Crystallographic alcove diagrams: the affine list realized by Weyl types A–G

By [F7] the finite crystallographic type is among the rows in (1). Define a candidate $t$ by the coefficient vector in its row. In the classical coordinate systems [F2], $t$ is respectively $e_1-e_{n+1}$, $e_1+e_2$, $2e_1$, and $e_1+e_2$ for $A,B,C,D$, so it is a root; expansion gives the displayed coefficients, including $(1,2)$ for $B_2$ and $(2,1)$ for $C_2$. For simply laced $A,D,E$, put $L^2=\mathrm{\parallel }{\alpha }_i{\mathrm{\parallel }}^2$. Their Gram matrix gives $(t,{\alpha }_i)=L^2(2c_i-{\sum }_{j\sim i}{c}_j)\mathrm{/}2$. Substituting the stated coefficients in the graphs gives bracket $1$ at the two $A_n$ endpoints ($n\ge 2$), $2$ at the sole $A_1$ node, $1$ at node $2$ of $D_n$, and $1$ at node $2,1,8$ of $E_6,E_7,E_8$ respectively, and zero elsewhere. Therefore $(t,{\alpha }_i)\ge 0$ and $\mathrm{\parallel }t{\mathrm{\parallel }}^2={\sum }_i{c}_i(t,{\alpha }_i)=L^2$ in every simply laced row. For $B,C$, the same coordinate calculations give nonnegative simple pairings and squared norm $M$; the $F_4,G_2$ Gram data [F3] give respectively $\mathrm{\parallel }t{\mathrm{\parallel }}^2=2$, $(t,{\alpha }_i)=(1,0,0,0)$ and $\mathrm{\parallel }t{\mathrm{\parallel }}^2=6$, $(t,{\alpha }_i)=(0,3)$ in the stated normalizations. All candidates have nonnegative integral coefficients, squared norm $M$, and nonnegative simple pairings.

Membership for the exceptional simply laced candidates follows locally by descent, without a highest-root table. Let $z={\sum }_i{d}_i{\alpha }_i$ have nonnegative integral coefficients and $\mathrm{\parallel }z{\mathrm{\parallel }}^2=L^2$. If it is not a simple root, $L^2={\sum }_i{d}_i(z,{\alpha }_i)>0$ supplies a positive pairing. For that $i$, $k=2(z,{\alpha }_i)\mathrm{/}{L}^2=2d_i-{\sum }_{j\sim i}{d}_j$ is a positive integer. Since $\mathrm{\parallel }z-{\alpha }_i{\mathrm{\parallel }}^2=L^2(2-k)\ge 0$, one has $k\le 2$; equality forces $z={\alpha }_i$. Thus for a nonsimple $z$, $k=1$ and $d_i\ge 1$ (otherwise its pairing is nonpositive). The reflection $s_{i}z=z-{\alpha }_i$ keeps all coefficients nonnegative and the norm unchanged while decreasing their sum by one. Repetition ends at a simple root; reversing this finite reflection word proves that $z$ is a root by reflection invariance. Apply this to the $E$ candidates from Step 1.1. For $F_4$, the coefficient vector $(2,3,4,2)$ is carried by successive reflections at nodes $1,2,3,2,1,4,3$ through $(1,3,4,2),(1,2,4,2),(1,2,2,2),(1,1,2,2),(0,1,2,2),(0,1,2,0),(0,1,0,0)$; the reflection coefficients are the row pairings with [F3]'s Cartan matrix. Reversing the word from ${\alpha }_2$ proves membership. For $G_2$, reflections at nodes $2,1$ carry $(3,2)$ to $(3,1)$ and $(0,1)$, again a simple root. This proves membership of every candidate, preserving its exact coefficients.

Each candidate $t$ is a positive root of squared norm $M$ and has $(t,{\alpha }_i)\ge 0$. If a root $t+\eta$ lay strictly above it in the root order, $\eta$ would be a nonzero nonnegative integral combination of simple roots. Positive definiteness then gives $\mathrm{\parallel }t+\eta {\mathrm{\parallel }}^2=M+2(t,\eta )+\mathrm{\parallel }\eta {\mathrm{\parallel }}^2>M$, contrary to the all-root length bound of [F7]. Thus $t$ is maximal. The unique-highest-root supplier [F1] identifies it with $\theta$. Every positive root lies below a maximal root by finiteness (extend an upward chain until it stops), and uniqueness makes that maximal root this candidate. This proves membership and the full highest-root assertion in (1), including the short-root orbits of $F_4,G_2$, without importing table maximality.

For simply laced $A,D,E$, the simple roots have one length $L$ by [F8], and all roots have that length by [F7]. Adjacent simple roots have angle $2\pi \mathrm{/}3$, so their pairing is $-L^2\mathrm{/}2$. Thus $L^2=({\alpha }_i,{\alpha }_i)$ is independent of $i$, and for the coefficients $c_i$ just listed, $(\theta ,{\alpha }_i)=\frac{L^2}{2}(2c_i-{\sum }_{j\sim i}{c}_j)$. The bracket is $1$ at both endpoints and $0$ elsewhere in $A_n$ for $n\ge 2$, is $2$ for the single node of $A_1$, is $1$ at node $2$ and $0$ elsewhere in $D_n$, and is $1$ at node $2,1,8$ respectively and $0$ elsewhere in $E_6,E_7,E_8$. The highest root $\theta$ also has length $L$, since it is a root in the same simply laced system. Hence the normalized pairing $(\theta ,{\alpha }_i)\mathrm{/}(\mathrm{\parallel }\theta \mathrm{\parallel }\mathrm{\parallel }{\alpha }_i\mathrm{\parallel })$ is $1\mathrm{/}2$ at each displayed attachment and $0$ elsewhere, including value $1$ for the sole $A_1$ node.

In the standard $B_n$ coordinates, $(\theta ,{\alpha }_i)$ vanishes except at $i=2$, where it is $1$; $\mathrm{\parallel }\theta {\mathrm{\parallel }}^2=2$ and $\mathrm{\parallel }{\alpha }_2{\mathrm{\parallel }}^2=2$ for $n\ge 3$, while $\mathrm{\parallel }{\alpha }_2{\mathrm{\parallel }}^2=1$ for $B_2$. Thus the normalized pairing is $1\mathrm{/}2$ for $n\ge 3$ and $1\mathrm{/}\sqrt{2}$ for $B_2$. In $C_n$, only $(\theta ,{\alpha }_1)$ is nonzero and equals $2$; $\mathrm{\parallel }\theta {\mathrm{\parallel }}^2=4$ and $\mathrm{\parallel }{\alpha }_1{\mathrm{\parallel }}^2=2$, so the normalized pairing is $1\mathrm{/}\sqrt{2}$. In the Bourbaki $F_4$ basis, $\mathrm{\parallel }\theta {\mathrm{\parallel }}^2=2$ and the only nonzero simple-root pairing is $(\theta ,{\alpha }_1)=1$, so its normalized value is $1\mathrm{/}2$. For $G_2$, normalize $\mathrm{\parallel }{\alpha }_1{\mathrm{\parallel }}^2=2$, $\mathrm{\parallel }{\alpha }_2{\mathrm{\parallel }}^2=6$, and $({\alpha }_1,{\alpha }_2)=-3$; then $\theta =3{\alpha }_1+2{\alpha }_2$ has squared length $6$, pairs to $0$ with ${\alpha }_1$ and to $3$ with ${\alpha }_2$, so the normalized values are $0$ and $1\mathrm{/}2$. These model calculations give the same normalized pairings for $\mathrm{\Phi }$ by [F15], hence exactly the finite values in (1).

By [F5], the walls $H_{{\alpha }_i,0}$ and $H_{\theta ,1}$ are precisely the facets of the bounded simplex $A_{\mathrm{\Phi }}$, with inward unit normals $u_i={\alpha }_i\mathrm{/}\mathrm{\parallel }{\alpha }_i\mathrm{\parallel }$ and $u_0=-\theta \mathrm{/}\mathrm{\parallel }\theta \mathrm{\parallel }$. For $i,j\ge 1$, their Gram entries are $-\cos (\pi \mathrm{/}{m}_{ij}^{\mathrm{\Phi }})$ by the finite-type root angles. For $0,i$, one has $(u_0,u_i)=-(\theta ,{\alpha }_i)\mathrm{/}(\mathrm{\parallel }\theta \mathrm{\parallel }\mathrm{\parallel }{\alpha }_i\mathrm{\parallel })$; Steps 4.1–4.2 give $-1\mathrm{/}2,-1\mathrm{/}\sqrt{2},$ or $0$ in rank at least two. The corresponding walls intersect by [F12], so [F17] gives exact product orders $3,4,$ or $2$, respectively. In type $A_1$, $u_0=-u_1$, giving the entry $-1$; in the coordinate $a=(x,{\alpha }_1)$ the endpoint reflections are $a\mapsto -a$ and $a\mapsto 2-a$, whose product is translation by two and has infinite order. Hence the full Gram matrix is the cosine matrix of $m^{\mathrm{\Phi }}$.

Reading the types in Step 1.1 against the graph recipes in [F10] gives the listed affine diagram for each finite type. The bounds are explicit: $A_n$ covers $n=1$ and $n\ge 2$ separately; $B_2$ gives the path $(4,4)$ and $B_n$ for $n\ge 3$ gives the branch diagram; $C_n$ covers every $n\ge 2$; and $D_n$ covers every $n\ge 4$. The three exceptional simply laced vectors attach at the nodes giving arms $(2,2,2),(1,3,3),(1,2,5)$, while the $F_4,G_2$ pairings give the displayed labelled paths. The low-rank conventions [F16] realize ${\tilde{C}}_1,{\tilde{B}}_2,{\tilde{D}}_3,{\tilde{E}}_4,{\tilde{E}}_5$ via $A_1,B_2,A_3,A_4,D_5$, respectively.

Take any standard affine diagram $D$. By Step 5.2, including the aliases [F16], it is the affine diagram of a finite root system $\mathrm{\Phi }$ of rank $n$. Step 5.1 identifies its cosine matrix with the Gram matrix of the $n+1$ inward unit normals of $A_{\mathrm{\Phi }}$, so the matrix is positive semidefinite by [F14]. The finite simple-root normals $u_1,\dots ,u_n$ form a basis of $E$. For any coefficient vector $x$, $x$ lies in the Gram kernel exactly when ${\sum }_i{x}_i{u}_i=0$, since $x^{\mathsf{T}}Gx=\mathrm{\parallel }{\sum }_i{x}_i{u}_i{\mathrm{\parallel }}^2$; hence the Gram matrix has rank $n$ and a nonzero radical. The matrix graph is connected with nonpositive off-diagonal entries and diagonal entries $1$ by [F10, F16]. Apply [F9] to obtain a one-dimensional radical generated by a vector with every coordinate positive; the same clause gives positive definiteness of every nonempty proper principal submatrix, while the empty case is vacuous.

Let $G=⟨s_0,\dots ,s_n⟩$. By F11, $G=W_a$, and by F11 the abstract Coxeter group on the actual reflection-product matrix $m^{\mathrm{\Phi }}$ maps isomorphically onto it. This uses the proved local generation and presentation statements in every rank, including the endpoint reflections in rank one, whose product is a nonzero coroot translation. The identities [F4] give $W_a=Q^{\vee }⋊W$ directly: every affine generator $r_{\alpha ,k}=t_{k{\alpha }^{\vee }}{s}_{\alpha }$ lies in that semidirect product, while $s_{\alpha }=r_{\alpha ,0}$ and $t_{{\alpha }^{\vee }}=r_{\alpha ,1}{r}_{\alpha ,0}$ lie in $W_a$ for every root. The coroots generate $Q^{\vee }$, and $w({\alpha }^{\vee })=(w\alpha {)}^{\vee }$ shows that $W$ normalizes its translations. A translation and an element of $W$ agree only at the identity, since $W$ fixes the origin. Together with the alcove simplex and matrix computation, this proves (5).

The kernel and proper-minor claims are those established in Step 6.1 for the cosine matrix identified in Step 5.1. The coefficient and normalized-pairing computations of Steps 1.1–4.2 give the stated verification data, including the $A_1$ parallel-wall case and the distinct $B_2\mathrm{/}{C}_2$ root-length assignments.

To distinguish the labelled graphs without using the coincidence assertion in clause (7) of the definition, first note that ${\tilde{A}}_1$ is the only listed graph with an $\mathrm{\infty }$-edge, and the cycles ${\tilde{A}}_n$ for $n\ge 2$ have all degrees $2$ and are distinguished by vertex count. Among the remaining trees, ${\tilde{B}}_n$ has one degree-$3$ vertex and exactly one $4$-edge; ${\tilde{C}}_n$ is a path with exactly two $4$-edges; ${\tilde{F}}_4$ is a path with one interior $4$-edge; and ${\tilde{G}}_2$ is the three-vertex path with a $6$-edge. The all-$3$ diagrams are distinguished by degree data: ${\tilde{D}}_4$ has a degree-$4$ vertex, ${\tilde{D}}_n$ for $n\ge 5$ has two degree-$3$ vertices, and each $\tilde{E}$ diagram has one degree-$3$ vertex with its stated arm lengths, which distinguish ${\tilde{E}}_6,{\tilde{E}}_7,{\tilde{E}}_8$. Vertex counts distinguish successive members within each family. Thus only the explicit naming conventions in [F16] identify two family names. No Choice is used: all root-coordinate checks and graph invariants are finite and explicit.