Coxeter diagram

English

Noun

Coxeter diagram (plural Coxeter diagrams)

1. (geometry, algebra) A Coxeter-Dynkin diagram.
   • 1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups‎, page 358:

     It suffices to make use of the tables above and observe that the Coxeter diagram of a direct sum of C-matrices is the disconnected union of the Coxeter diagrams of the summands.
     For the description of C⁻-matrices satisfying condition (2) we shall use Coxeter diagrams with the additional stipulation that in case a_ij < −1 the vertices v_i and v_j are joined by a dotted line with index −a_ij. By the Coxeter diagram of a Coxeter polyhedron and a lattice generated by reflections we mean the Coxeter diagram of the corresponding C⁻-matrix.

   • 1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. Buekenhout, J. Doyen, F. de Clerck, J. A. Thas, J. W. P. Hirschfeld, editors, Finite Geometries and Combinatorics, 2nd International Conference, page 277:

     An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type M. We introduce admissible partitions of Coxeter diagrams in order to generalize this situation.

   • 2008, Peter Abramenko, Kenneth S. Brown, Buildings: Theory and Applications, page 259:

     The Coxeter diagrams of type E₇ and E₈ have no nontrivial automorphisms, so σ₀ is trivial in those cases. […] The Coxeter diagram of type F₄ has a unique nontrivial automorphism, but σ₀ is trivial in this case. One can see this by using the Dynkin diagram instead of the Coxeter diagram and noting that it does not have any nontrivial automorphisms.