In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge), that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3 (60 degrees), and each pair of nodes that is not connected by a branch at all (such as non-adjacent nodes) represents a pair of mirrors at order-2 (90 degrees).
Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.
Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.
Description
Branches of a Coxeter–Dynkin diagram are labeled with a rational number p, representing a dihedral angle of 180°/p. When the angle is 90° and the mirrors have no interaction, so the branch can be omitted from the diagram. If a branch is unlabeled, it is assumed to have , representing an angle of 60°. Two parallel mirrors have a branch marked with "∞". In principle, n mirrors can be represented by a complete graph in which all n( branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, so the corresponding branches are omitted.
Diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes and regular honeycombs. Plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs.
Schläfli matrix
Every Coxeter diagram has a corresponding Schläfli matrix
(so named after Ludwig Schläfli), with matrix elements where p is the branch order between the pairs of mirrors. As a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the limited cases of p = 2,3,4, and 6, which are not symmetric in general.
The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite (positive), affine (zero), indefinite (negative). This rule is called Schläfli's Criterion.
The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of finite type (all positive), affine type (all non-negative, at least one is zero), or indefinite type (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use the following definition: A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite (i.e. have positive determinants), and paracompact if all its subgroups are finite or affine (i.e. have nonnegative determinants).
Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact hyperbolic groups in 1950, and Koszul (or quasi-Lannér) for the paracompact groups.
Rank 2 Coxeter groups
For rank 2, the type of a Coxeter group is fully determined by the determinant of the Schläfli matrix, as it is simply the product of the eigenvalues: Finite type (positive determinant), affine type (zero determinant) or hyperbolic (negative determinant). Coxeter uses an equivalent bracket notation which lists sequences of branch orders as a substitute for the node-branch graphic diagrams. Rational solutions [p/q], , also exist, with gcd(p,q)=1, which define overlapping fundamental domains. For example, 3/2, 4/3, 5/2, 5/3, 5/4, and 6/5.