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Explicit D4 comparison functors

PDF p. 31: §7.2.2, paragraph beginning “The functor”

Example

Take the $D_{4}$ Coxeter system whose central vertex is $2$ and whose leaves are $1, 3, 4$ , and put $J = {2}$ . Let $W$ be its Coxeter group and $A$ its Artin–Tits group. The canonical homomorphism $π : A \to W$ sends each standard Artin generator to the corresponding simple reflection.

Let $D$ be the Deligne groupoid of the relative chamber arrangement: its objects are the relative chambers, and it is the groupoid completion of the directed chamber-path category modulo identifications of geodesic paths with the same endpoints. A geodesic is a path of minimum length among paths with those endpoints. A chamber label $(x, I)$ consists of $x \in W$ and a subset $I$ with $W_{J} x = x W_{I}$ , where $W_{I}$ is generated by the simple reflections indexed by $I$ and $x$ has minimum length in $x W_{I}$ . The universal wall groupoid $\tilde{B H}$ has these labels as objects and exactly one arrow from $(x, I)$ to $(y, K)$ , labelled $x^{- 1} y$ .

Let $R$ be the reduced ribbon groupoid. Its objects are the associates of $J$ , meaning subsets $I$ for which $W_{I}$ is conjugate to $W_{J}$ ; its arrows are endpoint-tagged Artin-group elements generated by elementary ribbons and their inverses. The Brink–Howlett groupoid $B H$ has the same objects, with an arrow $I \to K$ labelled by an element of $W$ carrying the simple-root set indexed by $K$ onto that indexed by $I$ . In this example the objects of $R$ are ${1}, {2}, {3}, {4}$ .

There is a functor $H : D \to \tilde{B H}$ that is the identity on objects. For a directed chamber path $p : (x, I) \to (y, K)$ , it sends the represented arrow to the unique arrow labelled $x^{- 1} y = v_{p}$ , where $v_{p}$ is the ordered product of the simple wall-crossing elements along $p$ . On formal inverses, $H$ takes the inverse arrow. There is also a functor $P : R \to B H$ , identity on objects, induced on arrow labels by $π : A \to W$ .

Example references

D4 diagram and generator conventions

Positive arrangement category and its groupoid completion

Chamber paths, concatenation and wall words

Universal wall groupoid on relative chambers

Brink–Howlett groupoid

Reduced ribbon groupoid

D4 Deligne and ribbon graphs without quadratic relations

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Classification of finite Coxeter systems, including the H and dihedral families

Facts & Assumptions

[F1]

For a finite-rank Coxeter system and fixed $J$ , the natural functor from the arrangement groupoid to the universal wall groupoid is identity on chamber objects, sends a directed chamber path from $(x, I)$ to $(y, K)$ to the arrow labelled $v_{p} = x^{- 1} y$ , and sends formal inverses to inverses. Natural functors to the universal wall groupoid

[F2]

In spherical type, meaning that $W$ is finite, the canonical homomorphism $A \to W$ induces an identity-on-objects functor from the reduced ribbon groupoid to the Brink–Howlett groupoid. Commutative square of arrangement, ribbon and Brink–Howlett groupoids

[F3]

The specified $D_{4}$ example has the stated Deligne and reduced ribbon groupoids, and the reduced ribbon groupoid has the four singleton objects listed above. D4 Deligne and ribbon graphs without quadratic relations

[F4]

For a Coxeter system on a finite set with connected diagram, its group is finite if its labelled diagram is a standard $D_{n}$ diagram with $n \geq 4$ : one degree-three vertex with arms of $1, 1, n - 3$ vertices and all edges labelled $3$ . Classification of finite Coxeter systems, including the H and dihedral families

Verification

technique · direct

Given: The $D_{4}$ data, $J$ , and groupoids in the Example.

Goal: Verify both comparison functors and the stated formulas.

1.1F1given

The $D_{4}$ system is finite-rank, so [F1] applies to its arrangement groupoid. It supplies $H$ and its identity action on objects. For a directed chamber path with endpoints $(x, I)$ and $(y, K)$ , [F1] gives the unique target arrow with label $x^{- 1} y$ , equal to its wall word $v_{p}$ ; it also gives the action on formal inverses.

2.1F2F3F4step 1.1given∎

The diagram in the Example is connected on four vertices, with central vertex $2$ , three arms of one vertex each, and all three edges labelled $3$ . It is therefore the $n = 4$ case of [F4], so $W$ is finite. This supplies the spherical hypothesis required for the reduced ribbon groupoid and for [F2]. Thus [F2] supplies the functor $P$ , identity on objects and induced by $π$ on the endpoint-tagged arrow labels. The four-object description is [F3]. Together with step 1.1, this proves the example.

Source notes

The article citation attributes the example. The verification uses the completed suppliers, not the cited article paragraph as a premise.

Depends on

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 31, §7.2.2, paragraph beginning ‘The functor’