Commutative square of arrangement, ribbon and Brink–Howlett groupoids

PDF p. 3: (1.9); PDF p. 4: (1.9); PDF p. 25: Remark 6.5

Let $(W,S)$ be a finite-rank Coxeter system and $J\subseteq S$. Write $W_J$ for the subgroup generated by $J$ and $N(W,J)={\mathrm{Norm}}_W(W_J)\mathrm{/}{W}_J$. The relative arrangement groupoid $\mathcal{D}=\mathcal{D}(J)$ has relative chambers as objects and is generated by directed crossings of adjacent chambers and their inverses, with geodesic paths having the same endpoints identified. The universal wall groupoid $\widetilde{\mathcal{B}\mathcal{H}}$ has chamber labels $(x,I)$ as objects and exactly one arrow between any ordered pair, labelled $x^{-1}y$ from $(x,I)$ to $(y,K)$. The Brink–Howlett groupoid $\mathcal{B}\mathcal{H}$ has the associates of $J$ as objects, where $I$ is an associate when $W_I$ is conjugate to $W_J$, and arrows $I\to K$ are the elements carrying the simple-root set of $K$ onto that of $I$.

For arbitrary Coxeter type, the natural functors $H:\mathcal{D}\to \widetilde{\mathcal{B}\mathcal{H}}$ and $\mathcal{F}:\widetilde{\mathcal{B}\mathcal{H}}\to \mathcal{B}\mathcal{H}$ exist; $H$ is the identity on chamber objects and $\mathcal{F}$ sends $(x,I)$ to $I$ and the arrow labelled $x^{-1}y$ to the same element of $W$.

Suppose now that $W$ is finite, also called spherical type. Let $A$ be the associated Artin–Tits group, $A_I$ its standard parabolic subgroup, and $\pi :A\to W$ the homomorphism sending each Artin generator to its corresponding simple reflection. The reduced ribbon groupoid $\mathcal{R}$ has the associates of $J$ as objects and is generated by elementary ribbons ${\mu }_{a,I}={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_{I\cup \{a\}}$ and their inverses, where $a\in S\setminus I$ and ${\mathrm{\Delta }}_L$ is the positive lift of the longest element of $W_L$. The functor $\mathcal{G}:\mathcal{D}\to \mathcal{R}$ sends $(x,I)$ to $I$ and each directed crossing indexed by $a$ at $(x,I)$ to ${\mu }_{a,I}$.

The homomorphism $\pi$ induces a functor $P:\mathcal{R}\to \mathcal{B}\mathcal{H}$, identity on objects, sending ${\mu }_{a,I}$ to the corresponding wall-crossing element $v_{a,I}$. The square commutes: $P\mathcal{G}=\mathcal{F}H$. Both vertical functors $\mathcal{G}$ and $\mathcal{F}$ are normal coverings with deck group isomorphic to $N(W,J)$. A covering has a unique lift from each specified source object for each arrow starting at its image; it is normal when its deck transformations act transitively on a fibre, and a deck transformation is an automorphism over the covering.

At the chamber labelled $(1,J)$ and its image $J$, the respective vertex groups of $\mathcal{D}$, $\widetilde{\mathcal{B}\mathcal{H}}$, $\mathcal{R}$ and $\mathcal{B}\mathcal{H}$ are isomorphic to ${\pi }_1({\mathcal{X}}_J^{\mathbb{C}},x)$, the trivial group, $N(A,J)={\mathrm{Norm}}_A(A_J)\mathrm{/}{A}_J$, and $N(W,J)$. Here $x$ is a chamber basepoint corresponding to $(1,J)$ under the arrangement-groupoid equivalence, and ${\mathcal{X}}_J^{\mathbb{C}}$ is the relative complexified hyperplane complement.

Artin–Tits group, pure subgroup and positive lifts

Positive arrangement category and its groupoid completion

Universal wall groupoid on relative chambers

Brink–Howlett groupoid

Reduced ribbon groupoid

Assignment from wall crossings to ribbon generators

Relative complexified hyperplane complement

Normalisers and their quotients

Root-permuting normaliser complement

The article reference attributes diagram (1.9) and Remark 6.5. The argument above uses the stated completed suppliers; the article citation is not used as a proof premise.