The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere

If $S=\mathrm{\varnothing }$ then $W=\{1\}$, $V=V^{\ast }=\{0\}$, $\mathrm{\Phi }=\mathrm{\varnothing }$, $C=C^{\circ }=\{0\}=\overline{C_{\mathrm{\varnothing }}}=C_S$ and the coset face poset is empty, so (1)-(3) hold in the vacuous form described in the statement and (4) is exactly its stated convention for $S=\mathrm{\varnothing }$; assume $S\mathrm{\ne }\mathrm{\varnothing }$ from now on, so that $n\ge 1$.

By [F1] the root system $\mathrm{\Phi }$ is finite, so $\mathrm{Neg}(f)\subseteq {\mathrm{\Phi }}_{+}$ is finite for every $f\in V^{\ast }$; the criterion [F6] therefore gives $V^{\ast }\subseteq U$, and $U\subseteq V^{\ast }$ holds by definition, so $U=V^{\ast }$ and every $f\in V^{\ast }$ lies in some chamber $wC$.

By [F8] the $B$-dual basis $(v_s{)}_{s\in S}$ exists, with $B(v_s,e_t)={\delta }_{st}$ and $B(x,e_t)={\mu }_t$ for $x={\sum }_t{\mu }_t{v}_t$; hence $C=\{{\sum }_s{\lambda }_s{v}_s:{\lambda }_s\ge 0\}$, $\overline{C_I}=\{{\sum }_{s\mathrm{\notin }I}{\lambda }_s{v}_s:{\lambda }_s\ge 0\}$ and $C_I=\{{\sum }_{s\mathrm{\notin }I}{\lambda }_s{v}_s:{\lambda }_s>0\}$ for every $I\subseteq S$, so $C_S=\{0\}$ and $C_I\mathrm{\ne }\mathrm{\varnothing }$ for $I⊊S$; and the cone $C$ has extreme rays ${\mathbb{R}}_{\ge 0}{v}_s$, because $v_s=x+y$ with $x=\sum {\mu }_t{v}_t$, $y=\sum {\nu }_t{v}_t$ in $C$ forces ${\mu }_t+{\nu }_t=0$ for $t\mathrm{\ne }s$ and ${\mu }_s+{\nu }_s=1$ with $\mu ,\nu \ge 0$, hence $x,y\in {\mathbb{R}}_{\ge 0}{v}_s$; any vector with at least two positive coefficients splits into two nonproportional vectors of $C$, so it spans no extreme ray.

For $g\in C^{\circ }$ and $\alpha \in \mathrm{\Phi }$ one has $\alpha \in {\mathrm{\Phi }}_{+}$ or $\alpha \in {\mathrm{\Phi }}_{-}$, and $g(\alpha )\mathrm{\ne }0$ by [F2]; for every $w\in W$ the point $\rho (w)g$ lies in $w{C}^{\circ }$ and $B(\rho (w)g,\alpha )=B(g,\rho (w{)}^{-1}\alpha )\mathrm{\ne }0$ because $\mathrm{\Phi }$ is stable under $\rho$, so $\rho (w)g\mathrm{\notin }{H}_{\alpha }$ and $w{C}^{\circ }\cap H_{\alpha }=\mathrm{\varnothing }$.

$C^{\circ }={\bigcap }_{s\in S}\{f:f(e_s)>0\}$ is open, as a finite intersection of preimages of the open half-line under the coordinate functionals $f\mapsto f(e_s)$, which are continuous for the topology of [F5]; it is convex, because those functionals are linear; and in the coordinates $V^{\ast }\cong {\mathbb{R}}^S$ it is a nonempty convex subset, hence path-connected and connected by [F10]. Each $w{C}^{\circ }=\rho (w)C^{\circ }$ is the image of $C^{\circ }$ under the linear isomorphism $\rho (w)$ [F1], whose coordinate matrix and inverse give Lipschitz maps for the metric of [F5], so $w{C}^{\circ }$ too is open and connected.

From step 1.3: $\overline{C_I}\cap \overline{C_J}=\overline{C_{I\cup J}}$, and $\overline{C_I}\subseteq \overline{C_J}$ if and only if $J\subseteq I$; hence $\overline{C_I}=\overline{C_J}$ if and only if $I=J$. Moreover $C_I$ is the relative interior of $\overline{C_I}$ in its affine span, which is $\mathrm{span}\{v_s:s\mathrm{\notin }I\}$, because by step 1.3 the relative interior consists exactly of the combinations with all coefficients positive.

For every $I\subseteq S$ each $u\in W_I$ fixes $\overline{C_I}$ pointwise and fixes each $v_s$ with $s\mathrm{\notin }I$: by [F7] $u$ is a product of elements of $I$, so it suffices to check the generators; for $s\in I$ and $x\in \overline{C_I}\subseteq C$ one has $B(x,e_s)=0$, hence $r_s(x)=x-2B(x,e_s)e_s=x$ by the reflection formula, while for $t\in I$ and $s\mathrm{\notin }I$ one has $t\mathrm{\ne }s$ and $B(v_s,e_t)={\delta }_{st}=0$, hence $r_t(v_s)=v_s$.

Every $W$-orbit in $V$ meets $C$ in exactly one point, by the strict-fundamental-domain clause [F4] together with $U=V^{\ast }$ (step 1.2); and the map $w\mapsto w{C}^{\circ }$ is injective: if $w{C}^{\circ }=v{C}^{\circ }$ and $g\in C^{\circ }$ (nonempty by step 1.3), then $g=u\cdot h$ for some $h\in C^{\circ }$, where $u:=w^{-1}v$, and the collision rule [F4] applied to $g\in C$ and $h\in C$ gives $h=g$ and $u\in W_{S(g)}=W_{\mathrm{\varnothing }}=\{1\}$, so $w=v$.

Let $s\mathrm{\ne }t$, put $\theta :=\pi \mathrm{/}m(s,t)$, $c:=\cos \theta$, and $P:=\mathbb{R}{e}_s+\mathbb{R}{e}_t$. The order convention in Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups and finiteness of $W$ give $m(s,t)<\mathrm{\infty }$, so $0<\theta \le \pi \mathrm{/}2$. In the $B$-orthonormal coordinates $a=e_s$, $b=(e_t+c{e}_s)\mathrm{/}\sin \theta$ of $P$, one has $e_t=-ca+\sin \theta b$, using the Gram entries and [F13]. Orthogonal projection onto $P$ preserves the two values $B(v,e_s)$ and $B(v,e_t)$, because its kernel is $P^{\perp }=H_{e_s}\cap H_{e_t}$; every point of the resulting sector in $P$ is its own projection. Write a point of $P$ as $xa+yb$. The sector is $x\ge 0$, $-cx+\sin \theta y\ge 0$, bounded by the rays through $b$ and $\sin \theta a+cb$. These are unit vectors with inner product $c$, and the nonnegative sector between them has principal angle $\arccos c=\theta$ by [F13]. This proves the stated dihedral-angle formula.

$V\setminus {\bigcup }_{\alpha \in \mathrm{\Phi }}{H}_{\alpha }={⨆}_{w\in W}w{C}^{\circ }$, these sets are exactly the connected components of $V\setminus {\bigcup }_{\alpha }{H}_{\alpha }$, and $\overline{w{C}^{\circ }}=wC$: the inclusion $\supseteq$ holds by step 1.4; for $\subseteq$ take $f\mathrm{\notin }{\bigcup }_{\alpha }{H}_{\alpha }$, so $f\in wC$ for some $w$ by step 1.2, and $(w^{-1}\cdot f)(e_s)=f(\rho (w)e_s)\mathrm{\ne }0$ for every $s$ because $\rho (w)e_s\in \mathrm{\Phi }$ [F2], whence $w^{-1}\cdot f\in C^{\circ }$ and $f\in w{C}^{\circ }$; the sets $w{C}^{\circ }$ are pairwise disjoint by [F4], open and connected by step 2.1 and nonempty by step 1.3, so each is a component, since a connected set meeting two of them would be separated by the partition into $w{C}^{\circ }$ and its complement, both of which are open in $V$ minus the hyperplanes; finally each $wC$ is closed, being the intersection of the finitely many closed half-spaces $\{v:B(v,\rho (w)e_s)\ge 0\}$, and every $x\in wC$ is a limit of points $\rho (w)(w^{-1}x+\epsilon u)\in w{C}^{\circ }$ with $u={\sum }_s{v}_s$ and $\epsilon >0$, because $B(w^{-1}x+\epsilon u,e_s)=B(w^{-1}x,e_s)+\epsilon >0$ and $B(u,e_s)=1$; hence $\overline{w{C}^{\circ }}=wC$ and the closed chambers are exactly the sets $wC$.

If $w{W}_I=v{W}_J$ with $I,J\subseteq S$, then $u:=w^{-1}v$ satisfies $u{W}_J=W_I$, so $1=uj$ for some $j\in W_J$ (because $1\in W_I$), giving $u=j^{-1}\in W_J$ and $W_I=u{W}_J=W_J$, so $I=W_I\cap S=W_J\cap S=J$ by [F7]; then $u\in W_I$ fixes $\overline{C_I}$ pointwise by step 2.3, so $w\overline{C_I}=v{u}^{-1}\overline{C_I}=v\overline{C_I}=v\overline{C_J}$, and the assignment $w{W}_I\mapsto w\overline{C_I}$ is well defined.

For all $w,v\in W$ and $I,J\subseteq S$ the intersection formula $w\overline{C_I}\cap v\overline{C_J}=w\overline{C_{I\cup J\cup S(v^{-1}w)}}$ holds: if $x$ lies in the intersection, then $f:=w^{-1}x\in \overline{C_I}$ and $g:=v^{-1}x\in \overline{C_J}$ are points of $C$ with $(v^{-1}w)\cdot f=g$, so [F4] gives $f=g$ and $v^{-1}w\in W_{S(f)}$; then $I\subseteq S(f)$ and $J\subseteq S(f)$ by [F3] and $S(v^{-1}w)\subseteq S(f)$ by [F7], so $f\in \overline{C_{I\cup J\cup S(v^{-1}w)}}$ and $x\in w\overline{C_{I\cup J\cup S(v^{-1}w)}}$; conversely if $x=wf$ with $f\in \overline{C_{I\cup J\cup S(v^{-1}w)}}$, then $S(v^{-1}w)\subseteq S(f)$ and hence $v^{-1}w\in W_{S(f)}$ by [F7], so $v^{-1}w$ fixes $\overline{C_{S(f)}}$ pointwise by step 2.3, whence $v^{-1}x=v^{-1}wf=f\in \overline{C_J}$ and $x\in w\overline{C_I}\cap v\overline{C_J}$.

The faces $w{C}_I$ with $I⊊S$ are nonempty by step 1.3, cover $V\setminus \{0\}$ and have pairwise disjoint relative interiors, so $V\setminus \{0\}=⨆w{C}_I$ over the cosets $w{W}_I$ with $I⊊S$; moreover $w{C}_I\cap v{C}_J\mathrm{\ne }\mathrm{\varnothing }$ forces $w{W}_I=v{W}_J$ and hence $w{C}_I=v{C}_J$: if $x$ lies in $w{C}_I\cap v{C}_J$, then $f:=w^{-1}x\in C_I$ and $g:=v^{-1}x\in C_J$ are points of $C$ with $(v^{-1}w)\cdot f=g$, so [F4] gives $f=g$ and $v^{-1}w\in W_{S(f)}=W_I$ by step 1.3, and then $v=w{u}^{-1}$ with $u:=v^{-1}w\in W_I$ and $v{C}_I=w{u}^{-1}{C}_I=w{C}_I$ by step 2.3; for the covering, let $x\mathrm{\ne }0$ and choose $w$ with $x\in wC$ by step 1.2, so that $w^{-1}x\mathrm{\ne }0$ and therefore $S(w^{-1}x)⊊S$ (else $w^{-1}x=0$ by step 1.3), giving $w^{-1}x\in C_{S(w^{-1}x)}$ by step 1.3 and $x\in w{C}_{S(w^{-1}x)}$.

For $f\in C$ one has ${\mathrm{Stab}}_W(f)=W_{S(f)}$: if $h\cdot f=f$, then the collision rule [F4] applied to the two points $f\in C$ gives $h\in W_{S(f)}$, and conversely every $u\in W_{S(f)}$ fixes $\overline{C_{S(f)}}\ni f$ pointwise by step 2.3; hence for $x\in w{C}_I$, writing $x=wf$ with $f\in C_I$, one has $S(f)=I$ by step 1.3 and ${\mathrm{Stab}}_W(x)=w{\mathrm{Stab}}_W(f)w^{-1}=w{W}_I{w}^{-1}$.

For every $s\in S$ one has $C\cap S^{n-1}\cap {\bigcap }_{t\mathrm{\ne }s}{H}_{e_t}=\{v_s\mathrm{/}\parallel v_s{\parallel }_B\}$: a point $x$ of that intersection equals ${\mu }_s{v}_s$ with ${\mu }_s\ge 0$ by step 1.3, and $\parallel x{\parallel }_B=1$ forces ${\mu }_s=1\mathrm{/}\parallel v_s{\parallel }_B\mathrm{\ne }0$, while each $v_s\mathrm{/}\parallel v_s{\parallel }_B$ does lie in the intersection; moreover $\rho (w)\overline{C_I}=\{{\sum }_{s\mathrm{\notin }I}{\lambda }_s\rho (w)v_s:{\lambda }_s\ge 0\}$ with the vectors $\rho (w)v_s$ $(s\mathrm{\notin }I)$ linearly independent because $\rho (w)$ is invertible [F1], and the walls of $wC$ are $w{H}_{e_s}=H_{\rho (w)e_s}$ by [F3].

For all $w,v\in W$ and $I,J\subseteq S$: (a) $w{W}_I=v{W}_J$ if and only if $w\overline{C_I}=v\overline{C_J}$, because the forward implication is step 3.2 and if $w\overline{C_I}=v\overline{C_J}$ then the intersection formula (step 3.3) gives $w\overline{C_{I\cup J\cup S(v^{-1}w)}}=w\overline{C_I}$, so $I=I\cup J\cup S(v^{-1}w)$ by step 2.2, whence $J\subseteq I$ and $v^{-1}w\in W_I$ by [F7], and symmetrically $I\subseteq J$, so $I=J$ and $w{W}_I=v{W}_I=v{W}_J$; (b) $w{W}_I\subseteq v{W}_J$ is equivalent to $v^{-1}w\in W_J$ and $I\subseteq J$: containment gives $w\in v{W}_J$ and, after multiplying by $w^{-1}$, $W_I\subseteq W_J$, whence $I\subseteq J$ by [F7]; conversely these conditions give containment. By step 3.3, $v\overline{C_J}\subseteq w\overline{C_I}$ is equivalent to $v\overline{C_{J\cup I\cup S(w^{-1}v)}}=v\overline{C_J}$, hence to $I\subseteq J$ and $S(w^{-1}v)\subseteq J$ by step 2.2, which is the same pair of conditions by [F7] and closure of $W_J$ under inverses; (c) $w{C}_I\cap v{C}_J\mathrm{\ne }\mathrm{\varnothing }$ implies $w{W}_I=v{W}_J$ by step 3.4, and conversely $w{W}_I=v{W}_J$ gives $I=J$ and $v^{-1}w\in W_I$ by (a), hence $v{C}_I=w{C}_I$ by step 2.3 and this face is nonempty by step 1.3. Consequently $w{W}_I\mapsto w\overline{C_I}$ is a bijection from the coset face poset onto the set of proper faces, and by (b) it reverses inclusions, so it is an isomorphism from the coset face poset ordered by reverse inclusion onto the face poset ordered by containment.

Define $\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}:=\{w{W}_{S\setminus \{s\}}:w\in W,s\in S\}$ and $\sigma (w{W}_I):=\{w{W}_{S\setminus \{s\}}:s\mathrm{\notin }I\}\subseteq \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}$ for $w\in W$, $I⊊S$, and put $\widehat{u}(w{W}_{S\setminus \{s\}}):=\rho (w)v_s\mathrm{/}\parallel \rho (w)v_s{\parallel }_B$. The direction is well defined: if $w{W}_{S\setminus \{s\}}=w^{\mathrm{\prime }}{W}_{S\setminus \{s^{\mathrm{\prime }}\}}$, then step 3.2 gives $S\setminus \{s\}=S\setminus \{s^{\mathrm{\prime }}\}$ and $w\overline{C_{S\setminus \{s\}}}=w^{\mathrm{\prime }}\overline{C_{S\setminus \{s\}}}$, so $s=s^{\mathrm{\prime }}$ and, by step 3.6 applied to these one-dimensional cones, $\rho (w)v_s$ and $\rho (w^{\mathrm{\prime }})v_s$ are positive multiples of one another, whence $\parallel \rho (w)v_s{\parallel }_B=\parallel v_s{\parallel }_B$ because $B$ is $\rho$-invariant [F1] and the unit directions agree. The simplex is well defined: for $s\mathrm{\notin }I$ and $u\in W_I$ one has $W_I\subseteq W_{S\setminus \{s\}}$, hence $wu{W}_{S\setminus \{s\}}=w{W}_{S\setminus \{s\}}$ and $\sigma (wu{W}_I)=\sigma (w{W}_I)$. For $I⊊S$, the intersection of its vertex cosets is ${\bigcap }_{s\mathrm{\notin }I}w{W}_{S\setminus \{s\}}=w{W}_I$, since [F7] identifies the intersection of these subgroups with the support condition $S(u)\subseteq I$. Thus the vertex set determines the original coset. Finally a subset of $\sigma (w{W}_I)$ keeping precisely the indices $s\in T\subseteq S\setminus I$ is $\sigma (w{W}_{S\setminus T})$, so $K:=\{\mathrm{\varnothing }\}\cup \{\sigma (w{W}_I):w\in W,I⊊S\}$ is closed under subsets, contains the empty set and every singleton, and is finite.

For $x\in w{C}_I$ the setwise stabiliser of the face $w\overline{C_I}$ is $w{W}_I{w}^{-1}$: one has $w^{\mathrm{\prime }}w\overline{C_I}=w\overline{C_I}$ if and only if $w^{\mathrm{\prime }}w{W}_I=w{W}_I$ by step 4.1(a), that is, if and only if $w^{\mathrm{\prime }}\in w{W}_I{w}^{-1}$; this subgroup contains ${\mathrm{Stab}}_W(x)=w{W}_I{w}^{-1}$ from step 3.5.

Define $\phi (\alpha ):=({\sum }_{v\in \mathrm{supp}\alpha }\alpha (v)\widehat{u}(v))\mathrm{/}\parallel {\sum }_{v\in \mathrm{supp}\alpha }\alpha (v)\widehat{u}(v){\parallel }_B$ for $\alpha \in \mathrm{\mid }K\mathrm{\mid }$. This is well defined and takes values in $S^{n-1}$: on a simplex $\sigma (w{W}_I)$ containing $\mathrm{supp}\alpha$ the sum is ${\sum }_{s\mathrm{\notin }I}\alpha (w{W}_{S\setminus \{s\}})\rho (w)v_s\mathrm{/}\parallel v_s{\parallel }_B$, a combination with nonnegative coefficients, not all zero, of the linearly independent vectors $\rho (w)v_s$ of step 3.6, so the numerator does not vanish; and if $\alpha$ lies in two simplices, both contain the minimal simplex $\mathrm{supp}\alpha$, on which the formula is the same. On each closed simplex $\mathrm{\mid }\sigma (w{W}_I)\mathrm{\mid }$, which is compact and carries the Euclidean simplex topology of [F9], the barycentric coordinates $\alpha \mapsto \alpha (w{W}_{S\setminus \{s\}})$ are continuous, so the numerator is continuous as a map into $V$ (finite sums of scalar multiples of the fixed vectors, read in the coordinates of [F5]) and the denominator is a continuous positive real function by [F12], so $\phi {\mathrm{\mid }}_{\mathrm{\mid }\sigma (w{W}_I)\mathrm{\mid }}$ is continuous by [F12]; every simplex of $K$ is a face of one of the finitely many maximal simplices, whose traces are therefore continuous, and $\mathrm{\mid }K\mathrm{\mid }$ carries the weak topology of [F9], so $\phi$ is continuous.

The map $\phi$ is surjective: for $x\in S^{n-1}$ step 3.4 gives a coset $w{W}_I$ with $x\in w{C}_I$, so $x={\sum }_{s\mathrm{\notin }I}{\lambda }_s\rho (w)v_s$ with all ${\lambda }_s>0$ by step 1.3; putting $\mathrm{\Lambda }:={\sum }_{s\mathrm{\notin }I}{\lambda }_s\parallel v_s{\parallel }_B$ and $\alpha (w{W}_{S\setminus \{s\}}):={\lambda }_s\parallel v_s{\parallel }_B\mathrm{/}\mathrm{\Lambda }$ on the vertices of $\sigma (w{W}_I)$ and $\alpha :=0$ elsewhere defines a point of $\mathrm{\mid }\sigma (w{W}_I)\mathrm{\mid }\subseteq \mathrm{\mid }K\mathrm{\mid }$ whose numerator is $w({\sum }_{s\mathrm{\notin }I}({\lambda }_s\mathrm{/}\mathrm{\Lambda })v_s)$, so that $\phi (\alpha )=x\mathrm{/}\parallel x{\parallel }_B=x$. It is injective: if $\phi (\alpha )=x$ and $\sigma (w{W}_I)$ is the minimal simplex supporting $\alpha$, so that all $\alpha (w{W}_{S\setminus \{s\}})>0$ for $s\mathrm{\notin }I$, then the defining identity expresses $w^{-1}x$ as the positive multiple $y\mathrm{/}\parallel y{\parallel }_B$ of $y:={\sum }_{s\mathrm{\notin }I}(\alpha (w{W}_{S\setminus \{s\}})\mathrm{/}\parallel v_s{\parallel }_B)v_s$; since $w^{-1}x\in C_I$ is itself the combination ${\sum }_{s\mathrm{\notin }I}{\lambda }_s{v}_s$ with ${\lambda }_s>0$ (step 1.3) and $(v_s)$ is a basis, the coefficients satisfy $\alpha (w{W}_{S\setminus \{s\}})=\parallel y{\parallel }_B{\lambda }_s\parallel v_s{\parallel }_B$, and the coset $w{W}_I$ together with the numbers ${\lambda }_s$ is determined by $x$ (steps 3.4 and 1.3), summing the coefficients to $1$ gives $\parallel y{\parallel }_B=1\mathrm{/}({\sum }_{s\mathrm{\notin }I}{\lambda }_s\parallel v_s{\parallel }_B)$, so $\alpha$ is determined by $x$.

The realization $\mathrm{\mid }K\mathrm{\mid }$ is compact by [F9] and finiteness of $K$ (step 4.2), the sphere $S^{n-1}$ is Hausdorff, since distinct points are separated by the intersections with $S^{n-1}$ of disjoint metric balls in $V$ [F9], and $\phi$ is a continuous bijection by steps 5.2 and 5.3; hence $\phi$ is a homeomorphism by [F9] and $K$ is a triangulation of $S^{n-1}$, with the members of $\mathrm{\Sigma }$, the relative interiors and the face poset as described in steps 3.6, 3.4 and 4.1. The maximal simplices of $K$ are exactly the $\sigma (w{W}_{\mathrm{\varnothing }})=\{w{W}_{S\setminus \{s\}}:s\in S\}$, one for each $w\in W$: indeed $\sigma (w{W}_I)\subseteq \sigma (v{W}_J)$ holds if and only if $v{W}_J\subseteq w{W}_I$ (both sides are equivalent to the pair of conditions $J\subseteq I$ and $v^{-1}w\in W_I$, by the argument of step 4.1(b) applied to vertex sets), so a simplex is maximal exactly when $I=\mathrm{\varnothing }$, and $\sigma (w{W}_{\mathrm{\varnothing }})=\sigma (v{W}_{\mathrm{\varnothing }})$ forces $w=v$. Thus $K$ has exactly $\mathrm{\mid }W\mathrm{\mid }$ maximal simplices, indexed by the chambers $wC$, and the case $S=\mathrm{\varnothing }$ was disposed of in step 1.1.