Gelfand–Tsetlin graph
E928276
The Gelfand–Tsetlin graph is a combinatorial structure whose vertices encode interlacing patterns corresponding to representations of unitary groups, organizing the branching of these representations in a graded, graph-theoretic form.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gelfand–Tsetlin branching graph | 1 |
| Gelfand–Tsetlin graph canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411652 — 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: Gelfand–Tsetlin graph Context triple: [Gelfand–Tsetlin basis, associatedWith, Gelfand–Tsetlin graph]
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A.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
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B.
Young diagrams
Young diagrams are combinatorial diagrams consisting of left-justified rows of boxes that visually represent integer partitions and play a central role in the representation theory of symmetric and general linear groups.
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C.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
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D.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
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E.
Springer correspondence
The Springer correspondence is a fundamental result in geometric representation theory that links representations of Weyl groups to the geometry of nilpotent orbits in Lie algebras via the cohomology of Springer fibers.
- 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: Gelfand–Tsetlin graph Target entity description: The Gelfand–Tsetlin graph is a combinatorial structure whose vertices encode interlacing patterns corresponding to representations of unitary groups, organizing the branching of these representations in a graded, graph-theoretic form.
-
A.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
-
B.
Young diagrams
Young diagrams are combinatorial diagrams consisting of left-justified rows of boxes that visually represent integer partitions and play a central role in the representation theory of symmetric and general linear groups.
-
C.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
D.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
E.
Springer correspondence
The Springer correspondence is a fundamental result in geometric representation theory that links representations of Weyl groups to the geometry of nilpotent orbits in Lie algebras via the cohomology of Springer fibers.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Bratteli diagram
ⓘ
combinatorial structure ⓘ graded graph ⓘ infinite graph ⓘ |
| connectedTo |
Gel'fand–Tsetlin formulas for matrix elements
ⓘ
determinantal point processes on interlacing arrays ⓘ spectral measures of random Gelfand–Tsetlin patterns ⓘ |
| edgeDefinition |
an edge joins two patterns if one is obtained from the other by deleting the last row and interlacing holds
ⓘ
edges connect interlacing signatures on consecutive levels ⓘ |
| encodes |
branching of irreducible representations of unitary groups
ⓘ
interlacing patterns of signatures ⓘ |
| generalizes | branching graph of symmetric groups via analogous Young graph ⓘ |
| hasAlternativeName |
GT graph
ⓘ
Gelfand–Tsetlin branching graph ⓘ
linked to:
Gelfand–Tsetlin graph
|
| hasBoundary |
Martin boundary of the graph
ⓘ
Thoma-type boundary describing extreme characters of U(∞) ⓘ |
| hasCombinatorialModel | triangular arrays of integers with interlacing inequalities ⓘ |
| hasLevel0Vertex | empty signature at level 0 ⓘ |
| hasLocalFiniteProperty | each vertex has finitely many neighbors on adjacent levels ⓘ |
| hasNaturalOrientation | edges oriented from lower to higher levels ⓘ |
| hasOrigin | introduced in the context of constructing bases for representations of classical groups ⓘ |
| hasPathSpace | infinite paths correspond to coherent systems of measures on signatures ⓘ |
| hasSymmetry | invariance under simultaneous shifts of all coordinates of a signature ⓘ |
| hasVertexSetDescription |
vertices are Gelfand–Tsetlin patterns
ⓘ
vertices encode interlacing integer arrays ⓘ |
| isCountable | vertex set is countable ⓘ |
| isGradedBy | rank n of the unitary group U(n) ⓘ |
| levelStructure | n-th level corresponds to signatures of length n ⓘ |
| mathematicalDiscipline |
asymptotic combinatorics
ⓘ
combinatorics ⓘ probability theory ⓘ representation theory ⓘ |
| namedAfter |
Israel Gelfand
ⓘ
Mikhail Tsetlin NERFINISHED ⓘ |
| organizes | branching of representations U(1) ⊂ U(2) ⊂ U(3) ⊂ ⋯ ⓘ |
| relatedTo |
Gelfand–Tsetlin basis
ⓘ
Gelfand–Tsetlin patterns ⓘ
linked to:
Gelfand–Tsetlin modules
Young graph ⓘ branching graph of U(∞) ⓘ representation theory of classical groups ⓘ representation theory of unitary groups ⓘ |
| studiedIn |
asymptotic representation theory
ⓘ
probability on combinatorial structures ⓘ random matrix theory ⓘ |
| usedFor |
constructing probability measures on paths corresponding to characters
ⓘ
describing inductive limits of unitary groups ⓘ parametrizing irreducible characters of U(∞) ⓘ studying harmonic analysis on infinite-dimensional unitary groups ⓘ |
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: Gelfand–Tsetlin graph Description of subject: The Gelfand–Tsetlin graph is a combinatorial structure whose vertices encode interlacing patterns corresponding to representations of unitary groups, organizing the branching of these representations in a graded, graph-theoretic form.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gelfand–Tsetlin graph