Meaning of Coxeter-Dynkin diagram | Babel Free

Definitions

A graph with numerically labelled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

Examples

“A Coxeter-Dynkin diagram encodes the information in a Coxeter matrix, which in turn⟳ encodes the presentation of a Coxeter group.”

“Vertices of a Coxeter-Dynkin diagram represent⟳ generators of the corresponding Coxeter group. The generators (algebraic) in turn⟳ correspond to the reflecting hyperplanes (geometric). A pair of vertices which are not linked by an edge correspond to a pair of commuting generators. An unnumbered edge between a pair of vertices means that the braid relation between the corresponding generators has length three (e.g., aba = bab if the generators are a and b). An edge numbered ≥4 means that the braid relation between the corresponding generators has a length equal to that number⟳. For example, if the edge is numbered 4 then the braid relation is cdcd = dcdc if the generators are c and d. If a set⟳ of edges form⟳ a cycle then the Coxeter group can be shown to be infinite. If a tree in a Coxeter-Dynkin diagram has more than one numbered edge then the Coxeter group can be shown to be infinite. There are a few more such rules, which ensure that finite Coxeter groups have⟳ Coxeter-Dynkin diagrams with relatively simple shapes.”

“Let⟳ 𝒬 be an indecomposable root lattice and let⟳ Γ denote the Coxeter-Dynkin diagram of the underlying root system Δ.”

“There is a deep and only partially understood connection between the classification and structure⟳ of singularities and the Coxeter-Dynkin diagrams introduced by H. S .M. Coxeter for classification of reflection-generated groups and by E. B. Dynkin for classification of semisimple Lie⟳ algebras.”

“For 3 ≤ n ≤ 5, we will use⟳ Eₙ to denote the Coxeter-Dynkin diagrams of types A₁ + A₂(N = 3), A₄(N = 4) and D₅(N = 5).”

CEFR level