Coxeter–Dynkin diagrams
E412212
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.
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–Dynkin diagrams self-linksurface differs ⓘ
surface form:
Coxeter matrix
inner products of simple roots ⓘ relations in Coxeter presentations ⓘ |
| generalizes |
Coxeter–Dynkin diagrams
self-linksurface differs
ⓘ
surface form:
Coxeter graphs
Coxeter–Dynkin diagrams self-linksurface differs ⓘ
surface form:
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–Dynkin diagrams
self-linksurface differs
ⓘ
surface form:
Coxeter diagrams
Coxeter–Dynkin diagrams self-linksurface differs ⓘ
surface form:
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Coxeter–Dynkin diagrams Description of subject: 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.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dynkin diagram
this entity surface form:
Dynkin diagrams
this entity surface form:
Cartan matrix
this entity surface form:
Dynkin diagrams
this entity surface form:
Coxeter groups
this entity surface form:
Coxeter matrix
this entity surface form:
Dynkin diagrams
this entity surface form:
Coxeter diagrams
this entity surface form:
Dynkin diagrams
this entity surface form:
Coxeter graphs