Coxeter-Dynkin diagram

English

Some Coxeter-Dynkin diagrams

Etymology

After mathematicians H. S. M. Coxeter and Eugene Dynkin.

Noun

Coxeter-Dynkin diagram (plural Coxeter-Dynkin diagrams)

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

   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.

   • 1995 June, R. V. Moody, J. Patera, “Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices”, in Canadian Journal of Mathematics, page 597:

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

   • 2000, Andrei Gabrielov, “Coxeter-Dynkin diagrams and singularities”, in Evgeniĭ Borisovich Dynkin, A. A. Yushkevich, Gary M. Seitz, A. L. Onishchik, editors, Selected Papers of E. B. Dynkin with Commentary, page 367:

     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.

   • 2012, Igor V. Dolgachev, Classical Algebraic Geometry: A Modern View‎^[1], page 363:

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

Synonyms

• (graph with numerically labelled edges): Coxeter diagram, Coxeter graph

See also

• Dynkin diagram