The $A_1$ affine line: alcoves, translations, and the root versus coroot lattice

The set $\mathrm{\Phi }=\{1,-1\}$ is finite, spans $\mathbb{R}$, omits $0$, and meets the line $\mathbb{R}\alpha$ only in its two signs. The reflection in $0$ is $s_{\alpha }(x)=-x$, which exchanges the two roots, and the Cartan integers are $\pm 2$. Thus $\mathrm{\Phi }$ is a reduced crystallographic root system; its Weyl group is $\{1,s_{\alpha }\}$ by [F7].

For $\alpha =1$, $H_{\alpha ,k}=\{k\}$, while $H_{-\alpha ,k}=\{-k\}$; their union is $\mathbb{Z}$. By [F10], alcoves are the connected components of $\mathbb{R}\setminus \mathbb{Z}$, exactly the open intervals $A_n=(n,n+1)$: each such interval is connected and contains no integer, and any connected set meeting two of them would contain an intervening integer.

By [F3] and [F6], ${\alpha }^{\vee }=2$ and the affine reflection is $r_{\alpha ,k}(x)=x-(x-k)2=2k-x$. Composing two such reflections gives $x\mapsto x+2(k-l)$, so all translations by $2m$ occur. Every word is a composition of maps $x\mapsto 2k-x$; pairing consecutive factors shows it is either an even translation or a reflection of the same form. Conversely, $r_{\alpha ,m}=(x\mapsto -x+2m)$ and $r_{\alpha ,m}{r}_{\alpha ,0}=(x\mapsto x+2m)$, so [F9] gives exactly $W_a=\{x\mapsto x+2m,x\mapsto -x+2m:m\in \mathbb{Z}\}$; its translation subgroup is $2\mathbb{Z}=Q^{\vee }$. Also $Q=\mathbb{Z}$ by [F8].

The item Highest-root dominance and the fundamental alcove gives the fundamental factor $A=(0,1)$, with facet walls $\{0\}$ and $\{1\}$. Their reflections are $s_0(x)=-x$ and $s_1(x)=2-x$, and $s_1{s}_0(x)=x+2$.

Every element has one of the two forms $x\mapsto x+2m$ or $x\mapsto -x+2m$. The first sends $A$ to $(2m,2m+1)$, while the second sends it to $(2m-1,2m)$. These intervals are all distinct and exhaust the $A_n$, so each alcove is the image of $A$ under exactly one element; hence the action is simply transitive and ${\mathrm{Stab}}_{W_a}(A)=\{1\}$.

The left and right facets of $A_0$ have types $0$ and $1$, respectively, and $s_0,s_1$ exchange $A_0$ with $A_{-1},A_1$. By step 2.2 every interval is uniquely $g(A_0)$; applying $g$ to these two adjacencies gives every adjacent pair, since the maps $x\mapsto x+2m$ and $x\mapsto -x+2m$ send their shared endpoints through all integers. Thus right multiplication by a facet generator moves the interval index by $+1$ or $-1$, and [F12] guarantees that the shared endpoint has the same type from either interval. By [F5], each move crosses exactly its shared wall.

The relations $s_0^2=s_1^2=1$ reduce every word to an alternating word. Every nonempty alternating word of even length $2q$ is a nonzero translation by $2q$ or $-2q$; every alternating word of odd length is a reflection with slope $-1$. Hence no nonempty reduced alternating word is the identity, the only defining relations are the two involution relations, and $(s_1{s}_0{)}^q=t_{2q}\mathrm{\ne }1$ for $q>0$. Thus $m_{01}=\mathrm{\infty }$ and $W_a\cong ⟨s_0,s_1\mid s_0^2=s_1^2=1⟩$.

If $g(A)=A_n$, the walls separating $A_0$ and $A_n$ are the integer points between their intervals, and there are exactly $\mathrm{\mid }n\mathrm{\mid }$ of them: for $n>0$ they are $1,\dots ,n$, for $n<0$ they are $n+1,\dots ,0$, and for $n=0$ there are none. A word of length $m$ gives a gallery of $m$ adjacent intervals by step 3.1, so it must cross each separating wall and $m\ge \mathrm{\mid }n\mathrm{\mid }$. For $q\ge 0$, the maps $t_{2q}=(s_1{s}_0{)}^q$ and $t_{-2q}=(s_0{s}_1{)}^q$ send $A_0$ to $A_{2q}$ and $A_{-2q}$; the maps $t_{2q}{s}_1$ and $t_{-2q}{s}_0$ send it to $A_{2q+1}$ and $A_{-(2q+1)}$, respectively. These words have lengths equal to the absolute values of their indices, so the lower bound is attained for every $n$ and $\mathrm{\ell }(g)=\mathrm{\mid }\mathrm{Sep}(A,g(A))\mathrm{\mid }=\mathrm{\mid }n\mathrm{\mid }$.

The standard root lattice is $Q=\mathbb{Z}$ and the standard coroot translation lattice is $Q^{\vee }=2\mathbb{Z}$ by step 1.3. Under the dual-normal convention, the walls are $2x=k$; the coroot of ${\alpha }^{\vee }$ is $\alpha$, so reflection in $k\mathrm{/}2$ is $x\mapsto k-x$, and composing the reflections at $0$ and $1\mathrm{/}2$ gives translation by $1$. By [F13], ${\mathrm{\Phi }}^{\vee }$ meets the affine lemma's root-system hypotheses; applying [F11] to it gives translation subgroup $Q^{\vee }({\mathrm{\Phi }}^{\vee })=Q=\mathbb{Z}$ by [F6, F8]. Thus the alternative convention has translations by $Q$, with the two cosets $2\mathbb{Z}$ and $1+2\mathbb{Z}$ showing $[Q:Q^{\vee }]=2$. All computations use finite root lists and explicit integer arithmetic; no axiom of choice is used.