The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset

Let $(W,S)$ be a Coxeter system of finite type with $S$ finite, $n:=\mid S\mid$, and length function $\mathrm{\ell }$; thus $W$ is a finite group (Coxeter diagrams: edges, labels, components and finite type, clause (4), Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). On $V:={\mathbb{R}}^S$ let $B$ be the Coxeter form, let $r_a$ be the reflection with normal $a$ (defined when $B(a,a)\mathrm{\ne }0$), and let $\rho :W\to \mathrm{G}\mathrm{L}(V)$ be the canonical reflection homomorphism with root system $\mathrm{\Phi }=\{\rho (w)e_s:w\in W,s\in S\}\subset V$ and reflection set $T=\{ws{w}^{-1}:w\in W,s\in S\}\subset W$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Let $V^{\ast }$ be the algebraic dual with its dual action, its closed chamber $C$, its interior $C^{\circ }$ and its root hyperplanes $H_{\alpha }$ (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling).

Since $W$ is finite, $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite, clause (1), Disconnected diagrams, direct products, and comparison of invariant forms, clause (4)); hence $b:V\to V^{\ast }$, $b(v):=B(v,\cdot )$, is a linear isomorphism with $b(\rho (w)v)=w\cdot b(v)$ for all $w\in W$ and $v\in V$, because $B$ is $\rho$-invariant (Descent of the reflection representation, unit root norms, and conjugation of reflections, Invertible linear maps, linear isomorphisms, and inverse linear maps). Identify $V$ with $V^{\ast }$ by $b$ only now, and transfer $C$, $C^{\circ }$ and the $H_{\alpha }$ along $b^{-1}$; with the same letters

$C=\{v\in V:B(v,e_s)\ge 0\text{ for every }s\in S\},C^{\circ }=\{v\in V:B(v,e_s)>0\text{ for every }s\in S\},H_{\alpha }=\{v\in V:B(v,\alpha )=0\}\mathrm{.}$

Give $V$ the metric topology of the inner product $B$ (Real and complex inner-product spaces and their induced length, Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement); it is canonical because $B$ is determined by $(W,S)$, and $b$ is an isometry for $B$ and its transferred form, so topological notions may be moved across the identification.

(1) The finite reflection arrangement. Put $\mathcal{A}:=\{H_{\alpha }:\alpha \in \mathrm{\Phi }\}\mathrm{.}$ Each $\alpha \in \mathrm{\Phi }$ satisfies $B(\alpha ,\alpha )=1\mathrm{\ne }0$ (Descent of the reflection representation, unit root norms, and conjugation of reflections, clause (3)), so $H_{\alpha }$ is the kernel of the nonzero linear functional $B(\cdot ,\alpha )$ and in particular contains $0$; $\mathcal{A}$ is finite because $\mathrm{\Phi }$ is the image of the finite set $W\times S$ under $(w,s)\mapsto \rho (w)e_s$; and $\mathcal{A}$ is permuted by $\rho (W)$, since $\rho (w)H_{\alpha }=H_{\rho (w)\alpha }$ for all $w\in W$, $\alpha \in \mathrm{\Phi }$ (the form $B$ is $\rho$-invariant). A chamber of $\mathcal{A}$ is a connected component of $V\setminus {\bigcup }_{\alpha \in \mathrm{\Phi }}{H}_{\alpha }$, and a closed chamber is the closure of a chamber (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, Connected components, quasicomponents, and totally disconnected spaces, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). For $I\subseteq S$ put $\overline{C_I}:=\{v\in C:B(v,e_s)=0\text{ for }s\in I\},C_I:=\{v\in C:B(v,e_s)=0\text{ for }s\in I\text{ and }B(v,e_s)>0\text{ for }s\mathrm{\notin }I\},$ the closed face and the open face of $C$ of type $I$; for $w\in W$ put $wC:=\rho (w)C$, $w\overline{C_I}:=\rho (w)\overline{C_I}$ and $w{C}_I:=\rho (w)C_I$.

(2) The spherical chamber complex. The form $B$ is an inner product on $V$; let $\parallel v{\parallel }_B:=B(v,v{)}^{1\mathrm{/}2}$ and let $S^{n-1}:=\{v\in V:\parallel v{\parallel }_B=1\}$ be the unit sphere, with the subspace topology (The induced length is a norm, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$); the exponent is notation for the unit sphere of the $n$-dimensional inner-product space $(V,B)$, and no dimension theory of spheres is claimed here. The spherical chamber belonging to $w\in W$ is $wC\cap S^{n-1}$, and its spherical faces are the sets $w\overline{C_I}\cap S^{n-1}$.

(3) The coset face poset. For $I\subseteq S$ put $W_I:=⟨s:s\in I⟩\le W$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups) and let $w{W}_I:=\{wu:u\in W_I\}$ be a left coset (Left and right cosets $gH$ and $Hg$ of a subgroup). The coset face poset is the set $\{w{W}_I:w\in W,I⊊S\}$ of cosets of proper standard parabolics, ordered by reverse inclusion as subsets of $W$: $\sigma ⪯\tau$ if and only if $\sigma \supseteq \tau$.

(4) Abstentions. This definition asserts neither that the sets $w{C}^{\circ }$ are the connected components of $V\setminus {\bigcup }_{\alpha }{H}_{\alpha }$ nor that they cover it, neither that the cosets $w{W}_I$ index the faces $w\overline{C_I}$ nor that the spherical faces triangulate $S^{n-1}$; those assertions are proved in The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere ↗, the recorded justifier of this definition. Separately, the $B$-dual family of the basis $(e_s)$ is provided by The dual family $(b^{\ast }{)}_{b\in B}$ associated to a Hamel basis $B$, defined by $b^{\ast }(c)={\delta }_{bc}$, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis and The dual family of a finite basis is a basis of the dual space, with the same dimension, and no Choice is used: $S$ and $W$ are finite and all objects here are finite-dimensional or set-theoretic.