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The universal wall groupoid is generated by adjacent crossings

PDF p. 16: Proposition 4.4

Statement

Let $(W, S)$ be a finite-rank Coxeter system with vertex set $Γ_{0}$ , and fix $J \subseteq Γ_{0}$ . Let $Θ_{J} = {f : f (α_{j}) = 0 for every j \in J}$ , let $C o n e (J) = C o n e \cap Θ_{J}$ , and let $H^{J}$ be the set of distinct proper root-hyperplane traces on $Θ_{J}$ . Here $C o n e$ is the Tits cone, and $C o n e (J)^{\circ}$ is the interior in $Θ_{J}$ . A chamber is a connected component of the arrangement complement, and distinct chambers are adjacent when they share a wall.

Write $C h a m (J)$ for the pairs $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ of minimum length in $x W_{I}$ . The chamber-labelling bijection identifies these pairs with the chambers of $(C o n e (J)^{\circ}, H^{J})$ . Restriction from the full relative cone to its interior identifies the chambers and preserves adjacency. The universal wall groupoid $\tilde{B H} (J)$ has these labels as objects; for $A = (x, I)$ and $B = (y, K)$ its unique arrow is the endpoint-tagged triple $(A, B, x^{- 1} y)$ . Composition multiplies labels from left to right.

For any $A, B \in C h a m (J)$ there is a finite chamber path $p = ω_{a_{1}, I_{1}} * \dots * ω_{a_{m}, I_{m}} : A \to B$ in the relative-interior arrangement $(C o n e (J)^{\circ}, H^{J})$ , consisting of simple wall crossings between adjacent chambers. Its wall word is $v_{p} = v_{a_{1}, I_{1}} \dots v_{a_{m}, I_{m}} = x^{- 1} y \in W$ . Defining the tagged arrow $[p] = (A, B, v_{p})$ , we have ${Hom}_{\tilde{B H} (J)} (A, B) = {[p]}$ ; after taking arrow labels this singleton is ${v_{p}}$ . The path need not be unique. If $A = B$ , it may be the one-chamber path of length zero, with $v_{p} = 1$ . Thus the adjacent-crossing morphisms generate the groupoid.

Statement references

Algebraic chamber-label set

Chambers, walls and adjacency

Simple wall-crossing elements and maps

Universal wall groupoid on relative chambers

Chamber paths, concatenation and wall words

Finite wall-crossing connectivity

Tits cone intersections, relative roots and arrangements

Labelling theorem for relative chambers

Interior restriction preserves chambers and adjacency

Facts & Assumptions

[F1]

The relative-cone definition supplies $Θ_{J}$ , $C o n e (J)$ , its interior in $Θ_{J}$ , and the distinct proper root-hyperplane traces $H^{J}$ . Chambers are connected components of the complement; distinct chambers are adjacent when they share a wall. Tits cone intersections, relative roots and arrangements Chambers, walls and adjacency

[F2]

The algebraic set $C h a m (J)$ consists of the pairs $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ of minimum length in $x W_{I}$ . The chamber-labelling theorem identifies these pairs bijectively with the relative-interior chambers for finite rank and arbitrary $J$ , without a spherical-type hypothesis. Algebraic chamber-label set Labelling theorem for relative chambers

[F3]

Restriction from $C o n e (J)$ to $C o n e (J)^{\circ}$ identifies the chambers and preserves adjacency. Interior restriction preserves chambers and adjacency

[F4]

Any two relative chambers are connected by a finite sequence of simple wall crossings; the empty sequence is allowed for equal endpoints. Finite wall-crossing connectivity

[F5]

A chamber path is a finite nonempty sequence of chamber labels with consecutive chambers adjacent. Each crossing $ω_{a_{i}, I_{i}}$ has a defined element $v_{a_{i}, I_{i}} \in W$ ; its wall word is their ordered product, with empty product $1$ for the one-chamber path. Chamber paths, concatenation and wall words Simple wall-crossing elements and maps

[F6]

For $A = (x, I)$ and $B = (y, K)$ , the universal wall groupoid has the sole arrow $(A, B, x^{- 1} y)$ . Composition multiplies labels from left to right. For adjacent labels its unique arrow has their simple wall-crossing element as label. Universal wall groupoid on relative chambers

Proof

Given: Two labels $A = (x, I)$ and $B = (y, K)$ for the stated finite-rank Coxeter system and subset $J$ .

Goal: Construct the stated interior crossing path and identify its tagged composite, proving generation.

1.1F1F2F3

By [F2], identify $A$ and $B$ with their unique relative-interior chambers in the arrangement supplied by [F1]. By [F3], they are also chambers of the full relative arrangement, with the same adjacency relation. Thus the geometric connectivity assertion applies to these specified algebraic endpoints.

2.1F2F3F4F5step 1.1choose

Apply [F4] to the chambers in step 1.1 and choose a finite simple-crossing sequence between them. By [F3], every chamber and adjacency in the sequence is also in the relative-interior arrangement. By [F2], their labels give exactly the path $p$ of [F5], with endpoints $A, B$ and wall word $v_{p}$ . For equal endpoints choose the one-chamber path, whose word is $1$ .

3.1F5F6step 2.1algebra∎

Write the successive labels as $A_{i} = (x_{i}, I_{i})$ for $0 \leq i \leq m$ , with $A_{0} = A$ and $A_{m} = B$ . By [F6], each adjacent arrow is $(A_{i - 1}, A_{i}, x_{i - 1}^{- 1} x_{i})$ and its label is the corresponding crossing element. Their composite has label $v_{p} = (x_{0}^{- 1} x_{1}) \dots (x_{m - 1}^{- 1} x_{m}) = x^{- 1} y$ and is exactly $[p] = (A, B, v_{p})$ . For $m = 0$ the composite is $(A, A, 1)$ , the identity. Since [F6] makes the hom-set a singleton, it equals ${[p]}$ , with label set ${v_{p}}$ . Every ordered pair of objects admits the path constructed in step 2.1, so every arrow is a finite composite of adjacent-crossing arrows. No uniqueness of the path is required.

Source notes

The article reference attributes Proposition 4.4, including its interior-arrangement path assertion. The proof uses the declared geometric and algebraic suppliers and local composition, rather than the article proposition as a premise. Endpoint tags make explicit the arrow convention of the existing groupoid definition; the article's singleton notation is retained after taking labels.

Depends on

Used by · 5

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 16, Proposition 4.4