Finite wall-crossing connectivity

PDF p. 11: Lemma 3.20(3); Remark 3.22

Let $(W,S)$ be a finite-rank Coxeter system and let $J\subseteq S$. Let $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J)$ be the intersection of the Tits cone with ${\mathrm{\Theta }}_J=\{f:f({\alpha }_j)=0\text{ for all }j\in J\}$, and let ${\mathcal{H}}^J$ be the distinct proper traces on ${\mathrm{\Theta }}_J$ of root hyperplanes. Chambers are the connected components of the relative arrangement complement; distinct chambers are adjacent when their closures share a wall, a codimension-one intersection with an arrangement hyperplane.

Write $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$ for the pairs $(x,I)$ with $W_{J}x=x{W}_I$ and $x$ of minimum length in $x{W}_I$. For $a\mathrm{\notin }I$, let $D=[I,a]$ be the component containing $a$ in the Coxeter diagram induced on $I\cup \{a\}$. When $W_D$ is finite, the simple wall crossing is ${\omega }_{a,I}(x,I)=(x{w}_{D\setminus \{a\}}{w}_D,(I\cup \{a\})\setminus \{{\iota }_D(a)\})$, where $w_L$ is the longest element of the finite parabolic $W_L$ and $w_D{\alpha }_i=-{\alpha }_{{\iota }_D(i)}$.

Any two chambers are related by a finite sequence of simple wall crossings. In particular, this follows from local finiteness of the relative-interior arrangement and the identification of its chambers and adjacency relation with those of the full relative arrangement. The empty sequence is allowed when the two chambers coincide.

Tits cone intersections, relative roots and arrangements

Algebraic chamber-label set

Chambers, walls and adjacency

Simple wall-crossing elements and maps

Labelling theorem for relative chambers

The article supplies attribution for the statement and its local-finiteness observation. The proof uses the declared completed suppliers, not the article's assertion as a premise.