Crystallographic finite type: the Weyl types, reduced realizations and lattice stability

If some scaling of the geometry is crystallographic, then every edge label of $\mathrm{\Gamma }$ lies in $\{3,4,6\}$: by the label restriction every distinct pair satisfies $m(s,t)\in \{2,3,4,6\}$, and edges are exactly the pairs with $m(s,t)\ge 3$.

Since $W$ is finite, classification clauses (1)–(2) in [F5] give that every connected component of $\mathrm{\Gamma }$ is one of the standard diagrams: $A_n$, $B_n$, $D_n$, $E_6,E_7,E_8,F_4,H_3,H_4$ or $I_2(m)$; the diagrams $A_n$, $B_n$, $D_n$, $E_6,E_7,E_8,F_4$ and $I_2(m)$ with $m\in \{3,4,6\}$ are paths, stars or single edges, hence trees, and have all labels in $\{3,4,6\}$, while $H_3$ and $H_4$ contain a label $5$ and $I_2(m)$ has label $m$; and the coincidences (4) in [F6] give $I_2(3)=A_2$, $I_2(4)=B_2$, $I_2(6)=G_2$, so the Weyl-type list is $A_n,B_n,D_n,E_6,E_7,E_8,F_4,G_2$.

${\mathrm{\Phi }}_c$ is finite, contains the basis $\{a_s:s\in S\}$ of $V$ and hence spans $V$, omits $0$ because every $\beta =\rho (w)a_s$ has $B(\beta ,\beta )=B(a_s,a_s)=c_s^2\mathrm{\ne }0$, is closed under negation because $\rho (ws)a_s=-\rho (w)a_s$, and is $\rho (W)$-invariant because $\rho (w^{\mathrm{\prime }})\rho (w)a_s=\rho (w^{\mathrm{\prime }}w)a_s$.

For $\beta =\rho (w)a_s\in {\mathrm{\Phi }}_c$ the conjugation identity gives $r_{\beta }=\rho (w)r_s\rho (w{)}^{-1}=\rho (ws{w}^{-1})\in \rho (W)$, so every root reflection maps ${\mathrm{\Phi }}_c$ into itself and $W({\mathrm{\Phi }}_c)=⟨s_{\beta }:\beta \in {\mathrm{\Phi }}_c⟩\subseteq \rho (W)$; conversely $\rho (s)=r_s$ is the reflection $s_{a_s}$ in the root $a_s\in {\mathrm{\Phi }}_c$, so $\rho (W)\subseteq W({\mathrm{\Phi }}_c)$. Hence $W({\mathrm{\Phi }}_c)=\rho (W)$.

For all $\beta ,\alpha \in {\mathrm{\Phi }}_c$ one has $2B(\beta ,\alpha )\mathrm{/}B(\alpha ,\alpha )=B(\beta ,{\alpha }^{\vee })\in \mathbb{Z}$.

Let $S_1,\dots ,S_k$ be the components of $\mathrm{\Gamma }$ and $V_i=\mathrm{span}\{e_s:s\in S_i\}$, so $V=V_1\oplus \cdots \oplus V_k$; each generator $r_u$ fixes $e_v$ for $v$ outside the component of $u$, because $B(e_u,e_v)=0$ there by the vanishing criterion for $m=2$, and maps each $V_i$ into itself; hence every $\rho (w)$ preserves every $V_i$, and $a_s\in V_{S(s)}$.

Suppose $\mathrm{\Gamma }$ is connected, all edge labels lie in $\{3,4,6\}$, and $\{s,t\}$ is an edge of label $p\in \{4,6\}$. Then $\{s,t\}$ is the only edge of label $\ge 4$ and $\mathrm{\Gamma }$ contains no cycle; applying the tree construction with the root vertex on the $s$-side gives a crystallographic scaling $c$ with $c_t=2c_s\cos (\pi \mathrm{/}p)$, and applying it with the root on the $t$-side gives a crystallographic scaling $c^{\mathrm{\prime }}$ with $c_s^{\mathrm{\prime }}=2c_t^{\mathrm{\prime }}\cos (\pi \mathrm{/}p)$; these two scalings differ only by inverting the length ratio across $\{s,t\}$.

For every crystallographic scaling the conclusions of the last clause of the lemma hold: $r_s(a_t)=a_t-a_{ts}{a}_s$ and $r_s(a_t^{\vee })=a_t^{\vee }-a_{st}{a}_s^{\vee }$ with $r_s(Q)=Q$ and $r_s(Q^{\vee })=Q^{\vee }$; $Q$ and $Q^{\vee }$ are $\rho (W)$-stable free abelian groups of rank $\mathrm{\mid }S\mathrm{\mid }$; ${\mathrm{\Phi }}_c\subseteq Q$, $Q=\mathbb{Z}{\mathrm{\Phi }}_c$, $Q^{\vee }=\mathbb{Z}{\mathrm{\Phi }}_c^{\vee }$ with ${\mathrm{\Phi }}_c^{\vee }=\{2\beta \mathrm{/}B(\beta ,\beta ):\beta \in {\mathrm{\Phi }}_c\}$ and $Q\subseteq P$; every element of ${\mathrm{\Phi }}_c$ is an integral combination of the $a_s$ whose nonzero coefficients have one sign; and all pairings $B(\beta ,{\gamma }^{\vee })$ with $\beta ,\gamma \in {\mathrm{\Phi }}_c$ are integers.

Since $B$ is symmetric bilinear and positive definite, $(V,B)$ is a real inner product space, and for $\beta \in {\mathrm{\Phi }}_c$ one has $B(\beta ,\beta )\mathrm{\ne }0$ so that $r_{\beta }(x)=x-\frac{2B(x,\beta )}{B(\beta ,\beta )}\beta$ is the orthogonal reflection in $\beta$, coinciding for $\beta =a_s$ with the generator reflection $\rho (s)$ because $a_s=c_s{e}_s$ with $c_s>0$.

Let $\mathrm{\Psi }$ be a reduced crystallographic Euclidean root system in the real inner product space $E$ with base $\mathrm{\Delta }$, and let $(W,S)\cong (W(\mathrm{\Psi }),\{s_{\alpha }:\alpha \in \mathrm{\Delta }\})$ be an isomorphism of Coxeter systems; write ${\alpha }_s$ for the simple root corresponding to $s\in S$. Then $W(\mathrm{\Psi })$ is finite, so $W$ is finite and $B$ is positive definite. For distinct $s,t$ the roots ${\alpha }_s,{\alpha }_t$ are positive, hence nonproportional: ${\alpha }_s=\lambda {\alpha }_t$ with $\lambda >0$ forces $\lambda =1$ and ${\alpha }_s={\alpha }_t$ by reducedness, while $\lambda <0$ contradicts positivity. By the rank-two classification $({\alpha }_s,{\alpha }_t)\le 0$ and $n_{st}{n}_{ts}=4{\cos }^2\theta \in \{0,1,2,3\}$, where $n_{st}=2({\alpha }_t,{\alpha }_s)\mathrm{/}({\alpha }_s,{\alpha }_s)$ and $\theta$ is the angle between ${\alpha }_s$ and ${\alpha }_t$; hence $u:=\cos \theta$ satisfies $u\le 0$ and $u^2=n_{st}{n}_{ts}\mathrm{/}4\in \{0,1\mathrm{/}4,1\mathrm{/}2,3\mathrm{/}4\}$.

If every edge label of $\mathrm{\Gamma }$ lies in $\{3,4,6\}$, then by 1.2 every connected component of $\mathrm{\Gamma }$ is one of the Weyl-type diagrams $A_n,B_n,D_n,E_6,E_7,E_8,F_4,G_2$, each of which is a tree; so $\mathrm{\Gamma }$ is a forest with all edge labels in $\{3,4,6\}$ and the tree construction produces a crystallographic scaling of the geometry.

Assume $c$ crystallographic. If $\beta =\lambda \gamma$ with $\beta ,\gamma \in {\mathrm{\Phi }}_c$ and $\lambda >0$, then $2\lambda =B(\beta ,{\gamma }^{\vee })\in \mathbb{Z}$ and $2\mathrm{/}\lambda =B(\gamma ,{\beta }^{\vee })\in \mathbb{Z}$ by 1.5; writing $\lambda =p\mathrm{/}q$ in lowest terms, $q\mid 2$ and $p\mid 2$, so $\lambda \in \{1\mathrm{/}2,1,2\}$.

If $\beta =\rho (w)a_s$ and $\gamma =\rho (v)a_t$ are nonzero and proportional, then $s,t$ lie in one component by 1.6, and applying the ratio clause to that connected component gives ${\lambda }^2=B(\beta ,\beta )\mathrm{/}B(\gamma ,\gamma )=c_s^2\mathrm{/}{c}_t^2\in \{1,2,1\mathrm{/}2,3,1\mathrm{/}3\}$, since $\rho$ preserves $B$ and $B(a_s,a_s)=c_s^2$.

For the two scalings of 1.7 put ${\lambda }_{uv}:=c_u\mathrm{/}{c}_v$ and ${\lambda }_{uv}^{\mathrm{\prime }}:=c_u^{\mathrm{\prime }}\mathrm{/}{c}_v^{\mathrm{\prime }}$. Label-$3$ edges have equal lengths in both scalings, while the constructions invert the ratio across $\{s,t\}$, so ${\lambda }_{uv}^{\mathrm{\prime }}=1\mathrm{/}{\lambda }_{uv}$ for all $u,v$; since $a_{uv}=2{\lambda }_{uv}B(e_u,e_v)$ and $a_{uv}^{\mathrm{\prime }}=2{\lambda }_{uv}^{\mathrm{\prime }}B(e_u,e_v)$ one has $a_{uv}^{\mathrm{\prime }}=({\lambda }_{uv}^{\mathrm{\prime }}\mathrm{/}{\lambda }_{uv})a_{uv}=a_{uv}\mathrm{/}{\lambda }_{uv}^2$, and also $a_{vu}=2{\lambda }_{vu}B(e_v,e_u)=(1\mathrm{/}{\lambda }_{uv})\cdot 2B(e_u,e_v)=a_{uv}\mathrm{/}{\lambda }_{uv}^2$; hence $a_{uv}^{\mathrm{\prime }}=a_{vu}$ for all $u,v$, that is $A^{\mathrm{\prime }}=A^T$. These are the two dual length assignments, the $B_n\mathrm{/}{C}_n$ alternative on a label-$4$ path and the two orientations of $F_4$ and $G_2$, with the arrow pointing from the longer to the shorter root.

For a pair as in 1.10 put $P=\mathrm{span}({\alpha }_s,{\alpha }_t)$, $e_1={\alpha }_s\mathrm{/}\mathrm{\mid }{\alpha }_s\mathrm{\mid }$, $z={\alpha }_t-({\alpha }_t,e_1)e_1\mathrm{\ne }0$ and $e_2=z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$. Then $(e_1,e_2)$ is an orthonormal basis of $P$ and ${\alpha }_t=\mathrm{\mid }{\alpha }_t\mathrm{\mid }(u{e}_1+v{e}_2)$ with $u=\cos \theta$, $v=\mathrm{\mid }z\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\alpha }_t\mathrm{\mid }>0$ and $u^2+v^2=1$. The reflection formula gives $s_{{\alpha }_s}(e_1)=-e_1$, $s_{{\alpha }_s}(e_2)=e_2$, $s_{{\alpha }_t}(e_1)=-(2u^2-1)e_1-2uv{e}_2$ and $s_{{\alpha }_t}(e_2)=-2uv{e}_1+(2u^2-1)e_2$, so $R:=s_{{\alpha }_s}{s}_{{\alpha }_t}$ acts on $P$ by the matrix $\left(\begin{array}{cc}c& d\\ -d& c\end{array}\right)$ with $c=2u^2-1$, $d=2uv$, and fixes $P^{\perp }$ pointwise. Since $E=P\oplus P^{\perp }$, a power of $s_{{\alpha }_s}{s}_{{\alpha }_t}$ is the identity exactly when its restriction $R^k$ to $P$ is the identity. Products of such matrices add the pairs $(c,d)$ by $(c,d)(c^{\mathrm{\prime }},d^{\mathrm{\prime }})=(c{c}^{\mathrm{\prime }}-d{d}^{\mathrm{\prime }},c{d}^{\mathrm{\prime }}+d{c}^{\mathrm{\prime }})$, so by induction $R^k=\left(\begin{array}{cc}{C}_k& D_k\\ -D_k& C_k\end{array}\right)$ with $(C_1,D_1)=(c,d)$ and the same recursion; from $c^2+d^2=1$ one gets $C_2=c^2-d^2$, $D_2=2cd$, $C_3=4c^3-3c$ and $D_3=d(4c^2-1)$. Since $u\le 0$ and $u^2\in \{0,1\mathrm{/}4,1\mathrm{/}2,3\mathrm{/}4\}$, four cases occur: $u^2=0$ gives $c=-1$, $d=0$ and $R=-I$, of order $2$; $u^2=1\mathrm{/}4$ gives $c=-1\mathrm{/}2$, $d^2=3\mathrm{/}4$, hence $C_3=1$, $D_3=0$, so $R^3=I$ while $R\mathrm{\ne }I$ and $R^2\mathrm{\ne }I$, of order $3$; $u^2=1\mathrm{/}2$ gives $c=0$, $d^2=1$ and $R^2=-I$, of order $4$; and $u^2=3\mathrm{/}4$ gives $c=1\mathrm{/}2$, $d^2=3\mathrm{/}4$, hence $C_3=-1$, $D_3=0$, so $R^3=-I$ and $R^6=I$ while $R,R^2,R^3,R^4=-R,R^5=-R^2$ all differ from $I$, of order $6$.

Combining 1.1, 1.2 and 2.1: there exists a crystallographic scaling if and only if every edge label of $\mathrm{\Gamma }$ lies in $\{3,4,6\}$, if and only if every connected component of $\mathrm{\Gamma }$ is one of the Weyl types $A_n,B_n,D_n,E_6,E_7,E_8,F_4,G_2=I_2(6)$; in particular $H_3$, $H_4$ and $I_2(m)$ with $m\mathrm{\notin }\{2,3,4,6\}$ admit none, while $I_2(3)=A_2$, $I_2(4)=B_2$ and $I_2(6)=G_2$ do.

If $\beta =\lambda \gamma$ with $\beta ,\gamma \in {\mathrm{\Phi }}_c$ and $\lambda >0$, then 2.2 gives $\lambda \in \{1\mathrm{/}2,1,2\}$ and 2.3 gives ${\lambda }^2\in \{1,2,1\mathrm{/}2,3,1\mathrm{/}3\}$; hence $\lambda =1$ and $\beta =\gamma$.

By 2.5 the order of $s_{{\alpha }_s}{s}_{{\alpha }_t}$ lies in $\{2,3,4,6\}$ for every pair of distinct $s,t$; since $m(s,t)$ is the order of $st$ and the isomorphism carries $st$ to $s_{{\alpha }_s}{s}_{{\alpha }_t}$, the label $m(s,t)$ lies in $\{2,3,4,6\}$ whenever $s,t$ are distinct; in particular every edge label of $\mathrm{\Gamma }$ lies in $\{3,4,6\}$.

For every $\gamma \in {\mathrm{\Phi }}_c$ one has $\mathbb{R}\gamma \cap {\mathrm{\Phi }}_c=\{\gamma ,-\gamma \}$: if $\beta =\lambda \gamma \in {\mathrm{\Phi }}_c$ with $\lambda \mathrm{\ne }0$, then $-\gamma \in {\mathrm{\Phi }}_c$ by 1.3; if $\lambda >0$, step 3.2 gives $\beta =\gamma$, while if $\lambda <0$, applying step 3.2 to $\beta =(-\lambda )(-\gamma )$ gives $\beta =-\gamma$.

By 3.3 every edge label of $\mathrm{\Gamma }$ lies in $\{3,4,6\}$; since $W$ is finite, 1.2 now shows that every connected component of $\mathrm{\Gamma }$ is one of $A_n,B_n,D_n,E_6,E_7,E_8,F_4,G_2=I_2(6)$, so the type of $(W,S)$ is one of the types listed in 3.1, and all labels lie in $\{2,3,4,6\}$.

The $a_s$ are exactly the simple roots of the positive system of ${\mathrm{\Phi }}_c$ defined by a regular vector. By 1.8 every root is an integral combination ${\sum }_s{m}_s{a}_s$ whose nonzero coefficients have one sign, so $(V,{\mathrm{\Phi }}_c)$ satisfies the axioms of a reduced crystallographic Euclidean root system by 1.3, 1.4, 1.5 and 4.1. Because $B$ is positive definite, the map $v\mapsto (B(v,a_s){)}_s$ is injective on $V$ (a nonzero kernel vector would have $B(v,v)={\sum }_s{v}_{s}B(v,a_s)=0$) and hence an isomorphism onto ${\mathbb{R}}^S$ by [F38]; choose $v$ mapping to $(1,\dots ,1)$. Then $B(v,\beta )={\sum }_s{m}_{s}B(v,a_s)$ has the sign of the nonzero coefficients of $\beta$, so $v$ is regular and ${\mathrm{\Phi }}_c^{+}=\{\beta \in {\mathrm{\Phi }}_c:m_s\ge 0\}$, with simple roots ${\mathrm{\Delta }}_c$ by definition. Each $a_s$ is simple: a decomposition $a_s={\beta }^{\mathrm{\prime }}+{\gamma }^{\mathrm{\prime }}$ into positive roots would split the coordinate vector of $a_s$ into nonnegative integer coordinate vectors, forcing one summand to be $a_s$ and the other to be $0\mathrm{\notin }{\mathrm{\Phi }}_c$. By the basis theorem ${\mathrm{\Delta }}_c$ is a basis of $V$, so $\mathrm{\mid }{\mathrm{\Delta }}_c\mathrm{\mid }=\dim V=\mathrm{\mid }S\mathrm{\mid }=\mathrm{\mid }\{a_s:s\in S\}\mathrm{\mid }$; since $\{a_s\}\subseteq {\mathrm{\Delta }}_c$, equality ${\mathrm{\Delta }}_c=\{a_s:s\in S\}$ follows.

Therefore ${\mathrm{\Phi }}_c$ is a reduced crystallographic Euclidean root system in the inner product space $(V,B)$: it is finite, spans $V$ and omits $0$ (1.3), is closed under its root reflections (1.4), has integral Cartan integers (1.5) and is reduced (4.1). Its Weyl group is $W({\mathrm{\Phi }}_c)=\rho (W)$ (1.4), and $\rho :W\to W({\mathrm{\Phi }}_c)$ is an isomorphism because it is surjective by 1.4 and injective, carrying $s$ to $r_{a_s}$; the base is $\{a_s:s\in S\}$ (5.1), so the standard generators correspond to the reflections in a base, and by 3.1 every finite Coxeter system of the listed types is isomorphic to the Weyl group of such a system. Moreover the root, coroot and weight lattices of ${\mathrm{\Phi }}_c$ are the sets $Q,Q^{\vee },P$ of the scaling, its coroots are ${\alpha }^{\vee }=2\alpha \mathrm{/}B(\alpha ,\alpha )$, and its Cartan matrix relative to the base $\{a_s\}$ has entries $(a_t,a_s^{\vee })=a_{ts}$, the transpose of $A$.

All four clauses are established: (1) by 1.1, 2.1 and 3.1; (2) by 6.1 and 4.2; (3) by 1.8; and (4) by 1.7 and 2.4. No axiom of Choice is used. The construction in 2.1 selects a root vertex from each of the finitely many components of a finite forest; this finite selection follows by induction on the number of components, and no other non-unique selection is used.