Macdonald polynomials
E865108
family of symmetric functions
orthogonal polynomials
q-orthogonal polynomials
two-parameter symmetric functions
Macdonald polynomials are a family of orthogonal symmetric functions depending on two parameters that generalize several classical symmetric polynomials, such as Schur and Jack polynomials, and play a central role in algebraic combinatorics and representation theory.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Macdonald polynomials canonical | 4 |
| Haglund–Haiman–Loehr formula | 1 |
| Macdonald polynomials for arbitrary root systems | 1 |
| Q_λ(x;q,t) Macdonald polynomials | 1 |
| q-Whittaker functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462104 — 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: Macdonald polynomials Context triple: [Selberg integral, relatedTo, Macdonald polynomials]
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A.
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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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.
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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.
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.
Askey–Wilson algebra
The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
- 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: Macdonald polynomials Target entity description: Macdonald polynomials are a family of orthogonal symmetric functions depending on two parameters that generalize several classical symmetric polynomials, such as Schur and Jack polynomials, and play a central role in algebraic combinatorics and representation theory.
-
A.
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.
-
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.
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.
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.
Askey–Wilson algebra
The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
family of symmetric functions
ⓘ
orthogonal polynomials ⓘ q-orthogonal polynomials ⓘ two-parameter symmetric functions ⓘ |
| appearsIn | Macdonald’s book "Symmetric Functions and Hall Polynomials" ⓘ |
| dependsOn |
parameter q
ⓘ
parameter t ⓘ |
| domain |
algebraic combinatorics
ⓘ
representation theory ⓘ symmetric function theory ⓘ |
| generalizes |
Hall–Littlewood polynomials
ⓘ
Jack polynomials ⓘ Schur polynomials ⓘ
linked to:
Schur functions
monomial symmetric functions ⓘ q-Whittaker functions ⓘ
linked to:
Macdonald polynomials
|
| hasBasisProperty | form a basis of the ring of symmetric functions over Q(q,t) ⓘ |
| hasCombinatorialModel |
Haglund–Haiman–Loehr formula
ⓘ
linked to:
Macdonald polynomials
LLT polynomials as building blocks ⓘ |
| hasConjecture |
Macdonald positivity conjecture
ⓘ
linked to:
Macdonald positivity conjecture (proved)
|
| hasGeneralization |
Macdonald polynomials for arbitrary root systems
ⓘ
linked to:
Macdonald polynomials
|
| hasNormalization | usually normalized to have leading monomial x^λ ⓘ |
| hasOrthogonality | orthogonal with respect to a certain (q,t)-deformed scalar product ⓘ |
| hasType |
Macdonald polynomials associated to reduced root systems
ⓘ
type A Macdonald polynomials ⓘ |
| hasVariant |
P_λ(x;q,t) Macdonald polynomials
ⓘ
Q_λ(x;q,t) Macdonald polynomials ⓘ
linked to:
Macdonald polynomials
|
| indexedBy | partitions ⓘ |
| introducedBy | Ian G. Macdonald ⓘ |
| introducedIn | 1980s ⓘ |
| namedAfter | Ian G. Macdonald ⓘ |
| parameterSpecialization |
Hall–Littlewood polynomials at q = 0
ⓘ
linked to:
Hall–Littlewood polynomials
Jack polynomials at q = t^α, t → 1 ⓘ Schur polynomials at q = t ⓘ monomial symmetric functions at q = t = 0 ⓘ q-Whittaker functions at t = 0 ⓘ |
| relatedTo |
Cherednik operators
ⓘ
Demazure characters ⓘ Haiman’s n! theorem ⓘ Hilbert schemes of points on surfaces ⓘ
linked to:
Hilbert scheme theory
affine Hecke algebras ⓘ
linked to:
Hecke algebra
double affine Hecke algebras ⓘ
linked to:
Hecke algebra
|
| satisfies |
Cauchy identity for Macdonald polynomials
ⓘ
Macdonald positivity conjecture (proved) ⓘ Pieri-type rules ⓘ triangularity with respect to dominance order on partitions ⓘ |
| usedIn |
combinatorial representation theory of symmetric groups
ⓘ
representation theory of quantum groups ⓘ study of diagonal harmonics ⓘ theory of symmetric functions in infinitely many variables ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Macdonald polynomials Description of subject: Macdonald polynomials are a family of orthogonal symmetric functions depending on two parameters that generalize several classical symmetric polynomials, such as Schur and Jack polynomials, and play a central role in algebraic combinatorics and representation theory.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Macdonald polynomials
linked to: Macdonald polynomials
linked to: Macdonald polynomials
linked to: Macdonald polynomials