CoxeterDynkin

The term CoxeterDynkin refers to the Coxeter–Dynkin diagram, a combinatorial tool used in the study of reflection groups, root systems, and Lie algebras. It encodes the relations of a set of generating reflections in a Coxeter group by drawing vertices for each generator and connecting them with edges weighted by the order of the product of the corresponding reflections. An edge omitted between two vertices indicates that the associated reflections commute, while a single edge indicates a product of order three, a double edge a product of order four, and so on. A triple edge denotes order six, and an unweighted edge may be labeled with 2 for clarity when necessary. In more advanced contexts the diagram can be enhanced by labeling vertices with superscripts indicating the type of simple root in the corresponding root system.

Coxeter–Dynkin diagrams appear in the classification of finite Coxeter groups, crystallographic root systems, and simple Lie algebras. For example, the diagram A_n consists of n vertices in a straight line connected by single edges and represents the symmetric group S_{n+1}. The diagram D_n has a branching at one end, reflecting the symmetry of the corresponding root system. Similarly, the exceptional diagrams E_6, E_7, and E_8 encode the exceptional Lie algebras of these types. Affine extensions of these diagrams, such as \tilde{A}_n or \tilde{D}_n, correspond to infinite Coxeter groups and to the extended Dynkin diagrams used in the theory of affine Lie algebras.

In geometry, the Coxeter–Dynkin diagram describes regular polytopes and tessellations. For example, the diagram for a regular honeycomb in three-dimensional hyperbolic space can be read off from its diagram. In physics, the diagrams are used to analyze gauge symmetry breaking patterns in grand unified theories and in string theory compactifications. The compact notation of Coxeter–Dynkin diagrams allows mathematicians and physicists to classify structures that would otherwise require extensive algebraic data, making it a foundational element in group theory, geometry, and theoretical physics.