The $A_2$ and $B_2$ fundamental alcoves: coordinates, corner vectors, and facet types

In $A_2$, all six roots have squared norm $1$ and directions with angles that are multiples of $\pi \mathrm{/}3$. The set spans ${\mathbb{R}}^2$; if a root has direction $j\pi \mathrm{/}3$, its reflecting line is perpendicular to it and reflection sends a root direction $k\pi \mathrm{/}3$ to $(2j+3-k)\pi \mathrm{/}3$, so it permutes the six roots. For roots $\alpha ,\beta$, the Cartan number is $2B(\beta ,\alpha )$, an integer because the possible inner products are $1,-1,\frac{1}{2},-\frac{1}{2}$. The only roots on each root line are its two displayed signs, so the system is reduced. The positive roots for the chamber bounded by ${\alpha }_1,{\alpha }_2$ are ${\alpha }_1,{\alpha }_2,\theta$; the first two are simple, $\theta ={\alpha }_1+{\alpha }_2$ is the highest root, and each coroot is $2\alpha$.

In $B_2$, the roots span ${\mathbb{R}}^2$. Reflection with short-root normal $e_i$ changes the sign of coordinate $i$, while reflection with long-root normal $e_1-e_2$ swaps coordinates and reflection with normal $e_1+e_2$ sends $(x_1,x_2)$ to $(-x_2,-x_1)$; these signed coordinate maps preserve the displayed set. If $\alpha$ is short, ${\alpha }^{\vee }=2\alpha$ and $B(\beta ,{\alpha }^{\vee })\in \{0,\pm 2\}$; if $\alpha$ is long, ${\alpha }^{\vee }=\alpha$ and $B(\beta ,{\alpha }^{\vee })\in \{0,\pm 1,\pm 2\}$. Thus all Cartan numbers are integral, and the explicit root lines show reducedness. The positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2=e_1,{\alpha }_1+2{\alpha }_2=\theta$, so ${\alpha }_1,{\alpha }_2$ are simple and $\theta$ is highest. The coroot formula gives ${\alpha }_1^{\vee }={\alpha }_1$, ${\alpha }_2^{\vee }=2e_2$, and ${\theta }^{\vee }=\theta$.

For $A_2$, [F4] gives the region $B(x,{\alpha }_1)>0$, $B(x,{\alpha }_2)>0$, $B(x,\theta )<1$. Writing $x=(x_1,x_2)$, the vertices are the pairwise intersections of its three boundary lines: the two simple-root walls meet at $0$; $H_{{\alpha }_2,0}\cap H_{\theta ,1}$ gives $-\frac{1}{2}{x}_1+\frac{\sqrt{3}}{2}{x}_2=0$ and $x_1=1$, hence $v_1=(1,1\mathrm{/}\sqrt{3})$; $H_{{\alpha }_1,0}\cap H_{\theta ,1}$ gives $x_1=0$ and $x_2=2\mathrm{/}\sqrt{3}$, hence $v_2=(0,2\mathrm{/}\sqrt{3})$. By [F5], these three facets have types $1,2,0$. Since $B(0,\theta )=0$ and $B(v_1,\theta )=B(v_2,\theta )=1$, the origin is the unique such corner.

For $B_2$, [F4] gives $x_1-x_2>0$, $x_2>0$, and $x_1+x_2<1$. Intersecting the boundary pairs gives $0$ from $x_1=x_2=0$, $e_1=(1,0)$ from $x_2=0$ and $x_1+x_2=1$, and $(1\mathrm{/}2,1\mathrm{/}2)$ from $x_1=x_2$ and $x_1+x_2=1$. By [F5], the facets on $H_{{\alpha }_1,0},H_{{\alpha }_2,0},H_{\theta ,1}$ have types $1,2,0$.

The three $A_2$ facet-wall pairs meet at $0,v_1,v_2$. Their normals have pairwise inner products of absolute value $1\mathrm{/}2$, so the corresponding lines meet at acute angle $\pi \mathrm{/}3$. Translating the intersection to the origin turns each affine reflection pair into a pair of linear line reflections; their product rotates by $2\pi \mathrm{/}3$ or $-2\pi \mathrm{/}3$ and has order $3$. Since the coroots are twice the roots, the coroot pairing products are $B({\alpha }_1^{\vee },{\alpha }_2)B({\alpha }_2^{\vee },{\alpha }_1)=1$ and $B({\alpha }_i^{\vee },\theta )B({\theta }^{\vee },{\alpha }_i)=1$ for $i=1,2$. Thus [F6] gives $m_{01}=m_{02}=m_{12}=3$.

At $v_1$, the root pairings are $B(v_1,{\alpha }_2)=0$ and $B(v_1,{\alpha }_1)=B(v_1,\theta )=1$. Since the positive roots are exactly ${\alpha }_1,{\alpha }_2,\theta$, these give precisely the three walls $H_{{\alpha }_2,0},H_{{\alpha }_1,1},H_{\theta ,1}$. Their normal lines have the three directions modulo $\pi$ separated by $\pi \mathrm{/}3$, so the local arrangement has six sectors; [F7] identifies them with the six incident alcoves. The fundamental alcove has types $2$ and $0$ at this vertex, and [F7] gives the alternating six-panel cycle. Its boundary word, in a suitable orientation and starting point, is $({\sigma }_2{\sigma }_0{)}^3$; [F6] identifies the Coxeter generators with the facet generators, and $m_{20}=3$, so $(s_2{s}_0{)}^3=1$. The panel of the fundamental alcove along the ray from $v_1$ toward $v_2$ has type $0$; three positions later in the alternating cycle the opposite ray of the same wall $H_{\theta ,1}$ has type $2$. Thus the type belongs to a panel, not to the whole wall.

In $B_2$, $B({\alpha }_1,{\alpha }_2)=-1$, $B(\theta ,{\alpha }_1)=0$, and $B(\theta ,{\alpha }_2)=1$. Together with the coroots from 1.2, the products of Cartan pairings for pairs $(1,2),(0,1),(0,2)$ are respectively $(-1)(-2)=2$, $0$, and $(2)(1)=2$. The corresponding line angles are $\pi \mathrm{/}4,\pi \mathrm{/}2,\pi \mathrm{/}4$, respectively, so products of the intersecting affine reflections are rotations by $\pi \mathrm{/}2$ or $\pi$, with orders $4,2,4$. By [F6], the affine matrix therefore has $m_{12}=4$, $m_{01}=2$, and $m_{02}=4$. Both coordinate models and all their corner and endpoint calculations are finite and explicit, so no axiom of choice is used.