Alcove separation, facet reflections, panel types, and triviality of the fundamental alcove stabilizer

Let $O$ be a nonempty open subset of a finite-dimensional real affine space of positive dimension, and let $L_1,\dots ,L_m$ be finitely many proper affine subspaces. If $m=0$, any point of $O$ works. Otherwise let $V_j$ be the direction subspace of $L_j$; each is proper. By [F5] choose $v\mathrm{\notin }{\bigcup }_j{V}_j$, so $v\mathrm{\ne }0$. Choose $p\in O$; openness gives an interval $(-\epsilon ,\epsilon )$ with $p+tv\in O$. The line $p+\mathbb{R}v$ meets each $L_j$ in at most one point, since two intersections would imply $v\in V_j$. The interval is infinite and only finitely many parameters are excluded, so some $p+tv$ lies in $O\setminus {\bigcup }_j{L}_j$. In dimension zero every proper affine subspace is empty, so the same avoidance conclusion holds.

Let $V_i$ be the finite vertex set of $\overline{A_i}$. The componentwise simplex descriptions imply $\bar{A}={\prod }_i\overline{A_i}$, and this product is the convex hull of the finite product $V={\prod }_i{V}_i$: write each component point in barycentric coordinates and multiply the finitely many coordinate weights to obtain a convex combination of product vertices. For any alcove $C$, choose $y\in C$ and set $K=\mathrm{co}(V\cup \{y\})$. By [F6], $K$ is compact; it contains $\bar{A}$, $y$, and every segment joining $y$ to a point of $A$. By [F9], only finitely many walls meet $K$.

Every connected alcove lies strictly on one side of each wall, because it is connected and avoids that wall. Let $F$ be a facet of $\bar{C}$ on $H$, and choose a nonempty relatively open set $O\subseteq H$ in the relative interior of $F$. Pick $p_0\in O$. By [F10] and [F11], choose $r>0$ so the closed ball $K=\bar{B}(p_0,r)$ is compact; its open ball meets $O$ in a nonempty relatively open subset of $H$. By [F9] only finitely many walls meet $K$. For each such wall $H^{\mathrm{\prime }}\mathrm{\ne }H$, the intersection $H^{\mathrm{\prime }}\cap H$ is either empty or a proper affine subspace of $H$; step 1.1 therefore gives $p\in O\cap B(p_0,r)$ on no other wall. Since every wall through $p$ would meet $K$, the finite list contains all walls relevant near $p$. By [F12] and the affine equations of those finitely many walls, a smaller open ball about $p$, contained in $K$, misses every wall except $H$. Its two half-balls lie in the two adjacent alcoves, and reflection $r_H$ exchanges them; hence $r_H(C)$ is the other alcove adjacent across $F$. For any other wall $H^{\mathrm{\prime }}$, that ball misses $H^{\mathrm{\prime }}$, so the two adjacent alcoves lie on the same side of $H^{\mathrm{\prime }}$, while $H$ separates them. Thus $\mathrm{Sep}(C,r_H(C))=\{H\}$.

Among the finitely many walls meeting $K$ from step 1.2, consider each distinct intersecting pair $H,H^{\mathrm{\prime }}$ and put $P=H\cap H^{\mathrm{\prime }}$. Distinct affine hyperplanes that intersect have codimension-two intersection, so $\mathrm{aff}(\{y\}\cup P)$ is a proper affine subspace: $y$ lies on no wall because $y\in C$. By step 1.1 choose $z\in A$ outside the finite union of these subspaces. The segment $[z,y]$ lies in $K$ and meets finitely many walls; it cannot meet two distinct walls at the same point, since that would put $z$ in one of the excluded affine spans. Neither endpoint lies on a wall, and a segment not contained in a hyperplane crosses it at most once. Thus its crossings occur one at a time. At any crossing point $q$, the segment lies in a compact ball around $q$ by [F10] and [F11]. That ball meets finitely many walls by [F9], and $q$ lies on only the crossed wall. By [F12], shrink to a neighborhood inside the ball that misses all the other walls. The two local sides belong to adjacent alcoves, so the successive components along the segment form a finite gallery.

For each wall, an alcove has one fixed sign with respect to its defining affine functional, by connectedness. Comparing these two signs wall by wall shows that a wall separates $C$ from $C^{\mathrm{\prime }\mathrm{\prime }}$ exactly when it separates exactly one of the pairs $(C,C^{\mathrm{\prime }})$ and $(C^{\mathrm{\prime }},C^{\mathrm{\prime }\mathrm{\prime }})$. This is the symmetric-difference identity.

Let $G=⟨s_a:a\in J⟩\le W_a$. Start with $A$ and follow the gallery of step 3.1 to any alcove $C$. Inductively suppose the current alcove is $h(A)$ for $h\in G$. Each of its facets is $h(F_a)$ for a unique $a\in J$, since $h$ is an isometry and $\bar{A}$ has exactly these facets. If the next gallery step crosses the wall $h(H_a)$, reflection in it is $h{s}_a{h}^{-1}$; by step 2.1 the next alcove is $h{s}_a(A)\in G\cdot A$. Therefore $G$ acts transitively on all alcoves.

Every affine wall $H$ is a facet wall of some alcove. Take $p_0\in H$ and, by [F10] and [F11], a compact closed ball $K$ around it; only finitely many walls meet $K$ by [F9]. The intersections with $H$ of the listed walls other than $H$ are finitely many proper affine subspaces of $H$. Applying step 1.1 to their union in the relatively open set $H\cap \mathrm{int}(K)$ gives $p\in H$ on no other wall. The finite list includes every wall through $p$. By [F12], a smaller ball about $p$ contained in $K$ misses all listed walls other than $H$, hence meets the arrangement only in $H$. The two local alcoves on its sides share a relatively open subset of $H$ in their closures, so $H$ is a facet wall. By step 4.1 one such alcove is $h(A)$ for some $h\in G$, so $H=h(H_a)$ for a facet label $a$. The uniqueness of the Euclidean reflection gives $r_H=h{s}_a{h}^{-1}$. Hence every generator of $W_a$ lies in $G$, while $G\le W_a$ by definition; thus $W_a=G$.

Let $g\in W_a$ stabilize $A$, and write it as a word in the finite set of facet reflections with the least possible number $m$ of factors: $g=s_{a_1}\cdots s_{a_m}$. Put $h_0=1$, $h_k=s_{a_1}\cdots s_{a_k}$, and $H_k=h_{k-1}(H_{a_k})$. The gallery $h_0(A),\dots ,h_m(A)$ crosses $H_k$ at step $k$, and step 2.1 gives $\mathrm{Sep}(h_{k-1}(A),h_k(A))=\{H_k\}$. If $H_p=H_q$ for $p<q$, equality of their reflections gives, with $s=s_{a_p}$, $t=s_{a_q}$ and $b=s_{a_{p+1}}\cdots s_{a_{q-1}}$, the relation $s=sbt{b}^{-1}s$, hence $sbt=b$. Deleting the factors at positions $p,q$ shortens the word for $g$, a contradiction. Thus the crossed walls are pairwise distinct. Repeated use of step 3.2 now gives $\mathrm{Sep}(A,g(A))=\{H_1,\dots ,H_m\}$; since $g(A)=A$, this set is empty, so $m=0$ and $g=1$. Therefore ${\mathrm{Stab}}_{W_a}(A)=\{1\}$.

For $C\in W_a\cdot A$, existence of $g$ with $C=g(A)$ is the definition of the orbit. If also $C=g^{\mathrm{\prime }}(A)$, then $g^{-1}{g}^{\mathrm{\prime }}$ stabilizes $A$, so step 6.1 gives $g=g^{\mathrm{\prime }}$. The facets of $\bar{A}$ are precisely the $F_a$ for $a\in J$, each occurring once; applying $g$ gives a unique label for every facet of $\bar{C}$. Since $U=\mathrm{int}(\bar{A})=A$, its stabilizer is the same trivial stabilizer.

If adjacent alcoves $C=g(A)$ and $C^{\mathrm{\prime }}=g^{\mathrm{\prime }}(A)$ share a facet $F=g(F_a)$ on $H=g(H_a)$, step 2.1 says $C^{\mathrm{\prime }}=r_H(C)=g{s}_a(A)$. Uniqueness from step 7.1 yields $g^{\mathrm{\prime }}=g{s}_a$. The reflection $s_a$ fixes $F_a$ pointwise, so $F=g{s}_a(F_a)$ also has type $a$ as computed from $C^{\mathrm{\prime }}$. Thus the panel type is independent of the side; every alcove has one facet for each $a\in J$ because $A$ does; and $r_H=g{s}_a{g}^{-1}$.

If $\mathrm{\Phi }=\mathrm{\varnothing }$, the spanning condition gives $E=0$, the wall arrangement is empty, $A=\{0\}$, and $W_a=\{1\}$; all separation and panel claims are then vacuous and the stabilizer claim holds. For a single rank-one component, distinct walls are disjoint points, so the bad-pair family in step 3.1 is empty and its fundamental alcoves are intervals with endpoint facets. In reducible products of rank-one components, the general affine-subspace avoidance in step 3.1 also handles intersections between walls from different factors. All other selections above are finite: the generic point avoids finitely many affine subspaces, the compact hull supplies finitely many walls, and shortest word length is a least natural number. No axiom of choice is used.