Not to be confused with Longest element of a Coxeter group.
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 order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.[1]
^Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 978-0-8218-3722-1
In mathematics, a Coxeterelement is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the...
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...
the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Coxeter was...
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...
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...
mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes...
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...
the four-dimensional measure polytope, taken as a unit for hypervolume. Coxeter labels it the γ4 polytope. The term hypercube without a dimension reference...
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...
identity element of W) and W S = W {\displaystyle W_{S}=W} . The pair ( W I , I ) {\displaystyle (W_{I},I)} is again a Coxeter group. Moreover, the Coxeter group...
to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a spherical tiling, and projected...
=e_{1}e_{2}\cdots e_{n}.} This is both a Coxeterelement of sorts (product of reflections) and a longest element of a Coxeter group in the Bruhat order; this is...
polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end...
which preserves length ..." — Coxeter (1969) p. 29 3.11 Any two congruent triangles are related by a unique isometry.— Coxeter (1969) p. 39 Let T be a transformation...
Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter...
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