Full-dimensional J-chambers lie in the relative interior

Let $(W,S)$ be a finite-rank Coxeter system in its canonical real representation $V={\mathbb{R}}^S$, with basis $({\alpha }_s)$, dual $\mathrm{\Theta }$, root system $\mathrm{\Phi }$, contragredient action, and Tits cone $U$. For $I\subseteq S$ put $E_I=\{f:f({\alpha }_i)=0(i\in I)\}$ and $C_I=\{f\in E_I:f({\alpha }_s)>0(s\mathrm{\notin }I)\}$. Fix $J\subseteq S$ and put $E=E_J$, $T=U\cap E$, and $T^{\circ }={\mathrm{int}}_{E}T$. If $x\in W$, $I\subseteq S$, $x{C}_I\subseteq E$ and $\mathrm{\mid }I\mathrm{\mid }=\mathrm{\mid }J\mathrm{\mid }$, then $x{E}_I=E$, the set $x{C}_I$ is open in $E$, and $x{C}_I\subseteq T^{\circ }$. In particular every Iyama–Wemyss $J$-chamber lies in $T^{\circ }$.

Coordinate faces span their annihilator subspaces

Tits cone geometry with its exact interior boundary

Iyama–Wemyss, Definition 1.6(1), calls $x{C}_I$ a $J$-chamber precisely when $x{C}_I\subseteq E_J$ and $\mathrm{\mid }I\mathrm{\mid }=\mathrm{\mid }J\mathrm{\mid }$, with $x$ representing a coset in $W\mathrm{/}{W}_I$.

The external statement is used only as explicitly recorded above. It is an independent prerequisite source, not the article under study. The local topology and linear-algebra argument is authored here; the source proof has not been reviewed.