Coxeter–Dynkin diagrams

E412212

combinatorial structure graphical notation mathematical diagram

Coxeter–Dynkin diagrams are graphical representations that encode the structure of reflection groups and root systems, widely used in the classification of regular polytopes, Lie algebras, and symmetries.

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All labels observed (8)

Label	Occurrences
Dynkin diagrams	4
Coxeter–Dynkin diagrams canonical	2
Cartan matrix	1
Coxeter diagrams	1
Coxeter graphs	1
Coxeter groups	1
Coxeter matrix	1
Dynkin diagram	1

How this entity was disambiguated

This entity first appeared as the object of triple T4105494 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Coxeter–Dynkin diagrams
Context triple: [H. S. M. Coxeter, knownFor, Coxeter–Dynkin diagrams]

A. Penrose graphical notation
Penrose graphical notation is a diagrammatic method for representing and manipulating tensors using networks of shapes and lines, widely used in mathematics and theoretical physics.
B. Alexander–Briggs notation
Alexander–Briggs notation is a classical system for naming and classifying knots in knot theory, assigning each distinct knot a unique label based on its crossing number and order in knot tables.
C. Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
D. Hasse diagram (in lattice theory)
A Hasse diagram is a simplified graphical representation of a finite partially ordered set that shows the order relations by connecting elements with upward lines without drawing implied transitive relations.
E. Polytopes
Polytopes are large-scale multimedia architectural and musical installations created by Iannis Xenakis that combine sound, light, and spatial design into immersive, mathematically structured environments.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Coxeter–Dynkin diagrams
Target entity description: Coxeter–Dynkin diagrams are graphical representations that encode the structure of reflection groups and root systems, widely used in the classification of regular polytopes, Lie algebras, and symmetries.

A. Penrose graphical notation
Penrose graphical notation is a diagrammatic method for representing and manipulating tensors using networks of shapes and lines, widely used in mathematics and theoretical physics.
B. Alexander–Briggs notation
Alexander–Briggs notation is a classical system for naming and classifying knots in knot theory, assigning each distinct knot a unique label based on its crossing number and order in knot tables.
C. Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
D. Hasse diagram (in lattice theory)
A Hasse diagram is a simplified graphical representation of a finite partially ordered set that shows the order relations by connecting elements with upward lines without drawing implied transitive relations.
E. Polytopes
Polytopes are large-scale multimedia architectural and musical installations created by Iannis Xenakis that combine sound, light, and spatial design into immersive, mathematically structured environments.
F. None of above. chosen

Statements (51)

Predicate	Object
instanceOf	combinatorial structure ⓘ graphical notation ⓘ mathematical diagram ⓘ
appliesTo	Weyl groups of semisimple Lie algebras ⓘ affine Coxeter groups ⓘ crystallographic root systems ⓘ finite Coxeter groups ⓘ indefinite Coxeter groups ⓘ non-crystallographic root systems ⓘ
edgeLabelMeaning	multiplicity of bond corresponds to order of product of reflections ⓘ
encodes	Cartan matrix ⓘ Coxeter matrix ⓘ linked to: Coxeter–Dynkin diagrams inner products of simple roots ⓘ relations in Coxeter presentations ⓘ
generalizes	Coxeter graphs ⓘ linked to: Coxeter–Dynkin diagrams Dynkin diagrams ⓘ linked to: Coxeter–Dynkin diagrams
hasComponent	edge labels ⓘ edges ⓘ node decorations ⓘ nodes ⓘ
hasNotationConvention	absence of edge denotes commuting reflections (order 2) ⓘ arrow or different node sizes may indicate root length ratios in crystallographic cases ⓘ labeled edge m denotes order m of product of reflections ⓘ single edge usually denotes order 3 between reflections ⓘ
nodeMeaning	node corresponds to a generating reflection ⓘ node corresponds to a simple root ⓘ
originatedFrom	work of Eugene Dynkin ⓘ work of H. S. M. Coxeter ⓘ
relatedTo	Coxeter diagrams ⓘ linked to: Coxeter–Dynkin diagrams Dynkin diagrams ⓘ linked to: Coxeter–Dynkin diagrams
represents	angles between reflecting hyperplanes ⓘ orders of products of reflections ⓘ simple reflections ⓘ simple roots ⓘ
usedFor	classifying Lie algebras ⓘ classifying regular polytopes ⓘ classifying uniform polytopes ⓘ describing Kac–Moody algebras ⓘ describing Weyl groups ⓘ describing symmetries ⓘ encoding Coxeter groups ⓘ encoding reflection groups ⓘ encoding root systems ⓘ
usedIn	classification of finite reflection groups ⓘ classification of regular polytopes and honeycombs ⓘ classification of semisimple Lie algebras ⓘ classification of simple Lie algebras ⓘ theory of Kac–Moody algebras ⓘ theory of Lie algebras ⓘ theory of Lie groups ⓘ theory of buildings and Tits systems ⓘ

How these facts were elicited