Crystallographic Root Lattices and Weyl Group Interfaces

Let $S$ be a finite set, let $m$ be a Coxeter matrix on $S$, let $W$ be the presented Coxeter group with its universal property (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), let $V={\mathbb{R}}^S$ be the real vector space with its basis $(e_s{)}_{s\in S}$, let $B$ be the Coxeter form and let $\rho :W\to \mathrm{G}\mathrm{L}(V)$ be the canonical reflection homomorphism with its reflections $r_a$ and root system $\mathrm{\Phi }$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, The canonical reflection homomorphism, roots, reflections, and the positive cone). No definiteness or nondegeneracy of $B$ is assumed.

A scaling of this geometry is a family $c=(c_s{)}_{s\in S}$ of positive real numbers; with $a_s:=c_s{e}_s$ define $a_s^{\vee }:=\frac{2a_s}{B(a_s,a_s)}=\frac{2e_s}{c_s},a_{st}:=B(a_s,a_t^{\vee })=\frac{2B(a_s,a_t)}{B(a_t,a_t)}(s,t\in S)\mathrm{.}$

Since $c_s>0$ and $(e_s{)}_{s\in S}$ is a basis, the $a_s$ form a basis of $V$ and $B(a_s,a_s)=c_s^{2}B(e_s,e_s)=c_s^2\mathrm{\ne }0$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis); hence each $a_s^{\vee }$ is defined, and $a_s^{\vee }=\frac{2c_s{e}_s}{c_s^2}=\frac{2e_s}{c_s}$. The reflection $r_{a_s}$ of The real Coxeter form, its radical, reflections, and form-preserving maps is the generator reflection $r_{e_s}$ of the geometry, because the reflection formula depends only on the line spanned by the normal: for $\lambda \mathrm{\ne }0$ and $B(a,a)\mathrm{\ne }0$ substitution gives $r_{\lambda a}(v)=v-\frac{2B(v,\lambda a)}{B(\lambda a,\lambda a)}\lambda a=v-\frac{2B(v,a)}{B(a,a)}a=r_a(v)$, so $r_{\lambda a}=r_a$, and $a_s=c_s{e}_s$ with $c_s>0$. The number $a_{st}\in \mathbb{R}$ is the Cartan number of the ordered pair $(s,t)$, and $a_{ss}=B(a_s,a_s^{\vee })=B(a_s,\frac{2a_s}{B(a_s,a_s)})=2$.

The scaling $c$ is crystallographic when $a_{st}\in \mathbb{Z}$ for all $s,t\in S$, equivalently when $B(a_t,a_s^{\vee })\in \mathbb{Z}$ for all $s,t\in S$. In that case define the root lattice, coroot lattice and weight lattice of the scaling by $Q:={\sum }_{s\in S}\mathbb{Z}{a}_s,Q^{\vee }:={\sum }_{s\in S}\mathbb{Z}{a}_s^{\vee },P:=\{\lambda \in V:B(\lambda ,q^{\vee })\in \mathbb{Z}\text{ for all }{q}^{\vee }\in Q^{\vee }\},$ the scaled root set ${\mathrm{\Phi }}_c:=\{\rho (w)a_s:w\in W,s\in S\}$, and the scaled Cartan matrix $A:=(a_{st}{)}_{s,t\in S}$ of $c$.

Well-definedness, lattice provisos, and the interface with the published lattices

$a_s=c_s{e}_s$ and $a_s^{\vee }=2e_s\mathrm{/}{c}_s$ are nonzero scalar multiples of the basis vectors $e_s$, so both $(a_s{)}_{s\in S}$ and $(a_s^{\vee }{)}_{s\in S}$ are bases of $V$. Thus $Q$ and $Q^{\vee }$ are free abelian subgroups of rank $\mathrm{\mid }S\mathrm{\mid }$. The set $P$ is an additive subgroup, since each condition $B(\lambda ,q^{\vee })\in \mathbb{Z}$ is preserved by addition and negation. If $q={\sum }_{s\in S}{m}_s{a}_s\in Q$ and $q^{\vee }={\sum }_{t\in S}{n}_t{a}_t^{\vee }\in Q^{\vee }$ with $m_s,n_t\in \mathbb{Z}$, then $B(q,q^{\vee })={\sum }_{s,t\in S}{m}_s{n}_{t}B(a_s,a_t^{\vee })={\sum }_{s,t\in S}{m}_s{n}_t{a}_{st}\in \mathbb{Z}$ in the crystallographic case, so $Q\subseteq P$.

Here a lattice in $V$ means a discrete subgroup whose real span is $V$; this convention includes the rank-zero lattice $\{0\}$ when $V=0$. The map $\phi :V⟶{\mathbb{R}}^S,\phi (\lambda ):=(B(\lambda ,a_s^{\vee }){)}_{s\in S},$ is linear. Because $(a_s^{\vee }{)}_{s\in S}$ is a basis and $B$ is symmetric, $\ker \phi =\mathrm{rad}(B)$: vanishing against each $a_s^{\vee }$ is equivalent by linearity to vanishing against every vector of $V$. Hence $\phi$ is injective exactly when $B$ is nondegenerate; both its domain and codomain have dimension $\mathrm{\mid }S\mathrm{\mid }$, so the rank-nullity theorem (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$) makes this equivalent to $\phi$ being an isomorphism. If $\phi$ is an isomorphism then $P={\phi }^{-1}({\mathbb{Z}}^S)$ is the $\mathbb{Z}$-span of the real basis $({\omega }_s{)}_{s\in S}$ characterized by $B({\omega }_s,a_t^{\vee })={\delta }_{st}$, so it is a lattice. If $B$ is degenerate then $\mathrm{rad}(B)\subseteq P$, so $P$ contains a nonzero linear subspace and is not discrete. For instance for $S=\{s,t\}$, $m(s,t)=\mathrm{\infty }$, $c_s=c_t=1$ one has $B(e_s,e_t)=-1$ and $P=\{\lambda :{\lambda }_s-{\lambda }_t\in \frac{1}{2}\mathbb{Z}\}$, a union of parallel lines. In particular $P$ is a lattice in the positive definite setting of (2) of Cartan-number products, allowed edge labels, tree scalings and reflection stability and of Crystallographic finite type: the Weyl types, reduced realizations and lattice stability ↗; in the degenerate range the term "weight lattice" names $P$ without a discreteness claim.

When $B$ is positive definite and ${\mathrm{\Phi }}_c$ has been proved to be a reduced crystallographic Euclidean root system with base $\{a_s:s\in S\}$ (Crystallographic finite type: the Weyl types, reduced realizations and lattice stability ↗ (2)), the sets $Q$, $Q^{\vee }$ and $P$ are exactly the root lattice, coroot lattice and weight lattice of that root system in the sense of Root, coroot, weight, and coweight lattices, and the elements $a_s^{\vee }=\frac{2a_s}{B(a_s,a_s)}$ are its simple coroots in the sense of Coroot and dual root system. The definition is deliberately stated before that identification is available: it is a property declaration for the pair (geometry, scaling), and the paragraphs above justify only its own well-definedness.

This item asserts no existence of a crystallographic scaling, and in particular makes no claim about the non-crystallographic finite types $H_3$, $H_4$, or $I_2(m)$ with $m\mathrm{\notin }\{2,3,4,6\}$. It also does not assert that ${\mathrm{\Phi }}_c$ is a root system, that $Q=\mathbb{Z}{\mathrm{\Phi }}_c$, or that any pairing $B(\beta ,{\gamma }^{\vee })$ of non-simple elements of ${\mathrm{\Phi }}_c$ is an integer: for a crystallographic scaling these facts are proved in Cartan-number products, allowed edge labels, tree scalings and reflection stability and Crystallographic finite type: the Weyl types, reduced realizations and lattice stability ↗. No choice principle is used: $S$ is finite and every object above is defined from the given finite data.

Let $S$ be a finite set, $m$ a Coxeter matrix, $W$ the presented group, $V={\mathbb{R}}^S$, $B$ the Coxeter form, $\rho$ the canonical reflection homomorphism and $\mathrm{\Gamma }$ the diagram (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Coxeter diagrams: edges, labels, components and finite type), and let $c$ be a scaling with scaled simple roots $a_s$, coroots $a_s^{\vee }$ and Cartan numbers $a_{st}=B(a_s,a_t^{\vee })$ (Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices).

(1) Products. $a_{ss}=2$ for every $s$; for distinct $s,t$ with $m(s,t)<\mathrm{\infty }$, $a_{st}=-2\frac{c_s}{c_t}\cos \frac{\pi }{m(s,t)}\le 0,a_{st}{a}_{ts}=4{\cos }^2\frac{\pi }{m(s,t)},$ and for $m(s,t)=\mathrm{\infty }$ one has $a_{st}=-2c_s\mathrm{/}{c}_t$ and $a_{st}{a}_{ts}=4$. Moreover $a_{st}=0$ if and only if $m(s,t)=2$.

(2) Allowed labels and length ratios. Assume that $B$ is positive definite and that $c$ is crystallographic. Then for all distinct $s,t$ $0\le a_{st}{a}_{ts}<4,\text{so}{a}_{st}{a}_{ts}\in \{0,1,2,3\},$ and $m(s,t)\in \{2,3,4,6\}$, according to $a_{st}{a}_{ts}=4{\cos }^2(\pi \mathrm{/}m(s,t))=0,1,2,3$. If $\mathrm{\Gamma }$ is connected and has an edge of label $4$ (respectively $6$), then that is its only edge of label $\ge 4$, and $c_s^2\mathrm{/}{c}_t^2\in \{1,2,2^{-1}\}$ (respectively $\{1,3,3^{-1}\}$) for all $s,t\in S$; if $\mathrm{\Gamma }$ has no edge of label $\ge 4$, then $c_s=c_t$ for all $s,t$ in the same connected component.

(3) Realizations on trees. Let $\mathrm{\Gamma }$ be a forest (disjoint union of trees) all of whose edge labels lie in $\{3,4,6\}$. Choose a root vertex in each component, set $c_{s_0}:=1$ at each root, and for every edge $\{s,t\}$ with $s$ on the root side and $t$ the other endpoint set $c_t:=2c_s\cos (\pi \mathrm{/}m(s,t))$. Then $c$ is positive and crystallographic: on every edge $\{s,t\}$ with $s$ the root-side endpoint, $a_{st}=-1,a_{ts}=-4{\cos }^2\frac{\pi }{m(s,t)}\in \{-1,-2,-3\},$ while $a_{st}=0$ for non-adjacent $s,t$ and $a_{ss}=2$.

(4) Lattices, integrality and stability. Assume that $c$ is crystallographic. Then for all $s,t$ $r_s(a_t)=a_t-a_{ts}{a}_s,r_s(a_t^{\vee })=a_t^{\vee }-a_{st}{a}_s^{\vee },$ hence $r_s(Q)=Q$ and $r_s(Q^{\vee })=Q^{\vee }$. Consequently $Q$ and $Q^{\vee }$ are $\rho (W)$-stable lattices of rank $\mathrm{\mid }S\mathrm{\mid }$, ${\mathrm{\Phi }}_c\subseteq Q$, $Q=\mathbb{Z}{\mathrm{\Phi }}_c$, $Q^{\vee }=\mathbb{Z}{\mathrm{\Phi }}_c^{\vee }$, and $Q\subseteq P$. Here, for $\beta =\rho (w)a_s\in {\mathrm{\Phi }}_c$, write ${\beta }^{\vee }:=2\beta \mathrm{/}B(\beta ,\beta )$; this is defined because $\rho$ preserves $B$ and $B(a_s,a_s)=c_s^2>0$, and ${\mathrm{\Phi }}_c^{\vee }:=\{{\beta }^{\vee }:\beta \in {\mathrm{\Phi }}_c\}$. Moreover every element of ${\mathrm{\Phi }}_c$ is an integral linear combination of the $a_s$ whose nonzero coefficients all have the same sign, and $B(\beta ,{\gamma }^{\vee })\in \mathbb{Z}$ for all $\beta ,\gamma \in {\mathrm{\Phi }}_c$.

No Axiom of Choice is used. The forest in step 1.5 is finite, so its components form a finite family; choosing one vertex from each nonempty component is finite choice, provable by induction on the number of components. Every path and sum used in the proof is finite, and no arbitrary-index selection is made.

Let $S$ be a finite set, $m$ a Coxeter matrix, $W$ the presented group, $V={\mathbb{R}}^S$, $B$ the Coxeter form, $\rho$ the canonical reflection homomorphism and $\mathrm{\Gamma }$ the diagram, with the scaled data and Cartan numbers $a_{st}$ of Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices. Assume that $W$ is finite, equivalently that $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite).

(1) Criterion. 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 of type $A_n$ ($n\ge 1$), $B_n$ ($n\ge 2$), $D_n$ ($n\ge 4$), $E_6$, $E_7$, $E_8$, $F_4$, or $G_2=I_2(6)$. In particular the finite types $H_3$, $H_4$ and $I_2(m)$ with $m\mathrm{\notin }\{2,3,4,6\}$ admit no crystallographic scaling.

(2) Reduced realizations and Weyl groups. If $c$ is a crystallographic scaling with scaled root set ${\mathrm{\Phi }}_c$, then ${\mathrm{\Phi }}_c$ is a reduced crystallographic Euclidean root system in the inner product space $(V,B)$ (Reduced crystallographic Euclidean root system); its Weyl group $W({\mathrm{\Phi }}_c)=⟨s_{\beta }:\beta \in {\mathrm{\Phi }}_c⟩$ (Weyl group) equals $\rho (W)$, and $\rho$ is an isomorphism $W\to W({\mathrm{\Phi }}_c)$ carrying $s$ to the reflection $r_{a_s}$ in $a_s$. Consequently every finite Coxeter system of one of the types listed in (1) is isomorphic to the Weyl group of a reduced crystallographic Euclidean root system, with the standard generators corresponding to the reflections in a base. No other finite Coxeter system has this property: if $(W,S)$ is isomorphic to $(W(\mathrm{\Psi }),\{s_{\alpha }:\alpha \in \mathrm{\Delta }\})$ for a reduced crystallographic Euclidean root system $\mathrm{\Psi }$ with base $\mathrm{\Delta }$, then all labels of $\mathrm{\Gamma }$ lie in $\{2,3,4,6\}$ and the type is one of those listed in (1).

(3) Lattice stability. For every crystallographic scaling the lattices $Q=\mathbb{Z}{\mathrm{\Phi }}_c$ and $Q^{\vee }=\mathbb{Z}{\mathrm{\Phi }}_c^{\vee }$ are $\rho (W)$-stable of rank $\mathrm{\mid }S\mathrm{\mid }$ and $Q\subseteq P$; every root of ${\mathrm{\Phi }}_c$ is an integral combination of the scaled simple roots $a_s$ with coefficients of one sign, and all pairings $B(\beta ,{\gamma }^{\vee })$, $\beta ,\gamma \in {\mathrm{\Phi }}_c$, are integers.

(4) Dual length choices. Suppose $\mathrm{\Gamma }$ is connected, has edge labels in $\{3,4,6\}$ and has an edge of label $p\in \{4,6\}$. Then that is its only edge of label $\ge 4$, and the two scalings that differ only by inverting the length ratio across it (with $c_t=2c_s\cos (\pi \mathrm{/}p)$ at one end versus $c_s=2c_t\cos (\pi \mathrm{/}p)$, all other edge ratios as in Cartan-number products, allowed edge labels, tree scalings and reflection stability (3)) are both crystallographic and have mutually transposed scaled Cartan matrices $A^{\mathrm{\prime }}=A^T$. These are the two dual length assignments of the diagram: the $B_n\mathrm{/}{C}_n$ alternative for a label-$4$ path, and the two $F_4$ and $G_2$ orientations; the companion examples page verifies the identification explicitly for $B_2\mathrm{/}{C}_2$ and for $G_2$.