Simple wall-crossing elements and maps

PDF p. 10: (3.14)–(3.15)

Let $(W,S)$ be a Coxeter system indexed by the vertex set ${\mathrm{\Gamma }}_0$, and fix $J\subseteq {\mathrm{\Gamma }}_0$. Write $W_L$ for the subgroup generated by the simple generators indexed by $L\subseteq {\mathrm{\Gamma }}_0$. Let $(x,I)\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$, where $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$ is the algebraic chamber-label set: $W_{J}x=x{W}_I$ and $x$ has minimum length in $x{W}_I$.

Choose $a\in {\mathrm{\Gamma }}_0\setminus I$. Put $D=[I,a]$, the vertices of the connected component containing $a$ in the Coxeter diagram induced on $I\cup \{a\}$, and suppose $W_D$ is finite. For each finite standard parabolic $W_L$, write $w_L$ for its longest element and ${\iota }_L$ for the permutation of $L$ determined by $w_L{\alpha }_i=-{\alpha }_{{\iota }_L(i)}$ on its simple roots. Define the simple wall-crossing element and its associated pair by

$v_{a,I}=w_{D\setminus \{a\}}{w}_D,K=(I\cup \{a\})\setminus \{{\iota }_D(a)\},{\omega }_{a,I}(x,I)=(x{v}_{a,I},K)\mathrm{.}$

For fixed $J$, $I$ and $a$, this defines ${\omega }_{a,I}$ on the labels $(x,I)\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$ satisfying the stated finiteness condition, with values in $W\times \mathcal{P}({\mathrm{\Gamma }}_0)$, where $\mathcal{P}({\mathrm{\Gamma }}_0)$ denotes the set of all subsets of ${\mathrm{\Gamma }}_0$.

Connected component notation [I,a]

Longest parabolic elements and opposition permutation

Algebraic chamber-label set

The chamber-label definition supplies $W$, its standard parabolics, length and the set $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$. The connected-support definition makes $D$ a uniquely determined subset of $I\cup \{a\}$ containing $a$. Since $W_{D\setminus \{a\}}$ is a subgroup of the finite group $W_D$, it is finite. The longest-element definition therefore supplies both displayed longest elements and the permutation ${\iota }_D$; in particular ${\iota }_D(a)\in D\subseteq {\mathrm{\Gamma }}_0$. Thus $v_{a,I}\in W$, $K\subseteq {\mathrm{\Gamma }}_0$, and the displayed rule gives a unique pair in the stated codomain for every input. This justification does not assert that the output pair is a chamber label or that it represents an adjacent chamber.