Independent relative chamber labelling input

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 $T^{\circ }={\mathrm{int}}_{E}T$. Let $W_I=⟨I⟩$ and let $\mathrm{\ell }$ denote Coxeter length. The pairs $(x,I)$, with $x\in W$ and $I\subseteq S$, satisfying $W_{J}x=x{W}_I$ and $\mathrm{\ell }(x)=\min \{\mathrm{\ell }(y):y\in x{W}_I\}$ label the $J$-chambers bijectively by $(x,I)\mapsto x{C}_I$. These are exactly the chambers of $(T^{\circ },\mathcal{A})$, where $\mathcal{A}=\{E\cap H_{\alpha }:\alpha \in \mathrm{\Phi },E\subseteq ̸H_{\alpha }\}$, with traces missing $T^{\circ }$ omitted. No sphericity assumption on $J$ is imposed.

The full-cone and interior arrangements have identical chambers

Parabolic root subsystems and unique coset minima

Iyama–Wemyss, Definition 1.8 and Theorem 1.12, states that the pairs $(x,I)$ with $W_{J}x=x{W}_I$ and $\mathrm{\ell }(x)=\min \{\mathrm{\ell }(y):y\in x{W}_I\}$ are in bijection with their $J$-chambers, under $(x,I)\mapsto x{C}_I$.

The external statement is used only as explicitly recorded above. It is an independent prerequisite source, not the article under study. The local proof composes that imported bijection with the chamber-topology supplier; it does not reconstruct the source theorem. Root hyperplanes and geometric formulation conventions are supplied by the chamber-topology lemma, whose Statement defines $H_{\alpha }=\{f\in \mathrm{\Theta }:f(\alpha )=0\}$. The source proof has not been reviewed.