Reflecting hyperplanes, fundamental chamber and Tits cone

PDF p. 4: §2.2 before Notation 2.1; PDF p. 5: §2.2 before Notation 2.1

For the geometry in this item, let the vertex set ${\mathrm{\Gamma }}_0$ be finite. This is the finite-rank Coxeter-graph convention of Iyama–Wemyss, §§1.1–1.3; the Coxeter group $W$ may be infinite. Use the geometric representation, roots and algebraic dual $\mathrm{\Theta }=V^{\ast }$ of Coxeter-system, roots, length and weak-order conventions, specialized to this finite vertex set, and the finite-rank realization of Real Coxeter representation and contragredient notation.

The coordinate map $\mathrm{\Theta }⟶{\mathbb{R}}^{{\mathrm{\Gamma }}_0}$, $f⟼(f({\alpha }_i){)}_{i\in {\mathrm{\Gamma }}_0}$, is a linear isomorphism. Give $\mathrm{\Theta }$ its transported Euclidean topology, as in Locally finite arrangements and relative face terminology. For a subset $A\subseteq \mathrm{\Theta }$, its closure $\bar{A}$ consists of points every open neighbourhood of which meets $A$; its interior $A^{\circ }$ is the union of open subsets contained in $A$.

For each root $\alpha \in \mathrm{\Phi }$ define the reflecting hyperplane and the set of distinct such hyperplanes by $H_{\alpha }=\{f\in \mathrm{\Theta }:f(\alpha )=0\},\mathcal{H}=\{H_{\alpha }:\alpha \in \mathrm{\Phi }\}\mathrm{.}$ Define the open fundamental chamber and its closure by $C=\{f\in \mathrm{\Theta }:f({\alpha }_i)>0\text{ for every }i\in {\mathrm{\Gamma }}_0\},\bar{C}=\{f\in \mathrm{\Theta }:f({\alpha }_i)\ge 0\text{ for every }i\in {\mathrm{\Gamma }}_0\}\mathrm{.}$ The Tits cone and its interior in $\mathrm{\Theta }$ are $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}={\bigcup }_{w\in W}w\bar{C},{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ }={\mathrm{int}}_{\mathrm{\Theta }}(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e})\mathrm{.}$ The notation $w\bar{C}$ uses the contragredient action: $wf=f\circ \rho (w{)}^{-1}$. The family $\mathcal{H}$ is a set, so coincident hyperplanes are included only once.

The finite simple-root basis identifies each functional with its freely prescribed coordinates, giving the asserted coordinate isomorphism. The transported Euclidean topology and its neighbourhood and closure conventions are those of Locally finite arrangements and relative face terminology. Each coordinate evaluation is continuous, so the finite intersection defining $C$ is open. Its closure is contained in the nonnegative coordinate orthant. Conversely, for any functional in that orthant, adding $\epsilon >0$ to every coordinate gives a point of $C$ converging to it as $\epsilon \to 0$. This proves the stated formula for $\bar{C}$, including the empty-vertex case.

Every root is nonzero in the finite-rank representation of Real Coxeter representation and contragredient notation: by clause (3) of Descent of the reflection representation, unit root norms, and conjugation of reflections, its Coxeter-form norm is $B(\alpha ,\alpha )=1$. If a coordinate of $\alpha$ is nonzero, the corresponding coordinate functional evaluates nontrivially on $\alpha$. Thus $f\mapsto f(\alpha )$ is a nonzero linear functional on $\mathrm{\Theta }$, and its kernel is a linear hyperplane. The contragredient action is already well-defined by Coxeter-system, roots, length and weak-order conventions; its translates and their union therefore define subsets of $\mathrm{\Theta }$, whose interior is specified by the chosen topology. These definitions do not require $W$ to be finite. Convexity, fundamental-domain and local-finiteness assertions are separate results.

The article calls $C$ open in §2.2 and explicitly invokes a finite-dimensional vector space in the proof of Proposition 3.6. Its §2.2 follows Iyama–Wemyss for $\mathrm{\Theta }$, and §3 follows their construction. Iyama–Wemyss §1.1 explicitly starts with an $n\times n$ Coxeter matrix and the vertex set $\{1,\dots ,n\}$. The finite-rank geometric convention is made explicit here; it does not change the arbitrary-rank algebraic definitions in Coxeter-system, roots, length and weak-order conventions or assert an infinite-rank topological extension.