12.3 Macdonald Polynomials and Their Properties

Macdonald polynomials are a powerful family of symmetric functions that generalize several important bases. They depend on two parameters and encode crucial combinatorial and representation-theoretic information, making them a cornerstone of algebraic combinatorics.

These polynomials are uniquely characterized by their orthogonality and triangularity properties. Their study has led to significant developments in related fields, including quantum groups, Hecke algebras, and the geometry of Hilbert schemes.

Macdonald polynomials

Definition and significance

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• Macdonald polynomials are a family of symmetric functions indexed by partitions, depending on two parameters $q$ and $t$
  • They generalize several other important families of symmetric functions (Schur functions, Hall-Littlewood functions, Jack functions)
• Introduced by Ian G. Macdonald in 1988 as a tool for studying the representation theory of symmetric groups and related algebras
• The coefficients of Macdonald polynomials encode important combinatorial, geometric, and representation-theoretic information
  • They have connections to Hilbert schemes, quantum groups, and Hecke algebras
• The study of Macdonald polynomials has led to significant developments in algebraic combinatorics
  • This includes the theory of double affine Hecke algebras and the combinatorics of diagonal harmonics

Specializations and limiting cases

• Macdonald polynomials specialize to various well-known bases of the ring of symmetric functions
  • Schur functions ($q=t=0$)
  • Hall-Littlewood functions ($q=0$)
  • Jack functions ($q=t^\alpha$, $t\to1$)
• These specializations highlight the unifying nature of Macdonald polynomials in the study of symmetric functions
• Understanding the relationships between Macdonald polynomials and their specializations has led to important developments in algebraic combinatorics
  • For example, the theory of Macdonald positivity conjectures and the combinatorics of Garsia-Haiman modules

Orthogonality and triangularity

Orthogonality and scalar product

• Macdonald polynomials form an orthogonal basis with respect to a certain scalar product on the ring of symmetric functions, depending on the parameters $q$ and $t$
  • This scalar product generalizes the Hall inner product
• The orthogonality of Macdonald polynomials is a key property that distinguishes them from other bases of symmetric functions
  • It allows for the development of a rich theory analogous to the theory of orthogonal polynomials
• The orthogonality property has important consequences, such as the existence and uniqueness of Macdonald operators

Triangularity and characterization

• Macdonald polynomials are uniquely characterized by their orthogonality and a certain triangularity property with respect to the basis of monomial symmetric functions
• The triangularity property states that the transition matrix between Macdonald polynomials and monomial symmetric functions is upper unitriangular with respect to a certain partial order on partitions
• The triangularity property, combined with orthogonality, ensures the integrality of certain transition coefficients
  • This has important implications for the combinatorial and representation-theoretic aspects of Macdonald polynomials

Computing Macdonald polynomials

Recursive and determinantal formulas

• Macdonald polynomials can be computed using several different methods, each with its own advantages and challenges
• The original definition of Macdonald polynomials uses a recursive formula based on the orthogonality and triangularity properties
  • This method is theoretically important but computationally inefficient for large partitions
• Macdonald polynomials can also be computed using a determinantal formula involving the $q$-Kostka polynomials
  • The $q$-Kostka polynomials are the transition coefficients between Macdonald polynomials and Hall-Littlewood functions

Combinatorial formulas and algorithms

• The Haglund-Haiman-Loehr formula expresses Macdonald polynomials as a sum over fillings of the diagram of the indexing partition, with weights depending on the inversion and descent statistics of the fillings
  • This combinatorial formula has led to efficient algorithms for computing Macdonald polynomials
  • It also has connections to the geometry of Hilbert schemes
• Other methods for computing Macdonald polynomials include:
  • Interpolation formulas
  • Specializations of Cauchy-type identities
  • The use of Macdonald operators
• The development of efficient algorithms for computing Macdonald polynomials has been an active area of research in algebraic combinatorics

Macdonald polynomials vs symmetric functions

Schur, Hall-Littlewood, and Jack functions

• Macdonald polynomials are a unifying framework for studying various important classes of symmetric functions
  • These classes can be obtained as specializations or limiting cases of Macdonald polynomials
• Schur functions, which play a central role in the representation theory of symmetric groups and the geometry of Grassmannians, are obtained from Macdonald polynomials by setting $q=t=0$
• Hall-Littlewood functions, related to the representation theory of finite general linear groups and the geometry of flag varieties, are obtained from Macdonald polynomials by setting $q=0$
• Jack functions, with connections to the representation theory of the Virasoro algebra and the geometry of Hilbert schemes, are obtained from Macdonald polynomials by setting $q=t^\alpha$ and letting $t\to1$

Generalizations and variations

• Other classes of symmetric functions related to Macdonald polynomials include:
  • Non-symmetric Macdonald polynomials
  • Shifted Macdonald polynomials
  • Macdonald polynomials associated with root systems
• These generalizations and variations of Macdonald polynomials have led to further developments in algebraic combinatorics and related fields
• The study of the relationships between Macdonald polynomials and other symmetric functions continues to be an active area of research, with implications for various branches of mathematics