Coxeter-Dynkin diagram

English right|thumb|Some Coxeter-Dynkin diagrams Etymology After mathematicians Harold Scott MacDonald Coxeter H. S. M. Coxeter and Eugene Dynkin. Noun en geometry algebra A graph with numerically labelled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). en 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. en 1995 June R. V. Moody; J. Patera Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices w:Canadian Journal of Mathematics https://books.google.com.au/books?id=KoFIV53fTmkC&pg=PA597&dq=%22Coxeter-Dynkin+diagram%22%7C%22Coxeter-Dynkin+diagrams%22&hl=en&sa=X&ved=0ahUKEwiNseTziq_NAhXk34MKHaCkCu4Q6AEIUzAK#v=onepage&q=%22Coxeter-Dynkin%20diagram%22%7C%22Coxeter-Dynkin%20diagrams%22&f=false 597 Let 𝒬 be an indecomposable root lattice and let Γ denote the Coxeter-Dynkin diagram of the underlying root system Δ. en 2000 Andrei Gabrielov Coxeter-Dynkin diagrams and singularities Eugene Dynkin Evgeniĭ Borisovich Dynkin; A. A. Yushkevich; Gary M. Seitz; A. L. Onishchik Selected Papers of E. B. Dynkin with Commentary https://books.google.com.au/books?id=D9ZF5O_JH2gC&pg=PA367&dq=%22Coxeter-Dynkin+diagram%22%7C%22Coxeter-Dynkin+diagrams%22&hl=en&sa=X&ved=0ahUKEwiNseTziq_NAhXk34MKHaCkCu4Q6AEIfjAS#v=onepage&q=%22Coxeter-Dynkin%20diagram%22%7C%22Coxeter-Dynkin%20diagrams%22&f=false 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. en 2012 Igor V. Dolgachev Classical Algebraic Geometry: A Modern View 363 https://books.google.com.au/books?id=g9GLXmR9qg0C&pg=PA363&dq=%22Coxeter-Dynkin+diagram%22%7C%22Coxeter-Dynkin+diagrams%22&hl=en&sa=X&ved=0ahUKEwiNseTziq_NAhXk34MKHaCkCu4Q6AEIVzAL#v=onepage&q=%22Coxeter-Dynkin%20diagram%22%7C%22Coxeter-Dynkin%20diagrams%22&f=false 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). Synonyms graph with numerically labelled edges en Coxeter diagram, en Coxeter graph See also en Dynkin diagram en eponyms