Jack polynomials
E865107
Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jack polynomials canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462103 — 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: Jack polynomials Context triple: [Selberg integral, relatedTo, Jack polynomials]
-
A.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
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B.
Symanzik polynomials
Symanzik polynomials are graph-based polynomials that arise in the parametric representation of Feynman integrals in quantum field theory, encoding the topology and kinematic dependence of Feynman diagrams.
-
C.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
-
D.
Gegenbauer polynomials
Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
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E.
Askey scheme of hypergeometric orthogonal polynomials
The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
- 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: Jack polynomials Target entity description: Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
-
A.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
B.
Symanzik polynomials
Symanzik polynomials are graph-based polynomials that arise in the parametric representation of Feynman integrals in quantum field theory, encoding the topology and kinematic dependence of Feynman diagrams.
-
C.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
-
D.
Gegenbauer polynomials
Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
-
E.
Askey scheme of hypergeometric orthogonal polynomials
The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
family of symmetric polynomials
ⓘ
mathematical object ⓘ |
| appearsIn | Macdonald’s book "Symmetric Functions and Hall Polynomials" ⓘ |
| basisOf | ring of symmetric functions over ℚ(α) ⓘ |
| dependsOn | continuous parameter α ⓘ |
| developedBy | Ian G. Macdonald ⓘ |
| domain | symmetric functions in countably many variables ⓘ |
| eigenfunctionOf |
Calogero–Sutherland Hamiltonian
ⓘ
linked to:
Calogero–Sutherland model
|
| equalsAt |
Schur functions when α = 1
ⓘ
zonal polynomials for complex symmetric spaces when α = 1/2 ⓘ zonal polynomials when α = 2 ⓘ |
| field |
algebraic combinatorics
ⓘ
integrable systems ⓘ mathematical physics ⓘ random matrix theory ⓘ representation theory ⓘ symmetric function theory ⓘ |
| generalizes |
Hall–Littlewood polynomials
ⓘ
Schur functions ⓘ power-sum symmetric functions (via expansions) ⓘ zonal polynomials ⓘ |
| hasCombinatorialFormula |
Young diagram expansions
ⓘ
hook-length type formulas ⓘ |
| hasExpansion |
in monomial symmetric functions
ⓘ
in power-sum symmetric functions ⓘ |
| hasNormalization |
J-normalization (integral form)
ⓘ
P-normalization (monic with respect to dominance order) ⓘ |
| indexedBy | integer partitions ⓘ |
| introducedBy | Henry Jack ⓘ |
| limitOf | Macdonald polynomials as q → 1, t = q^α ⓘ |
| namedAfter | Henry Jack ⓘ |
| orderedBy | dominance order on partitions ⓘ |
| orthogonalityWithRespectTo | Jack inner product depending on α ⓘ |
| parameterNotation | α ⓘ |
| parameterRelation | β = 2/α in random matrix theory conventions ⓘ |
| relatedTo |
Calogero–Sutherland model
ⓘ
Macdonald polynomials ⓘ Selberg integrals ⓘ
linked to:
Selberg integral
|
| satisfies | orthogonality with respect to Jack inner product ⓘ |
| specialCaseAt |
α = 1
ⓘ
α = 1/2 ⓘ α = 2 ⓘ |
| structure | orthogonal basis depending rationally on α ⓘ |
| triangularWithRespectTo | monomial symmetric functions ⓘ |
| usedIn |
harmonic analysis on symmetric cones
ⓘ
study of β-ensembles in random matrix theory ⓘ theory of Jack characters ⓘ |
| variableType | commuting variables ⓘ |
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: Jack polynomials Description of subject: Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.