💁🏽Algebraic Combinatorics Unit 12 – Macdonald Polynomials & q-Analogues

Key Concepts and Definitions

• Macdonald polynomials generalize various classical symmetric polynomials (Hall-Littlewood polynomials, Jack polynomials, Schur functions)
• q-analogues extend classical combinatorial concepts by introducing a parameter q
  • Specialize to the classical case when q = 1
• Symmetric functions are polynomials invariant under permutations of the variables
  • Play a central role in algebraic combinatorics
• Partitions are non-increasing sequences of non-negative integers (λ₁ ≥ λ₂ ≥ ... ≥ λₖ)
  • Fundamental objects in the study of symmetric functions and Macdonald polynomials
• Young diagrams visually represent partitions as left-justified arrays of boxes
• Plethysm is a composition operation on symmetric functions generalizing multiplication
• Kostka numbers are the coefficients in the expansion of Schur functions in terms of monomial symmetric functions

Historical Context and Development

• Macdonald polynomials introduced by Ian G. Macdonald in the 1980s as a two-parameter family of symmetric functions
• Generalize and unify various classical symmetric functions (Hall-Littlewood polynomials, Jack polynomials, Schur functions)
• q-analogues have roots in the work of Euler and Gauss on hypergeometric series
  • Systematic study began with the work of Jackson in the early 20th century
• Development of Macdonald polynomials motivated by connections to representation theory and algebraic geometry
• Macdonald's conjectures on the positivity of transition coefficients sparked significant research
  • Proved using geometric and algebraic techniques (Haiman, 2001)
• Subsequent work has explored various generalizations and applications of Macdonald polynomials
  • Nonsymmetric Macdonald polynomials, affine Macdonald polynomials, Macdonald polynomials associated with root systems

Symmetric Functions and Their Properties

• Symmetric functions are formal power series in variables x₁, x₂, ... invariant under permutations of the variables
• Ring of symmetric functions Λ is a graded algebra with basis given by monomial symmetric functions mλ
  • Grading corresponds to the degree of the symmetric functions
• Elementary symmetric functions eₖ are the sums of all distinct k-fold products of the variables
• Complete homogeneous symmetric functions hₖ are the sums of all monomials of degree k
• Power sum symmetric functions pₖ are the sums of the k-th powers of the variables
• Schur functions sλ form an important basis of Λ with deep connections to representation theory
  • Defined as quotients of alternating polynomials
• Symmetric functions satisfy various algebraic identities and generating function relations
  • Fundamental involution ω maps eₖ to hₖ and sλ to sλ'

Introduction to Macdonald Polynomials

• Macdonald polynomials Pλ(x; q, t) are a two-parameter family of symmetric functions
  • Depend on a partition λ and two parameters q and t
• Specialize to various classical symmetric functions for specific values of q and t
  • Hall-Littlewood polynomials: Pλ(x; 0, t)
  • Jack polynomials: Pλ(x; α, α)
  • Schur functions: Pλ(x; 0, 0)
• Defined as the unique basis of the ring of symmetric functions satisfying certain triangularity and orthogonality conditions
• Macdonald's q-difference operators are a family of commuting operators diagonalized by the Macdonald polynomials
• Satisfy a Cauchy identity and a Pieri formula generalizing the corresponding identities for Schur functions
• Expand positively in terms of the monomial and Schur basis
  • Coefficients are rational functions in q and t with intricate combinatorial properties

q-Analogues: Basics and Applications

• q-analogues are q-deformations of classical combinatorial quantities obtained by introducing a parameter q
• q-integers $[n]_q = \frac{1-q^n}{1-q}$ generalize ordinary integers
  • Recover classical integers when q = 1
• q-factorial $[n]_q! = [n]_q [n-1]_q ... [1]_q$ generalizes the ordinary factorial
• q-binomial coefficients $\binom{n}{k}_q = \frac{[n]_q!}{[k]_q![n-k]_q!}$ generalize binomial coefficients
  • Satisfy q-analogues of classical binomial identities (q-binomial theorem)
• q-Pochhammer symbol $(a;q)_n = (1-a)(1-aq)...(1-aq^{n-1})$ appears in q-series and q-hypergeometric functions
• q-analogues have applications in various areas of mathematics
  • Combinatorics: q-series, partitions, permutations
  • Number theory: q-zeta functions, q-modular forms
  • Representation theory: quantum groups, Hecke algebras
  • Mathematical physics: quantum algebras, integrable systems

Algebraic Structures and Relationships

• Ring of symmetric functions Λ is a Hopf algebra with coproduct given by plethysm
• Macdonald polynomials are related to the double affine Hecke algebra (DAHA)
  • Nonsymmetric Macdonald polynomials are simultaneous eigenfunctions of the Cherednik operators
• Macdonald polynomials are connected to the geometry of the Hilbert scheme of points on the plane
  • Haiman's proof of the n! conjecture uses this connection
• Relationship between Macdonald polynomials and the Garsia-Haiman modules
  • Bigraded S_n-modules whose graded characters are given by modified Macdonald polynomials
• Macdonald polynomials have interpretations in terms of affine Lie algebras and quantum groups
  • Related to Demazure characters and Weyl modules
• Connections to the theory of Knizhnik-Zamolodchikov equations and affine flag varieties
• Categorification of Macdonald polynomials using Hilbert schemes and Cherednik algebras

Computational Techniques and Examples

• Combinatorial formulas for Macdonald polynomials in terms of fillings of Young diagrams
  • Involve statistics on fillings (inversions, coinversions, major index)
• Recursive algorithms for computing Macdonald polynomials using Pieri-type formulas
  • Exploit the triangularity properties of Macdonald polynomials
• Use of computer algebra systems (Sage, Maple, Mathematica) for symbolic computations
  • Packages for working with symmetric functions and Macdonald polynomials
• Example: Macdonald polynomial P₂₂₁(x; q, t) = m₂₂₁ + (q+t)m₃₁₁ + qt m₃₂ + (q²+qt+t²)m₄₁ + q²t² m₅
  • Expanded in the monomial basis with coefficients involving q and t
• Example: Macdonald polynomial P₃₁(x; q, t) = s₃₁ + (q+t)s₄ in the Schur basis
  • Coefficients are polynomials in q and t with positive integer coefficients
• Computational investigations of Macdonald polynomials and their properties
  • Verifying conjectures, discovering new identities and relations

Advanced Topics and Current Research

• Nonsymmetric Macdonald polynomials Eₐ(x; q, t) associated with compositions
  • Basis of the full polynomial ring with rich combinatorial properties
• Affine Macdonald polynomials associated with affine root systems
  • Related to affine Lie algebras and affine Hecke algebras
• Macdonald polynomials associated with other root systems (type B, C, D)
  • Involve additional parameters and exhibit new phenomena
• Generalized Macdonald polynomials depending on multiple sets of variables
  • Arise in the study of diagonal harmonics and Hilbert schemes
• Macdonald polynomials and the geometry of flag varieties and Schubert varieties
  • Connections to quantum cohomology and K-theory
• Macdonald polynomials and integrable systems
  • Relationship to Ruijsenaars-Schneider models and Macdonald operators
• Combinatorial expansions and positivity conjectures for Macdonald polynomials
  • Ongoing research on understanding the coefficients and their properties
• Categorical and geometric constructions related to Macdonald polynomials
  • Derived categories, perverse sheaves, matrix factorizations