Two-geodesic wall-crossing lemma

PDF p. 20: Lemma 4.23; PDF p. 21: Lemma 4.23; (4.24)

Let $(W,S)$ be a finite-rank Coxeter system with diagram vertex set ${\mathrm{\Gamma }}_0$, geometric root set $\mathrm{\Phi }$, and Coxeter length $\mathrm{\ell }$. Fix $J\subseteq {\mathrm{\Gamma }}_0$. For $L\subseteq {\mathrm{\Gamma }}_0$, let $W_L$ be the subgroup generated by its simple reflections and ${\mathrm{\Phi }}_L=\{w{\alpha }_i:w\in W_L,i\in L\}$. An associate of $J$ is a subset whose standard parabolic subgroup is conjugate to $W_J$. The Brink–Howlett groupoid $\mathcal{B}\mathcal{H}(J)$ has these associates as objects; its morphisms $I\to K$ are the elements $w\in W$ satisfying $\{w{\alpha }_k:k\in K\}=\{{\alpha }_i:i\in I\}$, with composition by multiplication from left to right.

In the relative arrangement $(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ },{\mathcal{H}}^J)$, write $(x,I)\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$ for a chamber label. A chamber path is a finite sequence of adjacent chambers; it is geodesic if its number of crossings is minimal among paths with the same endpoints. For $a\mathrm{\notin }I$, when the component $D=[I,a]$ containing $a$ in the diagram induced on $I\cup \{a\}$ has finite $W_D$, put $v_{a,I}=w_{D\setminus \{a\}}{w}_D$, where $w_L$ is the longest element of the finite parabolic $W_L$. Write ${\omega }_{a,I}$ for the associated simple wall crossing. A standard expression is a composable sequence of these wall-crossing elements whose Coxeter lengths sum to the length of their product.

Let distinct $a,b\in {\mathrm{\Gamma }}_0\setminus I$ satisfy $\mathrm{\mid }{\mathrm{\Phi }}_{I\cup \{a,b\}}\setminus {\mathrm{\Phi }}_I\mathrm{\mid }<\mathrm{\infty }$, and let $(x,I)\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$. Let $q=v_{a,b,I}:I\to K$ be the two-generator element supplied by Two-generator minimal common left multiple. Its two standard expressions lift to exactly two geodesic paths of equal length from $(x,I)$ to $(xq,K)$, beginning respectively with ${\omega }_{a,I}$ and ${\omega }_{b,I}$. If a chamber $C$ admits geodesics from $(x,I)$ beginning with each of these crossings, then some geodesic from $(x,I)$ to $C$ passes through $(xq,K)$.

The article reference attributes Lemma 4.23 and its translation method. The proof above uses the declared completed suppliers; the article claim is not used as its own proof premise.