Dihedral diagrams $I_2(m)$: Gram determinants, the infinite case, and the low-rank coincidences

(The matrix, the finite determinant and the infinite kernel vector.) Since $m(s,t)=r\ge 3$, or $r=\mathrm{\infty }$, the diagram $\mathrm{\Gamma }$ is the single edge $st$ labelled $r$ [F1], and in the basis $(e_s,e_t)$ the matrix of $B$ is $\left(\begin{array}{cc}1& -c\\ -c& 1\end{array}\right)$ with $c=\cos (\pi \mathrm{/}r)$ for finite $r$ and $c=1$ for $r=\mathrm{\infty }$ [F2]; this is (i). Its determinant is $1-c^2$. For finite $r$ the Pythagorean identity gives $1-c^2={\sin }^2(\pi \mathrm{/}r)$, which is positive because $0<\pi \mathrm{/}r\le \pi \mathrm{/}3<\pi$ [F6]; the $1\times 1$ principal minors are the diagonal entries $1>0$. For $r=\mathrm{\infty }$ we have $c=1$ and $\det B=0$, while $u=e_s+e_t\mathrm{\ne }0$ satisfies $B(u,u)=1+1-2\cdot 1\cdot 1=0$ [F2]; by [F5] the form is not positive definite, giving (iii) for the form.

(Finite case: positive definiteness, order $2r$.) Let $r<\mathrm{\infty }$. The leading principal minors of $B$ are $1$ and $1-c^2={\sin }^2(\pi \mathrm{/}r)>0$ by 1.1 [step 1.1], so $B$ is positive definite by [F5] and $W$ is finite by the criterion [F5]. By [F3] the product $\rho (s)\rho (t)=r_{e_s}{r}_{e_t}$ acts on $P=\mathbb{R}{e}_s+\mathbb{R}{e}_t$ as a rotation of exact order $r$ and fixes $P^{\perp }$ pointwise, so $\rho (s)\rho (t)$ has exact order $r$ on $V$. Put $A:=\rho (s)\rho (t)=\rho (st)$, of exact order $r$; every element of $W$ is of the form $A_0^k$ or $s{A}_0^k$ with $A_0:=st\in W$ and $k\in \mathbb{Z}$, because every alternating word in the two involutions is a power of $st$ possibly preceded by $s$. Hence $\rho (W)=\{A^k,\rho (s)A^k:0\le k<r\}$: the $A^k$ are distinct because $A$ has exact order $r$, the $\rho (s)A^k$ are distinct for the same reason, and $A^k\mathrm{\ne }\rho (s)A^l$ because $\det A^k=1$ while $\det (\rho (s)A^l)=-1$; so $\mathrm{\mid }\rho (W)\mathrm{\mid }=2r$ and injectivity of $\rho$ [F4] gives $\mathrm{\mid }W\mathrm{\mid }=2r$, $W$ being the dihedral group $I_2(r)$. For $r=\mathrm{\infty }$ the same factorisation with $\rho (s)\rho (t)$ of infinite order [F3] exhibits infinitely many distinct elements and $W$ is the infinite dihedral group, completing (ii) and (iii).

(The low-rank coincidences and the products (iv).) Reading the labelled graphs: $I_2(3)$ is a single edge labelled $3$, which is the one-edge path $A_2$; $I_2(4)$ is a single edge labelled $4$, which is $B_2$, also written $C_2$; $I_2(6)$ is a single edge labelled $6$, conventionally $G_2$; $m(s,t)=2$ draws no edge [F1], so the two-vertex diagram with $m=2$ is the disconnected diagram $A_1\times A_1$, which is the direct product of two one-vertex groups by the component statement [F7]; and $I_2(5)=H_2$ by the rank-two naming convention. The one-vertex diagram $A_1$ has $B=(1)$, which is positive definite [F5], and presentation $⟨s\mid s^2=1⟩$, so $W\cong \mathbb{Z}\mathrm{/}2$; the disconnected two-vertex diagram has group $(\mathbb{Z}\mathrm{/}2)\times (\mathbb{Z}\mathrm{/}2)$, of order $2\cdot 2=4$ [F7]. These coincidences and the group orders are (iv), and the group $I_2(r)$ of 2.1 [step 2.1] is the dihedral group appearing under both names in the classification.