Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices

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.