Coxeter group

plural Coxeter groups

1. geometryAny of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics.

   The above definitions, of Cayley graph, reflection, wall, half-space, folding, Bruhat order, etc., work for any Coxeter group.

   2007, Michael B. Smythe, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem, editors, Handbook of Spatial Logics, page 795

   2 more examples

   We may ask in general which Coxeter groups arise as subgroups of a given Coxeter group. This question is of course far too general. However, there are Coxeter groups which arise canonically as subgroups of a given Coxeter group.

   1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. de Clerck, editors, Finite Geometries and Combinatorics, page 277

   The problem of classifying all reflective lattices of given rank is also of interest in its own right from the perspectives of Coxeter groups and arithmetic subgroups of O(n,1).

   2012, Daniel Allcock, “The Reflective Lorentzian Lattices of Rank 3”, in Memoirs of the American Mathematical Society, number 1033, page vii