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Induced normaliser action on arrangement groupoids

PDF p. 14: Remark 3.30

Statement

Let $(W, S)$ be a finite-rank Coxeter system and let $J \subseteq S$ . Write $W_{J}$ for the subgroup generated by $J$ and $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ for its normaliser quotient. Let $C h a m (J)$ be the relative chamber set and let $D (J)$ be its arrangement groupoid. A simple wall crossing $ω_{a, I} : C \to C^{'}$ is the directed generating arrow across the indicated wall, where the chamber label of $C$ has second component $I$ and $a$ is the crossing index.

The action of $N (W, J)$ on chambers induces an action by automorphisms of $D (J)$ . For $g \in N (W, J)$ it sends $ω_{a, I} : C \to C^{'}$ to the simple wall crossing $ω_{a, I} : g C \to g C^{'}$ . It preserves geodesic paths and hence the defining relations of the positive arrangement category.

Statement references

Normaliser action on labels, orbits and adjacency

Positive arrangement category and its groupoid completion

Facts & Assumptions

[F1]

The quotient acts on relative chambers. It preserves adjacency and sends a simple crossing of index $a$ from a chamber with label component $I$ to the crossing with the same $a$ and $I$ between the image chambers. Normaliser action on labels, orbits and adjacency

[F2]

The positive arrangement category has one directed generating arrow for each ordered adjacent pair of chambers. Its relations identify geodesic paths with common endpoints, where a geodesic is a path of minimum length among paths with those endpoints. Its groupoid completion is universal for functors from the positive category into groupoids. Positive arrangement category and its groupoid completion

Proof

Given: The finite-rank Coxeter system, subset $J$ , and normaliser quotient in the Statement.

Proof technique: direct.

Goal: Construct the stated action on $D (J)$ and show that it preserves geodesics and the defining relations.

1.1F1F2given

Paths and geodesics. For $g \in N (W, J)$ , send each chamber to its image and each directed arrow to the arrow between the image chambers. By [F1] this is a map of the directed chamber graph, and the map for $g^{- 1}$ is its inverse. Applying it to each arrow gives a length-preserving bijection between paths from $C$ to $C^{'}$ and paths from $g C$ to $g C^{'}$ . A shortest path therefore maps to a shortest path: a shorter image path would pull back to a shorter path with the original endpoints. The crossing index and label component are preserved by [F1].

2.1F1F2step 1.1∎

Relations and completion. The map takes each pair of geodesics with common endpoints to another such pair, so it respects the generating relations and descends to an automorphism of the positive category. The object and arrow maps compose according to the group action, because there is only one generating arrow for each ordered adjacent pair. Compose this positive-category automorphism with the canonical functor into $D (J)$ and use the universal property in [F2] to extend it uniquely to a groupoid endofunctor. The extensions for $g$ and $g^{- 1}$ are inverse: their composites and the identity agree on the positive category and hence agree by uniqueness. The same uniqueness proves the group action law. Thus the specified assignment defines the claimed action and preserves the relations.

Source notes

Remark 3.30 of the article attributes the statement. The argument above uses the declared mathematical suppliers; it does not use the article's assertion as a premise.

Depends on

Used by · 4

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 14, Remark 3.30