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.
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" NERFINISHED ⓘ |
| dependsOn |
parameter q
ⓘ
parameter t ⓘ |
| domain |
algebraic combinatorics
ⓘ
representation theory ⓘ symmetric function theory ⓘ |
| generalizes |
Hall–Littlewood polynomials
NERFINISHED
ⓘ
Jack polynomials NERFINISHED ⓘ Schur polynomials NERFINISHED ⓘ monomial symmetric functions ⓘ q-Whittaker functions NERFINISHED ⓘ |
| hasBasisProperty | form a basis of the ring of symmetric functions over Q(q,t) ⓘ |
| hasCombinatorialModel |
Haglund–Haiman–Loehr formula
GENERATED
ⓘ
LLT polynomials as building blocks GENERATED ⓘ |
| hasConjecture | Macdonald positivity conjecture NERFINISHED ⓘ |
| hasGeneralization | Macdonald polynomials for arbitrary root systems NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| indexedBy | partitions ⓘ |
| introducedBy | Ian G. Macdonald NERFINISHED ⓘ |
| introducedIn | 1980s ⓘ |
| namedAfter | Ian G. Macdonald NERFINISHED ⓘ |
| parameterSpecialization |
Hall–Littlewood polynomials at q = 0
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Demazure characters ⓘ Haiman’s n! theorem NERFINISHED ⓘ Hilbert schemes of points on surfaces NERFINISHED ⓘ affine Hecke algebras NERFINISHED ⓘ double affine Hecke algebras NERFINISHED ⓘ |
| satisfies |
Cauchy identity for Macdonald polynomials
ⓘ
Macdonald positivity conjecture (proved) NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.