v \cdot s t = t v \cdot s t \cdot v$ , commute in $D^{+}$ and in $D$ .

The generating graph for $R$ has objects $1, 2, 3, 4$ . Its nonloop arrows have Coxeter-word labels $s t : 2 \to 1$ , $t s : 1 \to 2$ , $v t : 2 \to 4$ , $t v : 4 \to 2$ , $u t : 2 \to 3$ and $t u : 3 \to 2$ ; its loops are $u, v$ at $1$ , $s, u$ at $4$ , and $s, v$ at $3$ . Each word denotes its positive Artin lift, obtained by replacing each letter by its Artin generator. The defining relations are the images of the square and hexagon relations. With $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ , there is an isomorphism $D / N (W, J) ≅ R$ ; the quotient functor $D \to R$ is a normal covering with deck group $N (W, J)$ .

The two opposite generating arrows of a chamber edge are not mutual inverses in $D$ : their composite is nonidentity. Likewise, every positive two-cycle in the displayed ribbon graph, including going twice around a loop, is nonidentity in $R$ . In particular $s^{2} = e$ in $W$ , while $σ_{s}^{2} \neq e$ in the Artin group.

Example references

D4 diagram and generator conventions

Positive arrangement category and its groupoid completion

Reduced ribbon groupoid

D4 chamber labels and complete adjacency graph

D4 Brink–Howlett generating graph

Explicit D4 atomic braid relations and lifted paths

Reduced-word lifts and the prefix-order convention

Facts & Assumptions

[F1]

The labelled-chamber example gives all 32 chambers and their adjacencies. The atomic-relations example identifies all codimension-two relations here with square or hexagon arcs, and gives the two pairs of paths in (7.2), including the corrected second relation. D4 chamber labels and complete adjacency graph Explicit D4 atomic braid relations and lifted paths

[F2]

The positive arrangement category identifies all geodesics with common endpoints; its groupoid completion is $D$ . Any two such geodesics differ by finitely many codimension-two replacements. Positive arrangement category and its groupoid completion Geodesics differ by codimension-two relations

[F3]

The four-object quotient graph and its Coxeter labels are those listed above. The Deligne-to-ribbon functor sends a directed wall crossing to the positive lift of its Coxeter label. The orbit quotient is isomorphic to $R$ , and this functor is a normal covering with deck group $N (W, J)$ . D4 Brink–Howlett generating graph Well-defined Deligne-to-ribbon functor and geodesic images Reduced-word lifts and the prefix-order convention Reduced ribbon groupoid is the Deligne orbit quotient Normal Deligne–ribbon cover and its deck group

[F4]

An arrow of the reduced ribbon groupoid is tagged by its source and target and labelled by an Artin-group element; composition multiplies labels. The Artin group is presented by the standard generators and braid relations, whose two sides have equal word length. The corresponding Coxeter presentation additionally has $s_{i}^{2} = e$ . Reduced ribbon groupoid Artin monoid and Artin group presentations, and the canonical monoid-to-group map Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares D4 diagram and generator conventions

Verification

Given: The $D_{4}$ data and the chamber and ribbon graphs in the Example.

Goal: Establish the stated presentations, covering and nonidentity assertions.

1.1F1F2given

By [F1] the doubled adjacency graph is the directed chamber graph. By [F2] the positive category is its path category modulo all same-endpoint geodesic identifications. Every such identification follows from finitely many codimension-two replacements, while each replacement is itself an identification of geodesic arcs with the same endpoints. Thus the square and hexagon relations generate exactly the defining congruence. Groupoid completion gives the asserted presentation of $D$ . The two pairs of paths in [F1] are instances, so they commute in the positive category and remain equal after completion.

2.1F3step 1.1

The quotient graph and its positive Artin labels are as stated by [F3]. Taking the orbit quotient of the graph presentation in step 1.1 gives its presentation by the images of the polygon relations: a path in the quotient lifts from a chosen initial chamber, and an orbit of a relation is one of those images. The quotient and normal-cover assertions are precisely the applicable statements in [F3].

3.1F3F4step 2.1∎

Send every standard Artin generator to $1 \in Z$ . By [F4], each defining braid relation has equal total exponent on both sides, so the presentation gives a homomorphism $ε : A \to Z$ . Each listed positive arrow label is a nonempty word, hence has strictly positive $ε$ -value. Consequently a composite of two such arrows has positive value and cannot be an identity arrow, whose Artin label is the identity. This includes traversing a displayed loop twice. A chamber two-cycle maps under the functor of [F3] to such a positive ribbon two-cycle; if it were an identity in $D$ , its image would be an identity. Finally $s^{2} = e$ is a Coxeter relation, whereas $ε (σ_{s}^{2}) = 2$ , so $σ_{s}^{2} \neq e$ .

Source notes

The article citation is attribution for this example, not a premise of the verification. The corrected second atomic relation and its path interpretation are taken from the declared preceding example.

Depends on

Used by · 1

Explicit D4 comparison functors

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 30–31, §7.2.2 and (7.2)