Dynkin Diagram - Connection With Coxeter Diagrams

Connection With Coxeter Diagrams

Dynkin diagrams are closely related to Coxeter diagrams of finite Coxeter groups, and the terminology is often conflated.

Dynkin diagrams differ from Coxeter diagrams of finite groups in two important respects:

Partly directed

Dynkin diagrams are partly directed – any multiple edge (in Coxeter terms, labeled with "4" or above) has a direction (an arrow pointing from one node to the other); thus Dynkin diagrams have more data than the underlying Coxeter diagram (undirected graph).

At the level of root systems the direction corresponds to pointing towards the shorter vector; edges labeled "3" have no direction because the corresponding vectors must have equal length. (Caution: Some authors reverse this convention, with the arrow pointing towards the longer vector.)

Crystallographic restriction

Dynkin diagrams must satisfy an additional restriction, namely that the only allowable edge labels are 2, 3, 4, and 6, a restriction not shared by Coxeter diagrams, so not every Coxeter diagram of a finite group comes from a Dynkin diagram.

At the level of root systems this corresponds to the crystallographic restriction theorem, as the roots form a lattice.

A further difference, which is only stylistic, is that Dynkin diagrams are conventionally drawn with double or triple edges between nodes (for p = 4, 6), rather than an edge labeled with "p".

The term "Dynkin diagram" at times refers to the directed graph, at times to the undirected graph. For precision, in this article "Dynkin diagram" will mean directed, and the underlying undirected graph will be called an "undirected Dynkin diagram". Then Dynkin diagrams and Coxeter diagrams may be related as follows:

By this is meant that Coxeter diagrams of finite groups correspond to point groups generated by reflections, while Dynkin diagrams must satisfy an additional restriction corresponding to the crystallographic restriction theorem, and that Coxeter diagrams are undirected, while Dynkin diagrams are (partly) directed.

The corresponding mathematical objects classified by the diagrams are:

The blank in the upper right, corresponding to directed graphs with underlying undirected graph any Coxeter diagram (of a finite group), can be defined formally, but is little-discussed, and does not appear to admit a simple interpretation in terms of mathematical objects of interest.

There are natural maps down – from Dynkin diagrams to undirected Dynkin diagrams; respectively, from root systems to the associated Weyl groups – and right – from undirected Dynkin diagrams to Coxeter diagrams; respectively from Weyl groups to finite Coxeter groups.

The down map is onto (by definition) but not one-to-one, as the B_n and C_n diagrams map to the same undirected diagram, with the resulting Coxeter diagram and Weyl group thus sometimes denoted BC_n.

The right map is simply an inclusion – undirected Dynkin diagrams are special cases of Coxeter diagrams, and Weyl groups are special cases of finite Coxeter groups – and is not onto, as not every Coxeter diagram is an undirected Dynkin diagram (the missed diagrams being H₃, H₄ and I₂(p) for p = 5 p ≥ 7), and correspondingly not every finite Coxeter group is a Weyl group.