Assignment from wall crossings to ribbon generators

PDF p. 23: §6, opening construction of G and Q

Let $\mathrm{\Gamma }$ be a finite-rank spherical Coxeter diagram, meaning that its Coxeter group $W$ is finite, with vertex set ${\mathrm{\Gamma }}_0$, 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 $I$. A chamber label is a pair $(x,I)$ with $W_{J}x=x{W}_I$ and $x$ of minimum Coxeter length in $x{W}_I$. Identify these labels with the chambers of the relative arrangement. Let $\mathcal{D}=\mathcal{D}(J)$ be its arrangement groupoid: its objects are the chambers, its directed generating arrows are the simple crossings between adjacent chambers, and its positive-category relations identify geodesic paths with common endpoints. Let $\mathcal{R}=\mathcal{R}(J)$ be the reduced ribbon groupoid, whose objects are the subsets $I$ for which $W_I$ is conjugate to $W_J$ and whose elementary arrows are the ribbons ${\mu }_{a,I}$ for $a\in {\mathrm{\Gamma }}_0\setminus I$.

Define $\mathcal{G}$ on the directed chamber graph by $\mathcal{G}(x,I)=I$ on vertices and $\mathcal{G}({\omega }_{a,I})={\mu }_{a,I}$ on a generating arrow ${\omega }_{a,I}:(x,I)\to (y,K)$. Explicitly, with $T=I\cup \{a\}$, the target is $K=T\setminus \{{\iota }_T(a)\}$, where ${\iota }_T$ is the opposition permutation of the finite parabolic $W_T$. The elementary ribbon is ${\mu }_{a,I}={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_T:I\to K$, where ${\mathrm{\Delta }}_L$ is the positive reduced-word lift of the longest element of $W_L$.

Let $N(W,J)={\mathrm{Norm}}_W(W_J)\mathrm{/}{W}_J$ act on $\mathcal{D}$ through its action on relative chambers. Write $\mathcal{Q}:\mathcal{D}\to \mathcal{D}\mathrm{/}N(W,J)$ for the quotient functor: it sends each object and arrow to its orbit under this action. Composition is read from left to right.

Positive arrangement category and its groupoid completion

Algebraic chamber-label set

Labelling theorem for relative chambers

Simple wall-crossing elements and maps

Adjacent chambers and inverse wall-crossing elements

Reduced ribbon groupoid

Ribbon generators lift wall-crossing morphisms

Normaliser action on labels, orbits and adjacency

Induced normaliser action on arrangement groupoids

Free actions and quotient groupoids

Simplified wall crossing for a finite parabolic

Finite wall-crossing connectivity

The chamber-label definition and the chamber-labelling theorem identify every object of $\mathcal{D}$ with one such pair. Its second component $I$ is an object of $\mathcal{R}$ because $W_{J}x=x{W}_I$ implies $W_I=x^{-1}{W}_{J}x$. By the adjacent-wall formula, each directed generating arrow has a unique crossing index $a$ at its source. The simple-crossing definition uses the opposition permutation of the component containing $a$ in $T=I\cup \{a\}$. Since $W$ is finite, its subgroup $W_T$ is finite. The finite-parabolic wall-formula lemma therefore identifies the crossing target as $T\setminus \{{\iota }_T(a)\}$. The ribbon-generator lift lemma gives this same target $K$ for ${\mu }_{a,I}$, and the reduced ribbon groupoid contains this elementary arrow. Thus the displayed rule is a well-defined map of directed graphs, preserving sources and targets.

The label $(1,J)$ exists, so the chamber-labelling theorem gives an object of $\mathcal{D}$. The finite-wall-path lemma connects any two chamber objects by a finite sequence of simple crossings; their composite in $\mathcal{D}$ is an arrow between those objects. Thus $\mathcal{D}$ is nonempty and connected. It is small: the chamber labels form a subset of the set $W\times \mathcal{P}({\mathrm{\Gamma }}_0)$, the directed chamber graph has a set of arrows, and the arrangement-groupoid construction presents its morphisms as equivalence classes of finite composable words in these arrows and formal inverses. These words form a set, so their quotient is a set as well.

The normaliser chamber-action lemma gives an action free on chambers, and the induced-action result gives an action by automorphisms of $\mathcal{D}$. The connected small groupoid $\mathcal{D}$ therefore satisfies the hypotheses of the quotient-groupoid construction: orbit representatives can be aligned uniquely for composition, so taking object and arrow orbits defines the stated functor $\mathcal{Q}$.

The article's opening construction supplies attribution. The assertion that the graph assignment $\mathcal{G}$ respects the positive-category relations and extends to a functor $\mathcal{D}\to \mathcal{R}$ is the separate Lemma 6.1; it is not assumed in the graph assignment above.