Gelfand–Tsetlin basis
E270384
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Gelfand–Tsetlin patterns | 2 |
| Gelfand–Tsetlin basis canonical | 1 |
| Gelfand–Tsetlin integrable system | 1 |
| Gelfand–Tsetlin tableaux | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2475512 — 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 basis Context triple: [Israel Gelfand, knownFor, Gelfand–Tsetlin basis]
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A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
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B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
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C.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
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D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
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E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- 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 basis Target entity description: 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.
-
A.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
-
B.
Harish-Chandra character formula
The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
-
C.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
basis in representation theory
ⓘ
canonical basis ⓘ combinatorial basis ⓘ |
| appliesTo |
polynomial representations of GL(n)
ⓘ
representations with integral highest weights ⓘ |
| associatedWith |
Gelfand–Tsetlin graph
ⓘ
Gelfand–Tsetlin basis self-linksurface differs ⓘ
surface form:
Gelfand–Tsetlin patterns
branching rules for GL(n) ⓘ branching rules for unitary groups ⓘ |
| constructedAlong |
chain gl(1) ⊂ gl(2) ⊂ … ⊂ gl(n)
ⓘ
chain of groups GL(1) ⊂ GL(2) ⊂ … ⊂ GL(n) ⓘ |
| definedFor |
finite-dimensional irreducible representations of GL(n,ℂ)
ⓘ
finite-dimensional irreducible representations of gl(n,ℂ) ⓘ highest weight representations ⓘ |
| developedIn | mid 20th century ⓘ |
| field | mathematics ⓘ |
| gives |
simultaneous eigenbasis for a maximal commutative subalgebra of U(gl(n))
ⓘ
weight basis for representations of gl(n) ⓘ |
| hasConstructionMethod |
combinatorial construction
ⓘ
inductive construction along a chain of subalgebras ⓘ |
| hasProperty |
canonical up to normalization
ⓘ
compatible with restriction along the chain GL(1) ⊂ … ⊂ GL(n) ⓘ elements indexed by integer arrays satisfying interlacing inequalities ⓘ |
| namedAfter |
Israel Gelfand
ⓘ
Mikhail Tsetlin ⓘ |
| parameterizedBy |
Gelfand–Tsetlin basis
self-linksurface differs
ⓘ
surface form:
Gelfand–Tsetlin patterns
Gelfand–Tsetlin basis self-linksurface differs ⓘ
surface form:
Gelfand–Tsetlin tableaux
triangular arrays of integers or half-integers ⓘ |
| relatedTo |
Gelfand–Tsetlin algebra
ⓘ
Gelfand–Tsetlin basis self-linksurface differs ⓘ
surface form:
Gelfand–Tsetlin integrable system
Young diagrams ⓘ crystal bases ⓘ highest weight theory ⓘ |
| satisfies | interlacing conditions between rows of patterns ⓘ |
| subfield |
Lie theory
ⓘ
representation theory ⓘ |
| usedFor |
construction of Gelfand–Tsetlin integrable systems
ⓘ
explicit computation of Clebsch–Gordan coefficients ⓘ explicit computation of matrix elements ⓘ explicit description of representation branching ⓘ spectral analysis of commuting operators ⓘ |
| usedIn |
representation theory
ⓘ
representation theory of GL(n) ⓘ representation theory of Lie algebras ⓘ representation theory of Lie groups ⓘ representation theory of classical Lie algebras ⓘ representation theory of general linear groups ⓘ representation theory of unitary groups ⓘ |
How these facts were elicited
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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 basis Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gelfand–Tsetlin patterns
this entity surface form:
Gelfand–Tsetlin tableaux
this entity surface form:
Gelfand–Tsetlin patterns
this entity surface form:
Gelfand–Tsetlin integrable system