Reduced ribbon groupoid

PDF p. 22: Definition 5.5

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 and fix $J\subseteq {\mathrm{\Gamma }}_0$. For $I\subseteq {\mathrm{\Gamma }}_0$, write $W_I$ for the subgroup generated by the simple reflections indexed by $I$. An associate of $J$ is a subset $I$ for which $W_I$ is conjugate to $W_J$ in $W$.

Let ${\mathcal{R}}^{\mathrm{\prime }}(J)$ be the ribbon groupoid: its objects are the associates of $J$, and an arrow $I\to K$ is an element $\beta \in A$ with $A_I^{+}\beta =\beta A_K^{+}$, tagged with its source and target. Here $A_I^{+}$ is the submonoid generated by the standard Artin generators indexed by $I$. Composition is multiplication from left to right.

For an associate $I$ and $a\in {\mathrm{\Gamma }}_0\setminus I$, put $T=I\cup \{a\}$. Let $w_T$ be the longest element of $W_T$, let ${\iota }_T:T\to T$ be the opposition permutation defined by $w_T{\alpha }_i=-{\alpha }_{{\iota }_T(i)}$ on the simple roots, and put $K=T\setminus \{{\iota }_T(a)\}$. If ${\mathrm{\Delta }}_L$ denotes the positive reduced-word lift in $A$ of the longest element of $W_L$, the elementary ribbon is ${\mu }_{a,I}={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_T$, regarded as an arrow $I\to K$ of ${\mathcal{R}}^{\mathrm{\prime }}(J)$.

The reduced ribbon groupoid $\mathcal{R}(J)$ is the subgroupoid of ${\mathcal{R}}^{\mathrm{\prime }}(J)$ with all the same objects whose arrows are finite composable products of the arrows ${\mu }_{a,I}$ and their inverses, including empty products as identities. Equivalently, it is the smallest subgroupoid of ${\mathcal{R}}^{\mathrm{\prime }}(J)$ containing every object and every such elementary ribbon arrow.

Ribbon groupoid in spherical type

Associates of a parabolic subset

Positive elementary ribbon elements

Longest parabolic elements and opposition permutation

The associate condition and the ambient groupoid are supplied by Associates of a parabolic subset and Ribbon groupoid in spherical type. The longest elements, opposition permutations and elementary ribbon elements are well defined by Longest parabolic elements and opposition permutation and Positive elementary ribbon elements.

To apply Ribbon generators lift wall-crossing morphisms for an arbitrary associate $I$, choose $y\in W$ with $W_{J}y=y{W}_I$. Choose an element $x$ of minimum Coxeter length in the right coset $y{W}_I$. Since $x=yu$ for some $u\in W_I$, one still has $W_{J}x=x{W}_I$; thus $(x,I)$ satisfies the chamber-label hypotheses of the lemma. The lemma shows that $K$ is an associate of $J$ and that ${\mu }_{a,I}$ is an arrow $I\to K$ of ${\mathcal{R}}^{\mathrm{\prime }}(J)$.

The objects and arrows of ${\mathcal{R}}^{\mathrm{\prime }}(J)$ form sets. Finite composable products of the specified arrows and their inverses form a set of its arrows. Concatenation of such products is again such a product, reversing a product and inverting its factors gives its inverse, and the empty products give all identities. The inherited composition and inverses therefore make these arrows a subgroupoid. Any subgroupoid containing all objects and the specified arrows must also contain their inverses and every such product, proving the stated minimality.

Definition 5.5 of the cited article attributes this construction to its reference [17]. The article citation records attribution; the justification above uses the listed suppliers.