• In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the...
    19 KB (1,641 words) - 17:59, 20 November 2024
  • In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic...
    35 KB (3,763 words) - 03:21, 10 April 2025
  • Thumbnail for Harold Scott MacDonald Coxeter
    the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Coxeter was...
    17 KB (1,693 words) - 10:05, 30 June 2025
  • In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating...
    3 KB (382 words) - 00:53, 27 May 2025
  • Thumbnail for Coxeter–Dynkin diagram
    a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group...
    57 KB (3,233 words) - 15:31, 14 May 2025
  • Parabolic subgroup of a reflection group (category Coxeter groups)
    element of W) and W S = W {\displaystyle W_{S}=W} . The pair ( W I , I ) {\displaystyle (W_{I},I)} is again a Coxeter system. Moreover, the Coxeter group...
    31 KB (4,295 words) - 19:50, 1 July 2025
  • In group theory, the Todd–Coxeter algorithm, created by J. A. Todd and H. S. M. Coxeter in 1936, is an algorithm for solving the coset enumeration problem...
    7 KB (1,204 words) - 16:42, 28 April 2025
  • Thumbnail for Tesseract
    the four-dimensional measure polytope, taken as a unit for hypervolume. Coxeter labels it the γ4 polytope. The term hypercube without a dimension reference...
    26 KB (2,666 words) - 11:51, 4 June 2025
  • generated by some subset of S). A Coxeter matroid is a subset M of W/P that for every w in W, M contains a unique minimal element with respect to the w-Bruhat...
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  • =e_{1}e_{2}\cdots e_{n}.} This is both a Coxeter element of sorts (product of reflections) and a longest element of a Coxeter group in the Bruhat order; this is...
    20 KB (2,420 words) - 16:31, 27 March 2025
  • or 5-orthoplex. Coxeter, Regular Polytopes, sec 1.8 Configurations Coxeter, Complex Regular Polytopes, p.117 H.S.M. Coxeter: Coxeter, Regular Polytopes...
    10 KB (826 words) - 03:51, 20 April 2024
  • Thumbnail for 5-cell
    pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's α 4 {\displaystyle \alpha _{4}} polytope), the simplest possible convex...
    38 KB (3,699 words) - 19:35, 25 March 2025
  • Thumbnail for Affine symmetric group
    Affine symmetric group (category Coxeter groups)
    Coxeter element s 0 ⋅ s 1 ⋯ s n − 1 {\displaystyle s_{0}\cdot s_{1}\cdots s_{n-1}} in S ~ n {\displaystyle {\widetilde {S}}_{n}} is a Coxeter element...
    71 KB (10,250 words) - 02:22, 13 June 2025
  • Thumbnail for Weyl group
    corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to the longest element of a Coxeter group. There are a number...
    21 KB (3,256 words) - 23:36, 23 November 2024
  • mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes...
    7 KB (1,258 words) - 12:15, 10 February 2025
  • two reduced words of a Coxeter group to represent the same element. Sometimes, this is also called Matsumoto's lemma. A Coxeter group is a group that admits...
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  • Thumbnail for Point group
    n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram...
    69 KB (1,170 words) - 13:35, 16 April 2025
  • Thumbnail for 5-orthoplex
    alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211. It is a part of an infinite family of polytopes, called cross-polytopes...
    7 KB (505 words) - 17:44, 1 June 2025
  • of the group algebra of a Coxeter group. The Hecke algebra can also be viewed as a q-analog of the group algebra of a Coxeter group. Hecke algebras are...
    15 KB (2,097 words) - 12:45, 12 June 2025
  • Thumbnail for Regular 4-polytope
    Coxeter 1973, § 1.8 Configurations Coxeter, Complex Regular Polytopes, p.117 Conway, Burgiel & Goodman-Strauss 2008, p. 406, Fig 26.2 Coxeter, Star...
    27 KB (1,648 words) - 08:01, 15 October 2024
  • Thumbnail for Coxeter notation
    Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter...
    175 KB (6,423 words) - 17:51, 25 May 2025
  • Thumbnail for Pentellated 6-simplexes
    gotaf). H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by...
    55 KB (1,127 words) - 14:20, 3 June 2025
  • Thumbnail for Goldberg–Coxeter construction
    The Goldberg–Coxeter construction or Goldberg–Coxeter operation (GC construction or GC operation) is a graph operation defined on regular polyhedral graphs...
    21 KB (1,928 words) - 09:13, 13 March 2025
  • 5
    hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams...
    19 KB (1,821 words) - 05:47, 6 July 2025
  • Thumbnail for 7-demicube
    as HM7 for a 7-dimensional half measure polytope. Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches...
    9 KB (449 words) - 16:49, 9 April 2024
  • Thumbnail for Uniform 4-polytope
    Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher. 1986 Coxeter published a paper Regular...
    134 KB (4,315 words) - 11:59, 20 April 2025
  • Thumbnail for 5-demicube
    as HM5 for a 5-dimensional half measure polytope. Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1...
    9 KB (670 words) - 16:53, 9 April 2024
  • Thumbnail for Complexification (Lie group)
    BσB is determined by the length of σ as an element of W. The dimension is maximized at the Coxeter element and gives the unique open dense double coset...
    52 KB (7,216 words) - 14:30, 2 December 2022
  • Thumbnail for Dynkin diagram
    unoriented diagram (a special kind of Coxeter diagram), the Weyl group (a concrete reflection group), or the abstract Coxeter group. Although the Weyl group...
    78 KB (5,608 words) - 18:31, 28 June 2025
  • Thumbnail for ADE classification
    the simply laced finite Coxeter groups, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are...
    21 KB (2,644 words) - 14:56, 30 April 2025