Crystallographic Group

Crystallographic Group

A subgroup $G$ of $\mathrm{GL}(V)$ is said to be crystallographic if it stabilizes a lattice $L$ in $V$ (the $\mathbf{Z}$-span of a basis of $V$): $gL\subset L$ for all $g\in G$. (Since $G$ is a group, it is automatic that $gL=L$.)

The name comes from low-dimensional crystallography, where the classification of possible crystal structures depends heavily on the available symmetry groups. It turns out that ‘most’ finite reflection groups are crystallographic.

Proposition

If $W$ is crystallographic, then each integer $m(\alpha ,\beta )$ must be $2,3,4,$ or $6$ when $\alpha \ne \beta$ in $\Delta$.

Proof If $\alpha \ne \beta$, we know that $s_{\alpha }{s}_{\beta }\ne 1$ acts on the plane spanned by $\alpha$ and $\beta$ as a rotation through the angle $\theta :=2\pi /m(\alpha ,\beta )$, while fixing the orthogonal complement pointwise. Thus its trace relative to a compatible choice of basis for $V$ is $(n-2)+2\cos \theta$ ($n=\dim V$). So $\cos \theta$ must be a half-integer, while $0<\theta \le \pi$. The only possibilities are $\cos \theta =-1,-1/2,0,1/2$, corresponding to the cases $m(\alpha ,\beta )=2,3,4,6$. $\square$

Crystallographic Root System

We say that a root system $\Phi$ is if it satisfies the additional requirement: $\frac{2(\alpha ,\beta )}{(\beta ,\beta )}\in \mathbf{Z}\text{ for all }\alpha ,\beta \in \Phi .$ These integers are called Cartan integers. It is actually enough to require that the ratios be integers when $\alpha ,\beta \in \Delta$. The group $W$ generated by all reflections $s_{\alpha }$ ($\alpha \in \Phi$) is known as the Weyl group of $\Phi$.

Coroots

Setting ${\alpha }^{\vee }:=2\alpha /(\alpha ,\alpha )$, the set ${\Phi }^{\vee }$ of all coroots ${\alpha }^{\vee }$ ($\alpha \in \Phi$) is also a crystallographic root system in $V$, with simple system ${\Delta }^{\vee }:=\{{\alpha }^{\vee }\mid \alpha \in \Delta \}$. It is called the inverse or dual root system. The Weyl group of ${\Phi }^{\vee }$ is $W$, with $w{\alpha }^{\vee }=w(\alpha {)}^{\vee }$.

Note: In most cases ${\Phi }^{\vee }$ is isomorphic to $\Phi$; however, the root systems of types $B_n$ and $C_n$ are dual to each other. Short roots $\alpha$ in a system $\Phi$ of type $B_n$ give rise to long roots ${\alpha }^{\vee }$ in the system ${\Phi }^{\vee }$ of type $C_n$ (and vice versa).

Root Lattice & Coroot Lattice

The $\mathbf{Z}$-span $L(\Phi )$ of $\Phi$ in $V$ is called the root lattice; it is a lattice in the subspace of $V$ spanned by $\Phi$, which we can usually assume to be $V$ itself. Similarly, we define the coroot lattice $L({\Phi }^{\vee })$. Both lattices are $W$-stable.

Weight Lattice & Coweight Lattice $\hat{L}(\Phi ):=\{\lambda \in V\mid (\lambda ,{\alpha }^{\vee })\in \mathbf{Z}\text{ for all }\alpha \in \Phi \},$ and the coweight lattice $\hat{L}({\Phi }^{\vee }):=\{\lambda \in V\mid (\lambda ,\alpha )\in \mathbf{Z}\text{ for all }\alpha \in \Phi \}.$

Define the weight lattice

Then $\hat{L}(\Phi )$ contains $L(\Phi )$ as a subgroup of finite index $f$, and similarly $\hat{L}({\Phi }^{\vee })$ contains $L({\Phi }^{\vee })$ as a subgroup of index $f$. Here $f$ is just the determinant of the matrix of Cartan integers $(\alpha ,{\beta }^{\vee })$ ($\alpha ,\beta \in \Delta$). (It is called the index of connection in Lie theory: $\hat{L}/L$ is isomorphic to the fundamental group of a compact Lie group of adjoint type having $W$ as Weyl group; so $f$ is the order of the kernel of the associated map from the simply connected covering group.)