Alcove transitivity, the affine Coxeter presentation, and the length function

The induced inner-product norm is a norm. Addition is continuous because $\mathrm{\parallel }(x+y)-(x_0+y_0)\mathrm{\parallel }\le \mathrm{\parallel }x-x_0\mathrm{\parallel }+\mathrm{\parallel }y-y_0\mathrm{\parallel }$. Scalar multiplication is jointly continuous at $({\lambda }_0,x_0)$: if $\mathrm{\mid }\lambda -{\lambda }_0\mathrm{\mid }<\delta \le 1$ and $\mathrm{\parallel }x-x_0\mathrm{\parallel }<\delta$, then $\mathrm{\parallel }\lambda x-{\lambda }_0{x}_0\mathrm{\parallel }\le (\mathrm{\mid }{\lambda }_0\mathrm{\mid }+1)\delta +\mathrm{\parallel }{x}_0\mathrm{\parallel }\delta$, which is below any prescribed $\epsilon >0$ for sufficiently small $\delta >0$. Thus the norm topology on $E$ is a real topological vector space topology, so [F7] applies.

Let $O$ be a nonempty open subset of a finite-dimensional real affine space and let $L_1,\dots ,L_N$ be finitely many proper affine subspaces. If $N=0$, any point of $O$ works. Otherwise let $D_j$ be the direction subspace of $L_j$; each $D_j$ is proper. By [F6] choose a direction $v$ outside their union. For $p\in O$, openness gives an interval around $0$ with $p+tv\in O$. The line $p+\mathbb{R}v$ meets each $L_j$ in at most one point, so deleting the finitely many excluded parameters leaves a point in $O\setminus {\bigcup }_j{L}_j$. In dimension zero every proper affine subspace is empty.

Fix any alcove $C$ and follow the gallery from $A$ to $C$ supplied by [F4]. Start with $A=1(A)$. Suppose the current alcove is $g(A)$ with $g\in G$. The next shared panel is a facet $g(F_a)$ for some $a\in J$, since $A$ has exactly the listed facets and $g$ is an affine isometry. By [F3] reflection in its wall is $g{s}_a{g}^{-1}$, so the next alcove is $g{s}_a(A)$ and still lies in $G\cdot A$. Induction along the finite gallery gives $C\in G\cdot A$. Thus $G$ acts transitively on alcoves.

By [F12], each irreducible component of $\mathrm{\Phi }$ has one of the types listed in part (4), with the stated low-rank identifications. By [F11], the affine group defined here is the coroot-lattice semidirect product of that finite Weyl group, which is the standard affine Weyl group in the Euclidean root-system normalization. This comparison uses only the root-system type classification and the explicit Euclidean construction; it asserts no loop-algebra identification.

If $E=0$, there are no walls. If $\dim E=1$, each wall is a point and is an endpoint facet of the two adjacent interval alcoves. Assume $\dim E\ge 2$ and fix a wall $H$. Choose $p_0\in H$. Let $e_1,\dots ,e_n$ be any real basis of $E$, which exists by [F10]. For any $\epsilon >0$, let $S$ be the geometric simplex with vertices $p_0-\epsilon {\sum }_i{e}_i$ and $p_0+\epsilon e_i$ ($1\le i\le n$), using [F14]. Its edge vectors are $\epsilon (e_i+{\sum }_j{e}_j)$, and a linear relation among them has coefficients $c_i$ satisfying $c_i+{\sum }_j{c}_j=0$ for every $i$, hence all $c_i=0$. Thus $S$ is full-dimensional. Its barycenter is $p_0$, with all barycentric coordinates equal to $1\mathrm{/}(n+1)$, so $O:=\mathrm{int}(S)$ is nonempty. The simplex $S$ is compact by [F7]. By local finiteness [F1], only finitely many walls meet $S$. For each other wall $H^{\mathrm{\prime }}$, $H\cap H^{\mathrm{\prime }}$ is empty or a proper affine subspace of $H$. Apply the affine avoidance argument of 1.2 inside the relative open set $H\cap O$ to choose $p\in H$ on none of these other walls. Since $p$ lies on no other wall in that finite list, for each $H_{\beta ,l}\mathrm{\ne }H$ in the list Cauchy--Schwarz [F15] gives a positive-radius ball about $p$ missing $H_{\beta ,l}$: take radius less than $\mathrm{\mid }B(p,\beta )-l\mathrm{\mid }\mathrm{/}(2\mathrm{\parallel }\beta \mathrm{\parallel })$. Taking the minimum of these finitely many radii and the radius of a ball inside $O$ gives a ball meeting the arrangement only in $H$. Its two half-balls lie in two alcoves whose closures share a relatively open patch of $H$, so $H$ is a facet wall of either alcove. By 1.3 one such alcove is $g(A)$. Its facet is $g(F_a)$ for some $a\in J$, and [F3] gives $r_H=g{s}_a{g}^{-1}\in G$. Hence every wall reflection lies in $G$. Since $W_a$ is generated by all wall reflections by [F1] and $G\le W_a$ by [F13], we have $G=W_a$.

The matrix and homomorphism are those established in [F4], with the universal presentation and length convention of [F5]. Surjectivity of $\phi$ follows from $G=W_a$ in 2.1. If $w\in \ker \phi$, then $\phi (w)(A)=A$, so [F4] gives $w=1$; thus $\phi$ is injective. For any alcove $C$, transitivity gives $C=g(A)$. If $h(C)=C$, then $g^{-1}hg$ stabilizes $A$, which is trivial by [F3]; hence the action is free. Transitivity and freeness give exactly one group element carrying any $C$ to any $C^{\mathrm{\prime }}$.

Let $g=s_{a_1}\cdots s_{a_m}$ be any expression. The prefix alcoves $C_j=s_{a_1}\cdots s_{a_j}(A)$ form a gallery. Each step crosses one facet wall $H_j$, so [F3] gives $\mathrm{Sep}(A,C_j)=\mathrm{Sep}(A,C_{j-1})\mathrm{△}\{H_j\}$. Every wall in $\mathrm{Sep}(A,g(A))$ must therefore occur among $H_1,\dots ,H_m$, and $m\ge \mathrm{\#}\mathrm{Sep}(A,g(A))$. For the reverse inequality, if $E=0$ the claim is immediate. Otherwise take $x\in A$ and $y_0\in g(A)$. Let $S$ be a positive rescaling about $y_0$ of the finite simplex template from 2.1, chosen small enough that $S\subset g(A)$; this is possible because $g(A)$ is open. Let $O:=\mathrm{int}(S)$. Its finite vertices lie in $g(A)$, and $S$ is their convex hull by [F14]. The convex hull $K$ of $x$ and those vertices is compact by [F7], so by [F1] only finitely many walls meet $K$. Let $P_1,\dots ,P_N$ be the codimension-two intersections of distinct walls in this finite list. Since $x\in A$, $x\mathrm{\notin }{P}_j$, and each $\mathrm{aff}(\{x\}\cup P_j)$ is proper by [F9]. Also exclude $x+D$ for every codimension-two direction $D=\ker B(-,\alpha )\cap \ker B(-,\beta )$ with nonproportional roots $\alpha ,\beta$, and the singleton $\{x\}$. There are finitely many such directions because $\mathrm{\Phi }$ is finite, and all these affine sets are proper. Apply 1.2 to choose $y\in O$ outside the full excluded family. Then $[x,y]$ meets no codimension-two intersection, its nonzero direction $y-x$ is not parallel to any codimension-two direction, and every wall it crosses is met transversely and one at a time. For a defining affine functional $f_H$ of any wall $H$, $f_H$ has opposite signs at $x,y$ exactly when $H$ separates $A$ and $g(A)$; linearity along the segment shows that such a wall is crossed exactly once, while every other wall is not crossed. Thus the segment gives a gallery of exactly $\mathrm{\#}\mathrm{Sep}(A,g(A))$ steps. Inducting as in 1.3 gives $h\in G$ with $h(A)=g(A)$. By 2.1, $G=W_a$, and the trivial stabilizer of $A$ in [F3] then gives $h=g$. The minimum length therefore equals the separating-wall count.

The argument in 3.2 is carried out in the full product alcove $A={\prod }_i{A}_i$, so it already applies to reducible systems. More explicitly, every wall belongs to one irreducible factor, hence the separating-wall set is the disjoint union of the factorwise sets. Reflections from different orthogonal components act on separate summands and commute, and each component subgroup acts trivially on the other summands; hence $W_a$ is their direct product and the facet-generator set is their disjoint union. Any word has at least the sum of the factorwise minimum lengths, while concatenating factorwise minimum words attains that sum. For an $A_1$ factor the alcoves are intervals, and the straight segment crosses exactly the integer-level walls between the two intervals.

If $\mathrm{\Phi }=\mathrm{\varnothing }$, then $E=0$, $J=\mathrm{\varnothing }$, $W_a=\{1\}$, and all four claims reduce to the empty presentation and zero length. In rank one, the two endpoint reflections have infinite-order product by [F4], and the length count is the interval-gallery count in 4.1. In reducible systems the factor argument of 4.1 applies. The only nonunique choices in the proof were made from finite families: affine bad-set avoidance uses 1.2, and compact hulls and wall lists are finite. No axiom of choice is used.