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Universal wall graph as a groupoid Cayley graph

PDF p. 30: §7.2.1, final paragraph

Statement

Let $(W, S)$ be a finite-rank Coxeter system, with simple generators $s_{a}$ indexed by $a \in Γ_{0}$ , and fix $J \subseteq Γ_{0}$ . Let $Θ$ be the real dual reflection space, $Θ_{J} = {f \in Θ : f (α_{j}) = 0 (j \in J)}$ , and $(T_{J}, H^{J}) = (C o n e (J)^{\circ}, H^{J})$ the relative arrangement, with interior taken in $Θ_{J}$ . Relative chambers are the connected components of $T_{J} ∖ ⋃_{H \in H^{J}} H$ ; walls and adjacency use chamber closures in $Θ_{J}$ and the codimension-one wall convention. Identify these chambers with the algebraic labels $A = (x, I) \in C h a m (J)$ . Write $B H (J)$ for the Brink–Howlett groupoid, $\tilde{B H} (J)$ for its universal wall groupoid, and $F$ for the covering that sends $(x, I)$ to $I$ and an arrow to its label. Composition is read from left to right.

Let $L$ be the set of tagged base arrows $σ = v_{a, I} : I \to K$ realized as images under $F$ of adjacent wall crossings upstairs. The crossing element and its target subset depend only on $I, a$ . Define two labelled directed graphs with the same label set $L$ . The universal generating graph has vertex set $C h a m (J)$ and arc set consisting of $(A, σ, B)$ whenever $A, B$ are adjacent and their unique upstairs arrow maps to $σ$ . The right Cayley graph of $B H (J)$ based at $J$ has vertices all tagged arrows $g : J \to I$ , for all base objects $I$ , and arcs $(g, σ, g σ)$ for every $σ : I \to K$ in $L$ and every such $g : J \to I$ . A label-preserving directed graph isomorphism is a vertex bijection that preserves and reflects these labelled arc triples.

These graphs are isomorphic: the isomorphism sends $A$ to the image under $F$ of the unique arrow $(1, J) \to A$ .

When $J = \emptyset$ , the relative arrangement is the Coxeter arrangement: $(T_{\emptyset}, H^{\emptyset}) = ({C o n e}^{\circ}, H)$ . Moreover $B H (\emptyset)$ has one object and arrow group $W$ . The generating graph is the ordinary right Cayley graph of $W$ on the simple generators: its vertices are $x \in W$ and its arcs are $(x, s_{a}, x s_{a})$ . In particular $v_{a, \emptyset} = s_{a}$ .

Statement references

Tits cone intersections, relative roots and arrangements

Chambers, walls and adjacency

Reflecting hyperplanes, fundamental chamber and Tits cone

Algebraic chamber-label set

Universal wall groupoid on relative chambers

Brink–Howlett groupoid

Universal Brink–Howlett covering and its deck group

Set-sized groupoids, functors and composition conversion

Simple wall-crossing elements and maps

Labelled directed graphs, their underlying simple graphs, and label-preserving isomorphisms

Empty-subset wall crossings are simple reflections

Facts & Assumptions

[F1]

The universal wall groupoid has objects $(x, I) \in C h a m (J)$ , including $(1, J)$ , and exactly one arrow from $(x, I)$ to $(y, K)$ , labelled $x^{- 1} y$ . Its composition multiplies labels in order. For an adjacent pair its arrow has the crossing label $v_{a, I}$ for some $a \notin I$ . Universal wall groupoid on relative chambers Set-sized groupoids, functors and composition conversion

[F2]

The covering $F$ maps $(x, I)$ to $I$ and an arrow to its label. It is bijective on arrows with any prescribed source; adjacent crossings map to the corresponding Brink–Howlett generators. Its upstairs objects are the relative chambers in the finite-rank convention used here. Universal Brink–Howlett covering and its deck group

[F3]

The Brink–Howlett objects are the associates of $J$ ; its arrows $I \to K$ are the tagged elements of $W$ carrying the simple-root set of $K$ onto that of $I$ , with composition given by multiplication. Brink–Howlett groupoid

[F4]

The label set consists of pairs $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ minimal in $x W_{I}$ . Algebraic chamber-label set

[F5]

For $J = \emptyset$ , its only associate is $\emptyset$ , $W_{\emptyset} = {1}$ , $W_{{a}}$ has order two, and $v_{a, \emptyset} = s_{a}$ . Empty-subset wall crossings are simple reflections

[F6]

For a label $(x, I)$ and $a \notin I$ , if $W_{I \cup {a}}$ is finite, the finite wall-crossing theorem gives an adjacent chamber with first coordinate $x w_{I} w_{I \cup {a}}$ , where $w_{Q}$ denotes the longest element of the finite standard parabolic $W_{Q}$ . In particular, for $I = \emptyset$ this is $x s_{a}$ , with target subset again empty. Independent finite local wall-crossing formula Empty-subset wall crossings are simple reflections

[F7]

The normaliser action is transitive on the labels with any fixed second component. It preserves adjacency and transports a crossing at index $a$ to the crossing at the same index. The element $v_{a, I}$ and target subset depend only on $I, a$ . Normaliser action on labels, orbits and adjacency Simple wall-crossing elements and maps

[F8]

The relative definitions give $Θ_{J} = {f : f (α_{j}) = 0 (j \in J)}$ , $C o n e (J) = C o n e \cap Θ_{J}$ and $H^{J} = {H \cap Θ_{J} : H \in H, Θ_{J} \subseteq̸ H}$ . Every $H \in H$ is a proper hyperplane of $Θ$ . Relative interiors are taken in $Θ_{J}$ ; Coxeter-cone interiors are taken in $Θ$ . Chambers are connected components of the complement, and distinct chambers are adjacent when they share a wall of codimension one in the chamber's affine hull, with closure taken in the ambient space. Tits cone intersections, relative roots and arrangements Reflecting hyperplanes, fundamental chamber and Tits cone Chambers, walls and adjacency

[F9]

A labelled directed graph with label set $L$ has arc set contained in $V \times L \times V$ . An isomorphism is a vertex bijection preserving and reflecting membership of every labelled arc triple. Labelled directed graphs, their underlying simple graphs, and label-preserving isomorphisms

Proof

Given: The finite-rank Coxeter system and subset $J$ in the Statement, with arbitrary Coxeter type. Composition is read from left to right.

Proof technique: direct.

Goal: Construct the labelled graph isomorphism and establish both empty-subset specializations.

1.1F1F2

For each upstairs object $A = (x, I)$ let $u_{A}$ be the unique arrow $(1, J) \to A$ , and set $φ (A) = F (u_{A}) : J \to I$ . Given any base arrow $g$ with source $J$ , source-star bijectivity lifts it uniquely from $(1, J)$ ; its target is an object $A$ and its arrow is $u_{A}$ . Thus $φ$ is surjective. If $φ (A) = φ (B)$ as tagged arrows, uniqueness of that lift gives $u_{A} = u_{B}$ , hence $A = B$ . Therefore $φ$ is a vertex bijection.

1.2F8

If $J = \emptyset$ , the coordinate conditions defining $Θ_{J}$ are vacuous, so $Θ_{\emptyset} = Θ$ . Hence $C o n e (\emptyset) = C o n e$ , and their interiors agree because they are taken in the same space. Every $H \in H$ is proper in $Θ$ , so the condition $Θ_{\emptyset} \subseteq̸ H$ retains all of them, and $H \cap Θ_{\emptyset} = H$ . Thus $H^{\emptyset} = H$ , proving the equality of the two arrangement pairs. These arguments also include the empty vertex set.

2.1F1F2step 1.1

If $(A, σ, B)$ is an upstairs arc, write $h : A \to B$ for its crossing arrow. The singleton hom-set gives $u_{B} = u_{A} h$ . Functoriality yields $φ (B) = φ (A) σ$ , so $(φ (A), σ, φ (B))$ is a Cayley arc.

3.1F1F2F7F9step 1.1step 2.1

Conversely, take a Cayley arc $(g, σ, g σ)$ with $g : J \to I$ and $σ = v_{a, I} : I \to K$ . By the definition of $L$ , choose an adjacent crossing $A_{0} \to B_{0}$ realizing $σ$ , where $A_{0}$ has second component $I$ . Let $A = φ^{- 1} (g)$ , also with second component $I$ . By [F7], an element of the normaliser action takes $A_{0}$ to $A$ and takes $B_{0}$ to a chamber $B$ adjacent to $A$ , with the same crossing index and projected label $σ$ . Thus $(A, σ, B)$ is an upstairs arc. Step 2.1 gives $φ (B) = g σ$ ; step 1.1 makes this target unique. Consequently every Cayley arc lifts to an adjacent crossing. With step 2.1 and [F9], this proves that $φ$ is the claimed label-preserving directed graph isomorphism.

4.1F1F2F3F4F5F6step 3.1step 1.2∎

For $J = \emptyset$ , [F5] and [F3] give one base object and all elements of $W$ as its arrows, since both simple-root sets in the arrow condition are empty; composition is multiplication. By [F4] and [F5], every $x \in W$ gives the label $(x, \emptyset)$ , and these are all labels because any second component is a base object by [F2]. For every vertex $a$ , [F5] makes $W_{{a}}$ finite, and [F6] supplies the adjacent crossing $(x, \emptyset) \to (x s_{a}, \emptyset)$ . Conversely, [F1] gives some wall index $a$ for every adjacent pair, and [F5] makes its projected label $s_{a}$ ; the upstairs label formula in [F1] then forces its target to be $(x s_{a}, \emptyset)$ . Therefore $L$ consists exactly of the simple generators as arrows of this one-object groupoid, and the arc triples are exactly $(x, s_{a}, x s_{a})$ . Together with steps 3.1 and 1.2 this proves all assertions.

Source notes

The article's final paragraph of §7.2.1 supplies attribution for the graph interpretation and both empty-subset claims. The proof uses the covering, the normaliser action on adjacent crossings, the arrangement definitions and the finite rank-one crossing input. No article assertion is used as its own proof. The finite-rank qualification is the geometric convention of the suppliers; neither $W$ nor $W_{J}$ is required to be finite.

Depends on

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 30, §7.2.1, final paragraph