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 | 1 |
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" NERFINISHED ⓘ |
| basisOf | ring of symmetric functions over ℚ(α) ⓘ |
| dependsOn | continuous parameter α ⓘ |
| developedBy | Ian G. Macdonald NERFINISHED ⓘ |
| domain | symmetric functions in countably many variables ⓘ |
| eigenfunctionOf | Calogero–Sutherland Hamiltonian NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Schur functions NERFINISHED ⓘ power-sum symmetric functions (via expansions) ⓘ zonal polynomials ⓘ |
| hasCombinatorialFormula |
Young diagram expansions
GENERATED
ⓘ
hook-length type formulas GENERATED ⓘ |
| 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 NERFINISHED ⓘ |
| limitOf | Macdonald polynomials as q → 1, t = q^α ⓘ |
| namedAfter | Henry Jack NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Macdonald polynomials NERFINISHED ⓘ Selberg integrals NERFINISHED ⓘ |
| 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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.