12.4 Connections to Representation Theory and Geometry
3 min read•july 30, 2024
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 and intersection theory, making them indispensable tools for understanding complex geometric objects and their symmetries.
Macdonald Polynomials and Representation Theory
Macdonald Polynomials as Generalizations
Top images from around the web for Macdonald Polynomials as Generalizations
Operator Product Formula for a Special Macdonald Function View original
Is this image relevant?
Symmetric Functions of Generalized Polynomials of Second Order View original
Is this image relevant?
Representation Theory [The Physics Travel Guide] View original
Is this image relevant?
Operator Product Formula for a Special Macdonald Function View original
Is this image relevant?
Symmetric Functions of Generalized Polynomials of Second Order View original
Is this image relevant?
1 of 3
Top images from around the web for Macdonald Polynomials as Generalizations
Operator Product Formula for a Special Macdonald Function View original
Is this image relevant?
Symmetric Functions of Generalized Polynomials of Second Order View original
Is this image relevant?
Representation Theory [The Physics Travel Guide] View original
Is this image relevant?
Operator Product Formula for a Special Macdonald Function View original
Is this image relevant?
Symmetric Functions of Generalized Polynomials of Second Order View original
Is this image relevant?
1 of 3
Macdonald polynomials generalize various families of symmetric functions (, , )
Indexed by partitions and depend on two parameters q and t
q and t relate to the deformation of the 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
of of the
GL(n) spherical functions
Connections to Algebras and Coinvariants
Macdonald polynomials appear as the characters of certain representations of the ()
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
Geometric Interpretation of Macdonald Polynomials
Realizations and Expressions
Macdonald polynomials can be realized as the of certain modules over the rational Cherednik algebra
Related to the geometry of the of points in the plane
Macdonald polynomial Pλ(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 ()
Coefficients of Macdonald polynomials have a geometric interpretation in terms of the 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
Study the geometry of certain moduli spaces ( 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
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