Classification of affine Coxeter diagrams and their Euclidean simplex realization

For a connected $\mathrm{\Gamma }$ with positive-semidefinite corank-one $B$, [F2] enumerates the standard affine diagram. Conversely [F3] proves that each standard diagram has positive-semidefinite corank-one cosine form and occurs as a crystallographic alcove diagram; a relabelling permutes matrix rows and columns and preserves these properties. The aliases, nonisomorphism and label assertions are also [F3]. Finally [F1] proves that connected positive-semidefinite non-positive-definite is equivalent to corank one, without assuming it in advance.

Under these equivalent conditions, [F1] gives $\mathrm{rad}(B)=\mathbb{R}\delta$ with all ${\delta }_s>0$. Apply [F4] to obtain exactly the radical quotient, Euclidean slice, faithful action, coordinates, simplex, normals and intersection conclusions in (2), including rank one.

Choose the matching root system from [F5], using the actual affine type and aliases of [F3]. Its bounded fundamental alcove is supplied by [F13], and the inward unit normals have the same Gram matrix as the slice normals by [F3,F4]. The simplex similarity [F7] therefore matches their labelled facets and conjugates the corresponding reflections. By [F6], the fundamental reflections give the Coxeter group of this matrix and generate $W_a(\mathrm{\Phi })$. Thus this similarity intertwines the homomorphisms from $W$ to the two reflection groups and identifies $W$ with $W_a(\mathrm{\Phi })$, preserving its simple generators and lengths.

The finite criterion is [F8]. For affine $\mathrm{\Gamma }$, its form is not positive definite, so $W$ is infinite, while every proper standard parabolic is finite by [F1]. The converse fails already for the triangle with labels $(3,3,4)$. Put $c_3=\cos (\pi \mathrm{/}3)$. Since $0<\pi \mathrm{/}3<\pi \mathrm{/}2$ by [F22], strict decrease [F26] and [F25] give $c_3>0$; [F23] and [F24] give $2c_3^2-1=\cos (2\pi \mathrm{/}3)=-c_3$, so $(2c_3-1)(c_3+1)=0$ and $c_3=1\mathrm{/}2$. Put $c_4=\cos (\pi \mathrm{/}4)$. Again $0<\pi \mathrm{/}4<\pi \mathrm{/}2$ gives $c_4>0$, and [F23,F25] give $2c_4^2-1=0$; by [F27], $c_4=\sqrt{2}\mathrm{/}2$. Therefore the displayed Coxeter matrix [F19] has quadratic value $3+2(-c_3-c_3-c_4)=1-\sqrt{2}<0$ on $(1,1,1)$, while its value on each basis vector is $1$; the strict inequality follows from $(1-\sqrt{2})(1+\sqrt{2})=-1$ and $1+\sqrt{2}>0$. Every proper two-generator form has determinant $1-{\cos }^2(\pi \mathrm{/}m)>0$ for $m=3,4$, so its group is finite by [F8]; the whole group is infinite by that same criterion. Thus infiniteness together with finite proper parabolics cannot replace positive semidefiniteness.

Every point $x$ of the root-system Euclidean space lies in the closure of an alcove. By [F14] choose a compact closed ball $K$ about $x$; by [F15] only finitely many walls meet $K$. The finitely many walls in $K$ not containing $x$ have positive distance from $x$, so a smaller open ball $U$ about $x$ misses all of them. The direction hyperplanes of the finitely many walls through $x$, together with the zero subspace, are proper linear subspaces because $\dim E\ge 1$. By [F9] choose a nonzero direction $v$ outside their union. For all sufficiently small $\epsilon >0$, the point $x+\epsilon v$ lies in $U$ and lies on none of the walls through $x$, so the short ray lies in one connected component of the wall complement. Its closure contains $x$. By [F6], every alcove is a translate of $A_{\mathrm{\Phi }}$, so the alcove closures cover the space. Their interiors are disjoint because they are distinct connected components. Hence the transported slice closures tile $E$.

The slice closed alcove meets each orbit in exactly one point. Coverage gives existence. If $x,y\in \bar{A}$ and $y=wx$, then $y\in w\bar{A}\cap \bar{A}$; [F4]'s intersection formula gives $y(e_s)=0$ for every $s\in S(w)$. Every corresponding facet reflection fixes $y$, and [F12] gives $w\in W_{S(w)}$, so $wy=y$. Consequently $x=w^{-1}y=y$. This proves the strict closed fundamental-domain assertion, which transfers to $\overline{A_{\mathrm{\Phi }}}$. Simple transitivity on open alcoves and the separating-wall length formula transfer from [F6]; similarity sends the entire family of reflected walls bijectively to the slice walls, so it preserves which walls separate two alcoves.

By [F16], each affine Weyl element has a unique form $t_{q}u$ with $q\in Q^{\vee }$ and $u\in W(\mathrm{\Phi })$. If $(t_{q}u)K\cap K\mathrm{\ne }\mathrm{\varnothing }$ for a compact $K$, then $q\in K-uK$. There are finitely many $u$ by [F17]. A compact subset of a metric space is bounded (cover it by unit balls and take a finite subcover), and $u$ preserves the metric, so each $K-uK$ is bounded. By [F18], the simple coroots form a finite basis and integrally generate $Q^{\vee }$; after a finite enumeration, each basis-coordinate function is linear and continuous by [F20]. Its values on $K$ and on $uK$ are bounded: continuity gives a neighborhood of each point on which the coordinate differs by less than $1$ from its value at that point, and compactness gives a finite subcover and hence a finite bound. Since $u$ is an isometry, $uK$ is compact. Linearity then bounds the coordinate of each difference $x-uy$ by the sum of the two coordinate bounds, so the coordinates of $K-uK$ are bounded. Since $Q^{\vee }$ consists of integer coordinate tuples in this basis, only finitely many $q$ occur. Thus only finitely many affine Weyl elements meet $K$ back to itself, proving proper discontinuity. By [F13], $\overline{A_{\mathrm{\Phi }}}$ is a geometric simplex, hence the convex hull of finitely many vertices by [F10]; it is compact by [F11]. Coverage by its translates proves cocompactness. Both properties transfer under the similarity in Step 1.3.

If $\mathrm{\mid }S\mathrm{\mid }=1$, then $B=[1]$ is positive definite. No faithful realization by reflections in facets of a bounded Euclidean simplex is possible: a positive-dimensional bounded simplex has at least two facets, whereas a zero-dimensional simplex has no codimension-one reflecting hyperplane. Thus both sides of the stated equivalence are false in this case. Otherwise suppose the stated faithful Euclidean realization has a bounded fundamental Coxeter simplex of dimension $n\ge 1$, with its $n+1=\mathrm{\mid }S\mathrm{\mid }$ facets indexed by $S$ and labelled angles $\pi \mathrm{/}{m}_{st}$ (and disjoint endpoints in dimension one). Its inward unit-normal Gram matrix is the cosine matrix: in a two-dimensional normal section through a codimension-two face, the angle between inward normals is $\pi -\pi \mathrm{/}{m}_{st}$; in dimension one the two normals are opposite. The normals span the $n$-dimensional direction space, since otherwise a nonzero perpendicular direction would leave all facet inequalities unchanged and make the simplex unbounded. There are $n+1$ facets, so the Gram matrix is positive semidefinite of rank $n$, hence corank one. This proves the reverse Euclidean implication; Steps 1.2–3.2 prove the forward one. The facet-generated translation subgroup is exactly $Q^{\vee }$ by [F16]. Therefore adjoining translations in $P^{\vee }$ gives a strictly larger group if $P^{\vee }\mathrm{\ne }{Q}^{\vee }$, and gives the same group if the lattices are equal by [F21]. No twisted-diagram claim is made. All arguments use finite bases and finite avoidance, without Choice.