The finite-type Coxeter cell: exposed faces and normal cones

Since $C=\{y:B(y,e_s)\ge 0\text{ for all }s\}$ and $x\in C^{\circ }$, each $d_s=B(x,e_s)$ is positive; and by [F4] every $y\in V$ satisfies $y={\sum }_{s\in S}B(y,e_s)v_s$, so for $y\in C$ all coefficients $B(y,e_s)$ are $\ge 0$. Hence for $y\in C$ and any $z={\sum }_{s}z(s)e_s\in V_{+}$ with $z(s)\ge 0$, linearity gives $B(y,z)={\sum }_{s}B(y,e_s)z(s)\ge 0$.

We prove (1) by induction on $k=\mathrm{\ell }(v)$. For $k=0$ both sides vanish. For $k\ge 1$ write $v=v^{\mathrm{\prime }}{s}_k$ with $v^{\mathrm{\prime }}=s_1\cdots s_{k-1}$ reduced. Then $x-\rho (v)x=(x-\rho (v^{\mathrm{\prime }})x)+\rho (v^{\mathrm{\prime }})(x-\rho (s_k)x)$, the reflection formula [F1] gives $x-\rho (s_k)x=2B(x,e_{s_k})e_{s_k}=2d_{s_k}{e}_{s_k}$, and substituting the induction hypothesis for $v^{\mathrm{\prime }}$ and applying $\rho (v^{\mathrm{\prime }})$ produces the displayed sum. Each $\rho (s_1\cdots s_{i-1})e_{s_i}$ lies in ${\mathrm{\Phi }}_{+}$ by the root-length criterion, since $\mathrm{\ell }(s_1\cdots s_{i-1}{s}_i)=i>\mathrm{\ell }(s_1\cdots s_{i-1})$, and these are precisely the elements of $N(v^{-1})$ by [F3].

We record the convex-combination principle: if $\mathrm{\ell }$ is linear and $S_0\subseteq Wx$ is the set on which $\mathrm{\ell }$ attains its maximum over $Wx$, then the maximizer set of $\mathrm{\ell }$ on $P=\mathrm{conv}(Wx)$ is $\mathrm{conv}(S_0)$. Indeed every $p\in P$ is a convex combination $p={\sum }_i{\lambda }_i\rho (v_i)x$ with ${\sum }_i{\lambda }_i=1$, ${\lambda }_i\ge 0$; then $\mathrm{\ell }(p)={\sum }_i{\lambda }_i\mathrm{\ell }(\rho (v_i)x)\le {\max }_{Wx}\mathrm{\ell }$, and equality holds exactly when every $\rho (v_i)x$ with ${\lambda }_i>0$ lies in $S_0$; the set of such combinations is $\mathrm{conv}(S_0)$, which is nonempty.

If $v\mathrm{\ne }1$ then $k\ge 1$, so the expansion of [step 1.2] is a sum with all coefficients $2d_{s_i}$ positive of positive roots, which lie in $V_{+}\setminus \{0\}$ by [F2]; hence $x-\rho (v)x\in V_{+}\setminus \{0\}$. For $y\in C$ this gives $B(y,x-\rho (v)x)\ge 0$ by [step 1.1], that is $B(y,\rho (v)x)\le B(y,x)$: the maximum in (2) equals $B(y,x)$.

Equality case of (2). Let $y\in C$ and suppose $B(y,\rho (v)x)=B(y,x)$, i.e. $B(y,x-\rho (v)x)=0$. By [step 1.2] write $x-\rho (v)x={\sum }_{i=1}^{k}2d_{s_i}{\beta }_i$ with ${\beta }_i\in {\mathrm{\Phi }}_{+}\subseteq V_{+}$; by [step 1.1] every $B(y,{\beta }_i)\ge 0$, so $0={\sum }_{i}2d_{s_i}B(y,{\beta }_i)$ with positive coefficients forces $B(y,{\beta }_i)=0$ for all $i$. We prove $v\in W_{S(y)}$ by induction on $k$: for $k=0$ this is trivial; for $k\ge 1$, $i=1$ gives $B(y,e_{s_1})=0$, so $s_1\in S(y)$ and $\rho (s_1)y=y$ by [F1]. With $v^{\mathrm{\prime }}=s_2\cdots s_k$ one has $\rho (v)x=\rho (s_1)\rho (v^{\mathrm{\prime }})x$, hence $B(y,\rho (v^{\mathrm{\prime }})x)=B(\rho (s_1{)}^{-1}y,\rho (v^{\mathrm{\prime }})x)=B(y,\rho (v)x)=B(y,x)$, and $\mathrm{\ell }(v^{\mathrm{\prime }})=k-1$, so the induction hypothesis gives $v^{\mathrm{\prime }}\in W_{S(y)}$; with $s_1\in W_{S(y)}$ this gives $v\in W_{S(y)}$. Conversely if $v\in W_{S(y)}$ then every generator $s\in S(y)$ fixes $y$ by [F1], so $\rho (v)y=y$ and $B(y,\rho (v)x)=B(\rho (v{)}^{-1}y,x)=B(y,x)$. Hence the maximizer set in (2) is exactly $W_{S(y)}$.

For arbitrary $y=\rho (w)y_0$ with $y_0\in C$ [F4], the chamber representative $y_0$ is unique. If also $y=\rho (w^{\mathrm{\prime }})y_0$, then $w^{-1}{w}^{\mathrm{\prime }}\in {\mathrm{Stab}}_W(y_0)=W_{S(y_0)}$ by [F4], and conversely every such $w^{\mathrm{\prime }}$ gives the same $y$; thus the coset $w{W}_{S(y_0)}$ is independent of the representative. For any $v\in W$, put $u:=w^{-1}v$, so that $v=wu$; then $B(y,\rho (v)x)=B(\rho (w)y_0,\rho (w)\rho (u)x)=B(y_0,\rho (u)x)\le B(y_0,x)=B(y,\rho (w)x)$ by [step 2.1], with equality if and only if $u\in W_{S(y_0)}$ by [step 2.2], i.e. $v\in w{W}_{S(y_0)}$. So the orbit points maximizing $B(y,\cdot )$ on $Wx$ are exactly $\rho (w)W_{S(y_0)}x$, and by the principle of [step 1.3] the maximizer set on $P$ is $\mathrm{conv}(\rho (w)W_{S(y_0)}x)=\rho (w)\mathrm{conv}(W_{S(y_0)}x)$, since $\rho (w)$ is linear.

Every point of $Wx$ is an extreme point of $P$. Indeed put $y_0:={\sum }_{s\in S}{v}_s$, an element of $C^{\circ }$ because $B(y_0,e_s)=1>0$ for every $s$ by [F4]; thus $S(y_0)=\mathrm{\varnothing }$ and $W_{S(y_0)}=\{1\}$. For $w\in W$ apply [step 3.1] to $y:=\rho (w)y_0$: the maximizer set of the functional $B(y,\cdot )$ on $P$ is $\mathrm{conv}(\rho (w)W_{\mathrm{\varnothing }}x)=\{\rho (w)x\}$, a singleton, so the minimizer set of the continuous affine functional $-B(y,\cdot )$ is that same singleton. By [F7] applied to $-B(y,\cdot )$ this singleton is a face of the compact convex set $P$, and by [F6] a point whose singleton is a face is an extreme point of $P$. Hence $\rho (w)x$ is extreme in $P$.

Every face of $P$ is of the form $\mathrm{conv}(w{W}_{U}x)$. Let $F$ be a face and $A:=(Wx)\cap F$. Since $F$ is nonempty, choose $p_0\in F$ and express it as a convex combination of orbit points; induction on the number of positive coefficients and the face condition show that every orbit point with positive coefficient lies in $F$, so $A\mathrm{\ne }\mathrm{\varnothing }$. Applying the same argument to each $p\in F$ gives $F=\mathrm{conv}(A)$. If $A=Wx$ then $F=P=\mathrm{conv}(Wx)=\mathrm{conv}(W_{S}x)$, the case $w=1$, $U=S$. Otherwise put $A_{\mathrm{o}\mathrm{u}\mathrm{t}}:=(Wx)\setminus A$, which is nonempty, and choose $a_0\in A$. The finite set $D_0:=\{a-a_0:a\in A\}$ spans the direction space $L:=\mathrm{span}\{q-q^{\mathrm{\prime }}:q,q^{\mathrm{\prime }}\in F\}$ because $F=\mathrm{conv}(A)$. By [F11], choose a maximal linearly independent subset $d_1,\dots ,d_k$ of $D_0$; it spans $L$, and for each $i$ write $d_i=a_i-a_0$ with $a_i\in A$. Put $p:=(a_0+a_1+\cdots +a_k)\mathrm{/}(k+1)\in F$. We claim $P\cap (p+L)\subseteq F$. For $q\in P\cap (p+L)$ write $q-p={\sum }_{i=1}^k{t}_i{d}_i$. If $k=0$, then $q=p\in F$. If $k>0$, choose $\epsilon >0$ small enough that all coefficients in $r:=p-\epsilon (q-p)=(\frac{1}{k+1}+\epsilon {\sum }_i{t}_i)a_0+{\sum }_{i=1}^k(\frac{1}{k+1}-\epsilon t_i)a_i$ are positive; these coefficients sum to $1$, so $r\in F$. The identity $p=\frac{\epsilon }{1+\epsilon }q+\frac{1}{1+\epsilon }r$ and the face condition imply $q\in F$, proving the claim. Now set $D:=\mathrm{conv}(A_{\mathrm{o}\mathrm{u}\mathrm{t}})$, a nonempty compact convex set by [F9]. Let $\pi :V\to L^{\perp }$ send a vector to its $L^{\perp }$ component in [F12]; it is linear and $\ker \pi =L$. If $\pi (p)\in \pi (D)$, there is $q\in D$ with $q-p\in L$, so $q\in P\cap (p+L)\subseteq F$; writing $q$ as a convex combination of points of $A_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and using the face condition would put a positive-support orbit point in $A_{\mathrm{o}\mathrm{u}\mathrm{t}}\cap F$, a contradiction. Thus $\pi (p)\mathrm{\notin }\pi (D)$. Linearity gives $\pi (D)=\mathrm{conv}(\pi (A_{\mathrm{o}\mathrm{u}\mathrm{t}}))$, so [F9] makes $\pi (D)$ nonempty, compact, and closed; it is convex as well. Since the point is outside this nonempty set, $L^{\perp }$ is nonzero, and [F12] supplies orthonormal coordinates in which to apply [F8] in $L^{\perp }$; pulling the Euclidean normal back to $L^{\perp }$ gives $u\in L^{\perp }$, $u\mathrm{\ne }0$, and $b\in \mathbb{R}$ such that $B(u,\pi (q))\le b<B(u,\pi (p))$ for every $q\in D$. As $u\perp L$, the functional $B(u,\cdot )$ is constant on $p+L$, so its value on every point of $A\subseteq F\subseteq p+L$ equals $B(u,p)$, while every point of $A_{\mathrm{o}\mathrm{u}\mathrm{t}}$ has strictly smaller value. Therefore its maximizers on $Wx$ are exactly $A$, and [step 1.3] shows its maximizer face on $P$ is $\mathrm{conv}(A)=F$. Write $u=\rho (w)y_0$ with $y_0\in C$ by [F4] and apply [step 3.1]; then $F=\mathrm{conv}(w{W}_{S(y_0)}x)$, with $U:=S(y_0)$.

Fix $U\subseteq S$ and $w\in W$ and put $y_0:={\sum }_{s\mathrm{\notin }U}{v}_s$, an element of $C$ because $B(y_0,e_s)=1$ for $s\mathrm{\notin }U$ and $B(y_0,e_s)=0$ for $s\in U$ by [F4], so that $S(y_0)=U$. By [step 3.1] applied to $y=\rho (w)y_0$, the set $\mathrm{conv}(w{W}_{U}x)$ is the maximizer set of $B(y,\cdot )$ on $P$, hence a face of $P$: it is the minimizer set of the continuous affine functional $-B(y,\cdot )$, so [F7] applied to the compact convex $P$ gives the face and singles it out. Its extreme points are exactly $w{W}_{U}x$. Each such orbit point is extreme in $P$ by [step 4.1], and remains extreme in the contained convex subset $\mathrm{conv}(w{W}_{U}x)$. Conversely, if $z$ is extreme in this finite hull, express $z$ as a convex combination of its finite generating set; if a positive-weight term differs from $z$, grouping that term against the remaining terms gives a strict convex decomposition of $z$ into two distinct points of the hull, contradicting extremality. Thus $z$ is one of the generators. Finally the map $v\mapsto \rho (v)x$ is injective on $W$: if $\rho (v)x=\rho (v^{\mathrm{\prime }})x$, then $\rho (v^{-1}{v}^{\mathrm{\prime }})x=x$, so $v^{-1}{v}^{\mathrm{\prime }}$ lies in ${\mathrm{Stab}}_W(x)=W_{S(x)}=W_{\mathrm{\varnothing }}=\{1\}$ by [F4], whence $v=v^{\mathrm{\prime }}$.

The assignment is injective on cosets: if $\mathrm{conv}(w{W}_{U}x)=\mathrm{conv}(w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}x)$, then by [step 5.1] the vertex sets coincide, $w{W}_{U}x=w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}x$, and injectivity of $v\mapsto \rho (v)x$ from [step 5.1] gives $w{W}_U=w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}$ as subsets of $W$. The inclusion equivalence of (4) now follows: if $w{W}_U\subseteq w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}$ then $\mathrm{conv}(w{W}_{U}x)\subseteq \mathrm{conv}(w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}x)$; conversely, if the hulls are nested, each generator in the smaller vertex set $w{W}_{U}x$ is extreme in $P$ by [step 4.1] and therefore remains extreme in the larger convex hull, so [step 5.1] puts it in the larger vertex set $w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}x$. Thus $w{W}_U\subseteq w^{\mathrm{\prime }}{W}_{U^{\mathrm{\prime }}}$. Combining with [step 5.1] and [step 4.2], the map $w{W}_U\mapsto \mathrm{conv}(w{W}_{U}x)$ is a bijection from the coset poset onto the set of faces carrying containment on cosets to containment of faces, hence an isomorphism from the poset ordered by reverse inclusion onto the face poset ordered by reverse inclusion.

Dimension and stabilizer. Put $V_U:=\mathrm{span}\{e_s:s\in U\}$. For each $t\in U$ and $s\in U$, the reflection formula gives $\rho (t)e_s=e_s-2B(e_s,e_t)e_t\in V_U$, so $\rho (t)$ preserves $V_U$ and hence so does every $\rho (u)$ with $u\in W_U$. Also $\rho (t)x-x=-2d_t{e}_t\in V_U$; induction on a word in the generators of $W_U$ then gives $\rho (u)x-x\in V_U$ for every $u\in W_U$. Thus $w{W}_{U}x\subseteq \rho (w)x+\rho (w)V_U$ and its affine dimension is at most $\mathrm{\mid }U\mathrm{\mid }$. For the reverse inequality, its affine span contains $\rho (w)x$ and all $\rho (w)\rho (s)x=\rho (ws)x$ for $s\in U$, whose differences $\rho (w)(\rho (s)x-x)=-2d_s\rho (w)e_s$ are linearly independent because $d_s>0$, the $e_s$ are a basis subset and $\rho (w)$ is invertible. Hence $\dim \mathrm{conv}(w{W}_{U}x)=\mathrm{\mid }U\mathrm{\mid }$. For the setwise stabilizer, $g\cdot \mathrm{conv}(w{W}_{U}x)=\mathrm{conv}(w{W}_{U}x)$ if and only if $gw{W}_{U}x=w{W}_{U}x$ by the vertex-set description [step 5.1], if and only if $gw{W}_U=w{W}_U$ by injectivity in [step 5.1]. By the coset inclusion criterion [F5], equality implies $w^{-1}{g}^{-1}w\in W_U$; since $W_U$ is a subgroup and closed under inverses [F13], this is equivalent to $w^{-1}gw\in W_U$, and the reverse implication gives the same coset equality. Thus the stabilizer is $w{W}_U{w}^{-1}$.

Polyhedral description and the origin. First, $P\subseteq Q:=\{v:B(w{v}_s,v)\le c_s\text{ for all }w\in W,s\in S\}$: if $p={\sum }_i{\lambda }_i\rho (v_i)x\in P$, then for all $w,s$, linearity and $B$-invariance give $B(w{v}_s,p)={\sum }_i{\lambda }_{i}B(v_s,\rho (w^{-1}{v}_i)x)\le {\sum }_i{\lambda }_{i}B(v_s,x)=c_s$ by [step 2.1] applied to $v_s\in C$ and the orbit point $\rho (w^{-1}{v}_i)x$. Second, $Q\subseteq P$: suppose $z\in Q\setminus P$; since $P$ is compact (a finite point hull, [F9]) and convex, [F8] gives $u$ with $B(u,z)>{\max }_{P}B(u,\cdot )$. The set $Q$ is invariant under $\rho (W)$ because $B(\rho (w)v_s,\rho (g)z)=B(\rho (g{)}^{-1}\rho (w)v_s,z)$ and the $W$-orbit of each $v_s$ is permuted. Choose $g\in W$ with $\rho (g)u\in C$ by [F4], and replace both $z$ and $u$ by $\rho (g)z$ and $\rho (g)u$: the new point stays in $Q\setminus P$, and the strict inequality persists because $P$ is $\rho (W)$-invariant and $B$ is $\rho (W)$-invariant. Thus we may assume $u\in C$. Then $u={\sum }_s{a}_s{v}_s$ with $a_s=B(u,e_s)\ge 0$ and $B(u,z)={\sum }_s{a}_{s}B(v_s,z)\le {\sum }_s{a}_s{c}_s$, while ${\max }_{P}B(u,\cdot )=B(u,x)={\sum }_s{a}_{s}B(v_s,x)={\sum }_s{a}_s{c}_s$ by [step 2.1] and the dual-basis expansion; this contradicts the strict separation. Hence $P=Q$. Third, $0\in \mathrm{int}P$: the points $x$ and $\rho (s)x$ for $s\in S$ have differences $-2d_s{e}_s$, so if $\mathrm{\mid }S\mathrm{\mid }>0$ their convex hull is a full-dimensional simplex contained in $P$; if $S=\mathrm{\varnothing }$, then $V=P=\{0\}$ and its interior in $V$ is itself. Thus $P$ has nonempty interior. For any $p\in \mathrm{int}P$, each $\rho (w)p$ is also in $\mathrm{int}P$ since $\rho (w)$ is an invertible linear map with $\rho (w)P=P$; convexity of the interior [F9] then puts the average $\bar{p}:=\frac{1}{\mathrm{\mid }W\mathrm{\mid }}{\sum }_{w\in W}\rho (w)p$ in $\mathrm{int}P$. It is $W$-fixed, and the only $W$-fixed vector is $0$: if every $s$ fixes $v$, [F1] gives $B(v,e_s)=0$ for all $s$, so the dual-basis expansion [F4] gives $v={\sum }_{s}B(v,e_s)v_s=0$. Hence $0=\bar{p}\in \mathrm{int}P$. Fourth, $c_s>0$: if $c_s\le 0$ for some $s$, then for every $\epsilon >0$ the point $z=\epsilon v_s$ has $B(v_s,z)=\epsilon B(v_s,v_s)>0\ge c_s$, using positive definiteness [F4]; it violates the defining $w=1,s$ inequality of $Q=P$. Such points approach $0$ as $\epsilon \to 0$, contradicting $0\in \mathrm{int}P$. Thus each $c_s=B(v_s,x)$ is positive. Therefore $P=Q$ is presented by the finitely many affine inequalities $c_s-B(w{v}_s,\cdot )\ge 0$ in the Euclidean space $(V,B)$, is nonempty, bounded and compact, and contains $0$ in its interior; by [F10] it is a compact convex polyhedral cell with $0$ in its interior.

The five clauses are now proved: (1) is [step 1.2] with [step 2.1]; (2) is [step 2.1] with [step 2.2]; (3) is [step 3.1]; (4) is [step 4.1], [step 5.1], [step 4.2], [step 6.1], [step 6.2]; and (5) is [step 7.1]. No axiom of choice is used: each selection is a single witness or a finite selection from a finite set, the maximal independent subset in [step 4.2] is chosen from finitely many candidates, [F9] uses only finite choice proved in ZF, and strict separation [F8] follows from the nearest-point variational inequality within ZF.