Codimension-two relations

PDF p. 21: Definition 4.25

Let $(W,S)$ be a finite-rank Coxeter system with diagram vertex set ${\mathrm{\Gamma }}_0$ and geometric roots $\mathrm{\Phi }$. For $L\subseteq {\mathrm{\Gamma }}_0$, let $W_L$ be the subgroup generated by the simple reflections indexed by $L$, and put ${\mathrm{\Phi }}_L=\{w{\alpha }_i:w\in W_L,i\in L\}$. Fix $J\subseteq {\mathrm{\Gamma }}_0$ and a relative chamber label $(x,I)\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$, meaning that $W_{J}x=x{W}_I$ and $x$ has minimum length in $x{W}_I$.

A chamber path in the relative arrangement $(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ },{\mathcal{H}}^J)$ is a finite nonempty sequence of chambers with consecutive chambers adjacent; its length is the number of successive pairs. It is geodesic when its length is minimal among chamber paths with the same endpoints. For an eligible vertex $a\mathrm{\notin }I$, write ${\omega }_{a,I}$ for the simple wall crossing at $(x,I)$ as defined in Simple wall-crossing elements and maps.

Choose distinct $a,b\in {\mathrm{\Gamma }}_0\setminus I$ such that $\mathrm{\mid }{\mathrm{\Phi }}_{I\cup \{a,b\}}\setminus {\mathrm{\Phi }}_I\mathrm{\mid }<\mathrm{\infty }$. Let $q=v_{a,b,I}:I\to K$ be the two-generator element of Two-generator minimal common left multiple. The two-geodesic lemma supplies exactly two geodesic paths $p_a,p_b$ from $(x,I)$ to $(xq,K)$, beginning respectively with ${\omega }_{a,I}$ and ${\omega }_{b,I}$. The formal relation $p_a=p_b$ identifying these paths is called a codimension-two relation.

By Two-geodesic wall-crossing lemma, the stated hypotheses give exactly the two paths with the stated endpoints and initial crossings. Their ordered pair therefore determines the displayed formal identification; the definition does not assert that the two paths are equal as sequences of chambers.

The article attributes this terminology to [24, Definition 1.58]. Its Definition 4.25 is cited for attribution; the path existence and uniqueness used in the justification come from the declared two-geodesic lemma.