Coxeter-Dynkin diagram

plural Coxeter-Dynkin diagrams

1. algebraA 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.

   4 more examples

   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.

   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

   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.

   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

   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).

   2012, Igor V. Dolgachev, Classical Algebraic Geometry: A Modern View, page 363

   Synonyms: Coxeter diagram, Coxeter graph