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Reduced ribbon groupoid is the Deligne orbit quotient

PDF p. 24: Proposition 6.3

Statement

Let $(W, S)$ be a finite-rank spherical Coxeter system, meaning that $W$ is finite, and fix $J \subseteq S$ . Write $W_{I}$ for the subgroup generated by $I \subseteq S$ , and let $N = N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ . Let $D$ be the relative arrangement groupoid, whose objects are chamber labels $(x, I)$ , and let $R$ be the reduced ribbon groupoid. Its objects are the associates of $J$ : the subsets $I \subseteq S$ for which $W_{I}$ is conjugate to $W_{J}$ . The functor $G : D \to R$ sends $(x, I)$ to $I$ and a simple wall crossing of index $a$ to the elementary ribbon $μ_{a, I}$ . Let $Q : D \to D / N$ send objects and arrows to their orbits under the normaliser action.

The functor $G$ factors as $G = G^{'} \circ Q$ , where $G^{'} : D / N \to R$ is an isomorphism of groupoids. On objects it identifies the $N$ -orbits of chambers with the associates of $J$ by $[(x, I)] \mapsto I$ .

Statement references

Assignment from wall crossings to ribbon generators

Deligne-to-ribbon functor is a covering

Normaliser action on labels, orbits and adjacency

Induced normaliser action on arrangement groupoids

Free object actions give orbit coverings

Facts & Assumptions

[F1]

A chamber label $(x, I)$ satisfies $W_{J} x = x W_{I}$ , with $x$ the minimum-length element of $x W_{I}$ . The objects of $R$ are the associates of $J$ . The groupoid $D$ is generated by simple wall crossings and their inverses. The assignment on objects and crossings is as stated, and $Q$ sends each object and arrow to its orbit. Assignment from wall crossings to ribbon generators

[F2]

The assignment extends to a functor $G$ , and for each chamber $C$ and each arrow of $R$ with source $G (C)$ there is exactly one arrow of $D$ with source $C$ and that image. Deligne-to-ribbon functor is a covering

[F3]

The action of $N$ on chamber labels preserves their second components, is free, and is transitive on labels with a fixed second component. On $D$ it acts by automorphisms and sends a crossing of index $a$ at a label with second component $I$ to a crossing with the same indices $a, I$ . Normaliser action on labels, orbits and adjacency Induced normaliser action on arrangement groupoids

[F4]

For this free object action, objects and arrows of the quotient are orbits; every arrow orbit with source $[C]$ has a unique representative with source $C$ , and quotient composition is obtained by aligning consecutive representatives. Free object actions give orbit coverings

[F5]

Every right coset of a standard parabolic subgroup has a minimum-length representative. Parabolic root subsystems and unique coset minima

Proof

Given: The spherical Coxeter system and subset $J$ in the Statement.

Proof technique: direct.

Goal: Construct the factor functor and prove it is bijective on objects and on each hom-set.

1.1F1F2F3F4

For $n \in N$ , the action preserves the second component of every chamber label, so $G (n C) = G (C)$ . It also sends each generating crossing of index $a$ at second component $I$ to one with the same indices, and both have image $μ_{a, I}$ . Because the action and $G$ are functorial, the same equality holds for inverses and composites: $G (n u) = G (u)$ for every arrow $u$ . Define $G^{'} ([C]) = G (C)$ and $G^{'} ([u]) = G (u)$ . These are independent of representatives. Alignment of representatives in the quotient shows that they preserve composition and identities, so $G^{'}$ is a functor and $G = G^{'} \circ Q$ .

1.2F1F3F5

Every label has an associate as its second component, since $W_{I} = x^{- 1} W_{J} x$ . Conversely, let $I$ be an associate and choose $w$ with $w^{- 1} W_{J} w = W_{I}$ . Choose the minimum-length representative $x$ of $w W_{I}$ , so $w = x u$ for some $u \in W_{I}$ . Since $u$ normalises $W_{I}$ , also $x^{- 1} W_{J} x = W_{I}$ ; hence $(x, I)$ is a chamber label. Thus every object of $R$ is hit. Two chamber labels have the same image exactly when their second components agree, and by [F3] this holds exactly when they lie in the same $N$ -orbit. Therefore $G^{'}$ is bijective on objects.

2.1F2F4step 1.1step 1.2

Let $[C]$ and $[D]$ be quotient objects and let $r : G (C) \to G (D)$ be an arrow of $R$ . By unique lifting, there is an arrow $u : C \to E$ with $G (u) = r$ . Its endpoint satisfies $G (E) = G (D)$ , so $[E] = [D]$ by step 1.2. Hence $[u] : [C] \to [D]$ maps to $r$ , proving surjectivity on this hom-set.

3.1F2F4step 1.1step 1.2step 2.1∎

If two arrows from $[C]$ to $[D]$ have the same image under $G^{'}$ , choose their representatives with source $C$ . Their images under $G$ are equal by step 1.1. Uniqueness of the lift from $C$ makes these representatives equal, so their orbits are equal. Thus $G^{'}$ is injective on each hom-set. Its bijection on objects and hom-sets defines an inverse functor, since the unique preimages of identities and composites are identities and composites. Consequently $G^{'}$ is an isomorphism.

Source notes

The article reference attributes the statement and its original proof. The argument above uses the completed suppliers listed in Facts & Assumptions.

Depends on

Used by · 3

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 24, Proposition 6.3