Ribbon groupoid in spherical type

PDF p. 21: Definition 5.1

Let $\mathrm{\Gamma }$ be a finite-rank spherical Coxeter diagram, meaning that its Coxeter group $W$ is finite, with vertex set ${\mathrm{\Gamma }}_0$. Let $A$ be its Artin–Tits group, with generators ${\sigma }_i$ indexed by $i\in {\mathrm{\Gamma }}_0$, and let $A_I^{+}$ be the submonoid generated by the ${\sigma }_i$ for $i\in I$. Fix $J\subseteq {\mathrm{\Gamma }}_0$. An associate of $J$ is a subset $I\subseteq {\mathrm{\Gamma }}_0$ whose standard Coxeter parabolic subgroup $W_I=⟨s_i:i\in I⟩$ is conjugate to $W_J$ in $W$.

The ribbon groupoid ${\mathcal{R}}^{\mathrm{\prime }}(J)$ has the associates of $J$ as objects. For associates $I,K$, its arrows $I\to K$ are tagged by source and target and have labels

${\mathrm{Hom}}_{{\mathcal{R}}^{\mathrm{\prime }}(J)}(I,K)=\{\beta \in A:A_I^{+}\beta =\beta A_K^{+}\}\mathrm{.}$

Here equality is equality of subsets of $A$. For $\beta :I\to K$ and $\gamma :K\to L$, their composite, read left to right, has label $\beta \gamma$. The identity at $I$ has label $1\in A$.

Coxeter-system, roots, length and weak-order conventions

Positive Artin and parabolic monoids

Associates of a parabolic subset

Connected small groupoids and vertex groups

The positive parabolic monoids and the associate condition are supplied by Positive Artin and parabolic monoids and Associates of a parabolic subset. The objects form a set, and the arrows form a set of triples $(I,\beta ,K)$. If $A_I^{+}\beta =\beta A_K^{+}$ and $A_K^{+}\gamma =\gamma A_L^{+}$, associativity of multiplication in $A$ gives $A_I^{+}\beta \gamma =\beta A_K^{+}\gamma =\beta \gamma A_L^{+}$. The identity satisfies the defining equality, and multiplying that equality for $\beta$ by ${\beta }^{-1}$ on both sides gives $A_K^{+}{\beta }^{-1}={\beta }^{-1}{A}_I^{+}$. Associativity, identity and inverse laws therefore follow from the group laws of $A$. These are the groupoid and left-to-right composition conventions of Connected small groupoids and vertex groups.

The canonical reflection representation, its simple-root basis and positive-root convention are those of Coxeter-system, roots, length and weak-order conventions. It remains to justify connectedness. By Brink–Howlett groupoid, for any two associates $I,K$ there is $x\in W$ with $x\{{\alpha }_k:k\in K\}=\{{\alpha }_i:i\in I\}$ in the canonical reflection representation. For each $k\in K$, write $x{\alpha }_k={\alpha }_i$; the basis property makes $k\mapsto i$ a bijection from $K$ to $I$. The reflection formula and invariance of the Coxeter form in Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order show that $x{s}_k{x}^{-1}=s_i$ as linear transformations; faithfulness in The root-length criterion and faithfulness of the canonical reflection representation makes this an equality in $W$.

The same root-length criterion gives $\mathrm{\ell }(x{s}_k)=\mathrm{\ell }(x)+1$ because $x{\alpha }_k={\alpha }_i$ is positive. It also gives $\mathrm{\ell }(x^{-1}{s}_i)=\mathrm{\ell }(x^{-1})+1$ because $x^{-1}{\alpha }_i={\alpha }_k$ is positive; reversing words in the involutive Coxeter generators gives $\mathrm{\ell }(s_{i}x)=\mathrm{\ell }(x)+1$. The reduced-word lift ${\sigma }_x$ and its length-additive multiplication rule from Reduced-word lifts and the prefix-order convention therefore yield ${\sigma }_i{\sigma }_x={\sigma }_{s_{i}x}={\sigma }_{x{s}_k}={\sigma }_x{\sigma }_k$. Since the corresponding generators are in bijection, taking all finite positive products gives $A_I^{+}{\sigma }_x={\sigma }_x{A}_K^{+}$. Thus ${\sigma }_x$ labels an arrow $I\to K$. The object $J$ exists, so the groupoid is nonempty and connected.