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Coxeter group

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 mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not…

Products, Finite Coxeter groups & Definition

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Overview

Definition

An example

Abstraction of reflection groups

Finite Coxeter groups

Affine Coxeter groups

Hyperbolic Coxeter groups

Irreducible Coxeter groups

Partial orders

Homology

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Coxeter group

Nodes93
Edges92
Triples95
Avg. degree1.98
Density0.021505
Components1

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Coxeter group

Top relations

related to Partial orders · 12
Coxeter group → An, Bruhat, Cayley, Coxeter, For, François Bruhat, In, Indeed, S3, The, The Hasse, Using
related to Homology · 11
Coxeter group → Coxeter, For, Howlett, Ihara, In, Schur, Since, The Schur, This, Weyl, Yokonuma
related to Affine Coxeter groups · 9
Coxeter group → An, Coxeter, For, In, Stiefel, The, The Stiefel, These, Weyl
related to Abstraction of reflection groups · 8
Coxeter group → Coxeter, Each, Euclidean, However, In, On, Technically, The
related to External links · 8
Coxeter group → Cayley, Coxeter, EMS Press, Encyclopedia, Eric, Mathematics, MathWorldJenn, Weisstein
related to Symmetry groups of regular polytopes · 7
Coxeter group → An, Bn, Coxeter, Note, Sn, The, There
see also · 7
Coxeter group → Artin, Coxeter, Hecke, Lusztig, Shephard, Tits, Todd
related to Definition · 6
Coxeter group → Coxeter, For, Formally, Here, Note, The
is a · 5
Coxeter group → affine Weyl group of An, automatic group, direct product of finitely many of these irreducible groups, direct product of finitely many of these irreducible groups.Weyl groupsMany, direct product of the irreducible groups that correspond to the components of its Coxeter
related to An example · 5
Coxeter group → Any, Coxeter, Further, The, Therefore

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Important terminology

coxeter group groups displaystyle finite reflection generators symmetry weyl regular affine diagram polytopes every graph two order reflections ij one

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Coxeter groupis adirect product of finitely many of these irreducible groups.Weyl groupsMany0.90text
Coxeter groupis aautomatic group0.90text
Coxeter groupis adirect product of finitely many of these irreducible groups0.90text
Coxeter groupis aaffine Weyl group of An0.90text
Coxeter groupis adirect product of the irreducible groups that correspond to the components of its Coxeter0.90text
Coxeter grouprelated to Abstraction of reflection groupsCoxeter0.60section
Coxeter grouprelated to Abstraction of reflection groupsOn0.60section
Coxeter grouprelated to Abstraction of reflection groupsTechnically0.60section
Coxeter grouprelated to Abstraction of reflection groupsEach0.60section
Coxeter grouprelated to Abstraction of reflection groupsIn0.60section
Coxeter grouprelated to Abstraction of reflection groupsThe0.60section
Coxeter grouprelated to Abstraction of reflection groupsEuclidean0.60section

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