Coxeter–Dynkin diagram
Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.
Since the Coxeter matrix is symmetric, it can be viewed as the adjacency matrix of an edge-labeled graph that has vertices corresponding to the generators ri, and edges labeled with mi,j between the vertices corresponding to ri and rj.
Every Coxeter diagram has a corresponding Schläfli matrix (so named after Ludwig Schläfli), A, with matrix elements ai,j = aj,i = −2 cos(π/pi,j) where pi,j is the branch order between mirrors i and j; that is, π/pi,j is the dihedral angle between mirrors i and j.
All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized.
A is closely related to the Cartan matrix, used in the similar but directed graph: the Dynkin diagram, in the limited cases of p = 2,3,4, and 6, which are generally not symmetric.
The determinant of the Schläfli matrix is called the Schläflian;[citation needed] the Schläflian and its sign determine whether the group is finite (positive), affine (zero), or indefinite (negative).
We use the following definitions: Finite and affine groups are also called elliptical and parabolic respectively.
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; these define overlapping fundamental domains.
A mirror represents a hyperplane within a spherical, Euclidean, or hyperbolic space of given dimension.
For each, the Coxeter diagram can be deduced by identifying the hyperplane mirrors and labelling their connectivity, ignoring 90-degree dihedral angles (order 2; see footnote [a] below).
Here, domain vertices are labeled as graph branches 1, 2, etc., and are colored by their reflection order (connectivity).
Faces are generated by the repeated reflection of an edge eventually wrapping around to the original generator; the final shape, as well as any higher-dimensional facets, are likewise created by the face being reflected to enclose an area.
An unconnected diagram (subgroups separated by order-2 branches, or orthogonal mirrors) requires at least one active node in each subgraph.
All regular polytopes, represented by Schläfli symbol {p, q, r, ...}, can have their fundamental domains represented by a set of n mirrors with a related Coxeter–Dynkin diagram of a line of nodes and branches labeled by p, q, r, ..., with the first node ringed.
Uniform polytopes with one ring correspond to generator points at the corners of the fundamental domain simplex.
The special case of uniform polytopes with non-reflectional symmetry is represented by a secondary markup where the central dot of a ringed node is removed (called a hole).
These shapes are alternations of polytopes with reflective symmetry, implying that every other vertex is deleted.
There are 7 convex uniform polyhedra that can be constructed from this symmetry group and 3 from its alternation subsymmetries, each with a uniquely marked up Coxeter–Dynkin diagram.
The Wythoff symbol represents a special case of the Coxeter diagram for rank 3 graphs, with all 3 branch orders named, rather than suppressing the order 2 branches.
The Wythoff symbol is able to handle the snub form, but not general alternations without all nodes ringed.
The same constructions can be made on disjointed (orthogonal) Coxeter groups like the uniform prisms, and can be seen more clearly as tilings of dihedrons and hosohedra on the sphere, like this [6]×[] or [6,2] family: In comparison, the [6,3], family produces a parallel set of 7 uniform tilings of the Euclidean plane, and their dual tilings.
These extensions are usually marked by an exponent of 1,2, or 3 + symbols for the number of extended nodes.
The extending process can define a limited series of Coxeter graphs that progress from finite to affine to hyperbolic to Lorentzian.
The determinant of the Cartan matrices determine where the series changes from finite (positive) to affine (zero) to hyperbolic (negative), and ending as a Lorentzian group, containing at least one hyperbolic subgroup.
The determinant of the Schläfli matrix by rank are:[6] Determinants of the Schläfli matrix in exceptional series are: A (simply-laced) Coxeter–Dynkin diagram (finite, affine, or hyperbolic) that has a symmetry (satisfying one condition, below) can be quotiented by the symmetry, yielding a new, generally multiply laced diagram, with the process called "folding".
[10] Geometrically this corresponds to orthogonal projections of uniform polytopes and tessellations.
Coxeter–Dynkin diagrams have been extended to complex space, Cn where nodes are unitary reflections of period greater than 2.
Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed.
by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection.
