12.4 Connections to Representation Theory and Geometry

Macdonald polynomials bridge representation theory and geometry, generalizing symmetric functions and encoding key info about algebraic structures. They're like Swiss Army knives, popping up in various mathematical contexts and revealing deep connections between seemingly unrelated areas.

These polynomials shine in studying flag varieties and Hilbert schemes. They help crack tough problems in equivariant cohomology and intersection theory, making them indispensable tools for understanding complex geometric objects and their symmetries.

Macdonald Polynomials and Representation Theory

Macdonald Polynomials as Generalizations

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• Macdonald polynomials generalize various families of symmetric functions (Schur functions, Hall-Littlewood polynomials, Jack polynomials)
• Indexed by partitions and depend on two parameters $q$ and $t$
  • $q$ and $t$ relate to the deformation of the Weyl group action on the space of polynomials
• Specializations of Macdonald polynomials at certain values of $q$ and $t$ give rise to important objects in representation theory
  • Characters of irreducible representations of the symmetric group
  • $GL(n)$ spherical functions

Connections to Algebras and Coinvariants

• Macdonald polynomials appear as the characters of certain representations of the double affine Hecke algebra (DAHA)
  • DAHA generalizes the affine Hecke algebra
  • DAHA is a quotient of the group algebra of the extended affine Weyl group
  • Representation theory of DAHA closely tied to the geometry of the affine flag variety
• Coefficients of Macdonald polynomials encode the dimensions of certain spaces of coinvariants
  • Coinvariants related to the representation theory of the rational Cherednik algebra

Geometric Interpretation of Macdonald Polynomials

Realizations and Expressions

• Macdonald polynomials can be realized as the Frobenius characters of certain modules over the rational Cherednik algebra
  • Related to the geometry of the Hilbert scheme of points in the plane
• Macdonald polynomial $P_λ(x;q,t)$ can be expressed as a sum over certain tableaux
  • Each tableau corresponds to a fixed point of a torus action on the Hilbert scheme of points in the plane

Cohomology and Varieties

• Macdonald polynomials can be interpreted as the equivariant cohomology classes of certain subvarieties of the affine Grassmannian (affine Springer fibers)
• Coefficients of Macdonald polynomials have a geometric interpretation in terms of the intersection cohomology of certain varieties
  • Affine flag variety
  • Hilbert scheme of points in the plane
• Geometric interpretation of Macdonald polynomials has led to important developments in the study of the geometry of the affine Grassmannian and the affine flag variety

Applications of Macdonald Polynomials

Representation Theory

• Compute characters of irreducible representations of the symmetric group and $GL(n)$ spherical functions
• Study the representation theory of the double affine Hecke algebra and the rational Cherednik algebra
  • Applications to the geometry of the affine flag variety
  • Applications to the Hilbert scheme of points in the plane

Geometry and Cohomology

• Compute the intersection cohomology of certain varieties (affine Grassmannian, affine flag variety)
  • Applications to geometric representation theory
• Study the geometry of certain moduli spaces (moduli space of sheaves on a surface) via their connection to the Hilbert scheme of points

Algebraic Combinatorics

• Prove important results in algebraic combinatorics
  • Positivity of certain coefficients
  • Existence of certain bijections between combinatorial objects

Macdonald Polynomials in Flag Varieties and Hilbert Schemes

Affine Flag Variety

• Macdonald polynomials appear naturally in the study of the equivariant cohomology of the affine flag variety
  • Correspond to certain Schubert classes
• Coefficients of Macdonald polynomials encode information about the intersection cohomology of certain subvarieties of the affine flag variety (affine Springer fibers)

Hilbert Scheme of Points

• Closely related to the geometry of the Hilbert scheme of points in the plane
  • Parametrizes certain ideals in the polynomial ring
  • Important connections to representation theory and physics
• Macdonald polynomials can be used to study the geometry of certain moduli spaces (moduli space of sheaves on a surface) via their connection to the Hilbert scheme of points
  • Appear in the computation of certain generating functions that encode the Euler characteristics of these moduli spaces

Developments and Results

• Geometric interpretation of Macdonald polynomials has led to important developments in the study of the geometry of flag varieties and Hilbert schemes
  • Proof of the positivity of certain coefficients
  • Construction of certain canonical bases