In mathematics, a double affine hecke algebra, or cherednik algebra, is an algebra. Rational cherednik algebras of type a northeastern its. Specializations of nonsymmetric macdonaldkoornwinder polynomials. Exposition of cherednik algebras and nonsymmetric macdonald polynomials for all root systems, including a new proof of the duality theorem for cherednik algebras. A probabilistic interpretation of the macdonald polynomials. The polynomial representation of the type a n1 rational cherednik algebra in characteristic pn proves cherednik algebras. Pavel et al representations of rational cherednik algebras with minimal support and torus knots discuss cherednik algebras with characteristic 0. Rational cherednik algebras, hilbert schemes and combinatorics. These polynomials can be defined in a number of ways, including a definition via divided difference operators and a definition as specialized nonsymmetric macdonald polynomials.
As a corollary, we give a proof of a version of the agt conjecture concerning pure n2 gauge theory for the group sur. Till now, there hasnt been any interpretation of the nonsymmetric opdammacdonald polynomials as characters or generalized chearacters ek the equivalence of the spincs model and the a. Double weyl groups, braid groups and hecke algebras. Ivan cherednik, introduction to double hecke algebras, math. For example, q,t deformations of virasoro algebras and w algebras are related to macdonald polynomials, and the combinatorics of crystals are related to. Quasiharmonic polynomials for coxeter groups and representations of cherednik algebras. Since then, it has been realized that cherednik algebras are of great indepen. A ne lie algebras calogeromoser system nakajima quiver varieties hecke, schur algebras rational cherednik algebras resolutions of orbifold singularities double a ne hecke algebras combinatorics hilbert scheme.
Signatures of the contravariant form on representations of. Cherednik algebras, macdonald polynomials and combinatorics mark haiman. The elliptic hall algebra, cherednik hecke algebras and. Ams representation theory of the american mathematical society. Nonsymmetric macdonald polynomials and demazure characters ion, bogdan, duke mathematical journal, 2003. The ring gp is an algebraic avatar for halllittlewood polynomials in the sense that its graded character is one. Rational cherednik algebras, quiver schur algebras and. The notion of nonsymmetric macdonald polynomials arise naturally from the study of double a ne hecke algebras. Cherednik developed the theory of double a ne hecke algebras, and used it to prove macdonald s conjectures in a uniform manner for all reduced a ne root systems see 6 and 7. This article consists of an overview of the theory of cherednik algebras and non symmetric macdonald polynomials, followed by the combinatorial formula for. Algebraic combinatorics the extended shi arrangement combinatorial algebra rational cherednik algebras. The present notes are based on a course on cherednik algebras given by the. Colorful combinatorics and macdonald polynomials sciencedirect.
The rational cherednik algebra is a deformation of the algebra ct t owdepending on parameters t. Cheredniks basic representation of the affine hecke algebra see 7. Although the schur expansion of macdonald polynomial still eludes us, jim haglund made a breakthrough in 2004 by proposing a combinatorial formula for h. Cherednik hecke algebras and macdonald polynomials, compos. On connection matrices of quantum knizhnikzamolodchikov equations based on lie super algebras galleas, wellington and stokman, jasper v. Symmetric functions, macdonald polynomials, and diagonal harmonics a. The twoparameter macdonald polynomials are a central object of algebraic combinatorics and representation theory. Next, jeff defined a family of polynomials called demazure atoms.
Symmetric functions, macdonald polynomials, and diagonal harmonics. Download the ebook affine hecke algebras and orthogonal polynomials i. Lecture notes on cherednik algebras pavel etingof and xiaoguang ma contents 1. The elliptic hall algebra, cherednik hecke algebras and macdonald polynomials article in compositio mathematica 14701. Lecture notes on cherednik algebras mit opencourseware. The main idea in the study of representations of rational cherednik algebras at t 1 is to handle them like universal enveloping algebras of semisimple complex lie algebras and study in particular a \category o. Affine hecke algebras and orthogonal polynomials by i. Representations of rational cherednik algebras, infinitedimensional aspects of representation theory and applications. Suzuki, cylindrical combinatorics and representations of cherednik algebras of type a, arxiv. Kacmoody groups, infinite dimensional differential geometry and cities freyn, walter, asian journal of mathematics, 2012.
Alexey pakharev hilbert series of the representation of cherednik algebras vector space we can think of the coe cients on the monomials as coordinates. Cherednik algebras have been in uential in a wide range of subjects, having been used for instance to answer questions in real algebraic geometry, integrable systems, combinatorics, and symplectic quotient singularities. In the early 1990s cherednik 1 introduced the double af. Rational cherednik algebras of type a jos e simental march 26, 2014 1 rational cherednik algebras 1. They are also closely related to the hilbert scheme hilbnc2 of points in the plane, a connection that was. Knot invariants, hilbert schemes and macdonald polynomials joint with a. Tatyana chmutova, representations of the rational cherednik algebras of dihedral type, j. If t1 the macdonald polynomials become the sums over w orbits, which are the monomial symmetric functions when the root system has type a. Cherednik algebras, walgebras and the equivariant cohomology. Cherednik algebras, macdonald polynomials and combinatorics. These bres describe macdonald polynomials and have deep connections with many areas of representation theory and algebraic combinatorics.
In particular, we will explain how the spectrum of the centre of the rational cherednik algebra is diffeomorphic to a certain component of the hilbert scheme. This is a unique, essentially selfcontained, monograph in a new field of fundamental importance for representation theory, harmonic analysis, mathematical physics, and combinatorics. Nonsymmetric macdonald polynomials sage reference manual. Generalized weyl modules, alcove paths and macdonald polynomials. Double affine hecke algebras and macdonald s conjectures, ann. Interpolation macdonald polynomials, introduced by sahi and knop 20, 7 in the. Macdonald polynomials garsia, haiman, i hilbert series of some cohomology module in the theory of. This is an essentially selfcontained monograph in an intriguing field of fundamental importance for representation theory, harmonic analysis, mathematical physics, and combinatorics. Its representation theory plays a critical role in the proof of the macdonald conjectures. Ams representation theory of the american mathematical. Rather than using semistandard tableaux and cocharge, different statistics are associated to arbitrary fillings.
Cherednik algebras, macdonald polynomials and combinatorics by mark haiman topics. In mathematics, a double affine hecke algebra, or cherednik algebra, is an algebra containing the hecke algebra of an affine weyl group, given as the quotient of the group ring of a double affine braid group. Double affine hecke algebras by cherednik, ivan ebook. Ivan cherednik, double affine hecke algebras, london mathematical society lecture note series, vol. Cambridge core algebra affine hecke algebras and orthogonal polynomials by i. Rasmussen eugene gorsky university of california, davis university of southern california. From the fact that they are eigenfunctions, we get that nonsymmetric macdonald polynomials exist.
Pdf we prove that macdonald polynomials are characters of irreducible cherednik algebra modules. The quantum knizhnikzamolodchikov equation and nonsymmetric macdonald polynomials kasatani, masahiro. We are interested in ltered deformations of the algebra csymnc2 cc2 ns n ch h s n. Morse, recursions and explicit formulas for particular nvariable sahiknop and macdonald polynomials. The proof uses representation theory of rational cherednik algebras. Birs workshop on applications of macdonald polynomials. A combinatorial formula for macdonald polynomials sciencedirect. Nonsymmetric macdonald polynomials mathematics libretexts. Pdf macdonald polynomials as characters of cherednik. This result is analogous to how nonsymmetric macdonald polynomials. Infinitesimal cherednik algebras have significant implications. Orthogonal polynomials, algebraic combinatorics, and cherednik algebras 20 august guanajuato, mexico, mathematical congress of the americas macdonald polynomials as characters of rational cherednik algebra modules. Explicit representations of 3dimensional sklyanin algebras associated to a point of order 2 reich, daniel j. Macdonald polynomials as characters of cherednik algebra modules.
Find, read and cite all the research you need on researchgate. The major discovery of cherednik was that macdonald polynomials natu. We will discuss the connection between cyclotomic rational cherednik algebras at t0 and the hilbert scheme of points in the plane. The rational cherednik algebra h c h cw is an associative. Representation theory, algebraic combinatorics, hecke algebras, macdonald polynomials funding simons foundation, collaboration grants for mathematicians, \ combinatorics and geometry via quantum algebra representations 20192024 nsf dms1600653 \ combinatorics of koornwinder polynomials and stable double a ne. What follows is a summary of the two moderated problem sessions from the workshop. The deformations of ch h s n we are going to produce arise from. Cherednik developed the theory of double a ne hecke algebras, and used it to prove macdonalds conjectures in a uniform manner for all reduced a ne root systems see 6 and 7. It is a major source of general information about the double affine hecke algebra, also called cherednik s algebra, and its impressive applications. Double affine hecke algebras london mathematical society lecture note series series by ivan cherednik. A nonsymmetric version of okounkovs bctype interpolation.
Since then, it has been realized that cherednik algebras are of great independent interest. Affine hecke algebras and orthogonal polynomials i. Citeseerx cherednik algebras, macdonald polynomials and. Pavel etingof, lectures on calogeromoser systems, pdf. Ams transactions of the american mathematical society. Haiman, cherednik algebras, macdonald polynomials and combinatorics, proceedings of the international congress of.
Cherednik algebras, macdonald polynomials and combinatorics 845 of x is a re. Then, the x i are position vectors, d i are momenta, and the extra part is accounting for heisenbergs uncertainty principle. The combinatorics of knot invariants arising from the. Haiman, combinatorics, symmetric functions, and hilbert schemes. Double affine hecke algebra ivan cherednik download. Hilbert schemes and macdonald polynomials joint with a. Cherednik, nonsemisimple macdonald polynomials, arxiv. The rational cherednik algebra h c h cw is an associative algebra generated by. Macdonald in pdf or epub format and read it directly on your mobile phone, computer or any device. Algebraic combinatorics the extended shi arrangement. There is a similar algebraic avatar for macdonald polynomials, which was rst introduced by garsiahaiman. This chapter gives an expository account of some unexpected connections which have arisen over the last few years between macdonald polynomials, invariants of torus knots, and lattice path combinatorics.
Cylindrical combinatorics and representations of cherednik. Cherednik algebras and yangians nicolas guay abstract we construct a functor from the category of modules over the trigonometric resp. Unitary representations of rational cherednik algebras. The purpose of this article is to work out the details of the ramyip formula for nonsymmetric macdonaldkoornwinder polynomials for the double affine hecke algebras of notnecessarily reduced affine root systems. Vasserot skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. This is all relevant because much of the representation theory of rational cherednik algebras can be understood through the representation theory of cyclotomic hecke algebras via the kz functor. Cherednik algebras, w algebras and the equivariant cohomology of the moduli space of instantons on a 2.
Garsia gave an introduction to parking functions and macdonald polynomials, covering topics from the indispensability of pleythistic notation to the miracle of q. In appendix we prove the cherednikorr conjecture for the multiples of. The elliptic hall algebra, cherednik hecke algebras and macdonald polynomials volume 147 issue 1 o. A combinatorial formula for macdonald polynomials arun rama,b. Knot invariants, hilbert schemes and macdonald polynomials. In the early 1990s cherednik 1 introduced the double affine hecke algebra the daha and used it. Cherednik algebras, macdonald polynomi als and combinatorics. Hilbert series of the representation of cherednik algebras. Jeff presented two new characterizations of demazure atoms. Morse, sahiknop and macdonald polynomials related to qultraspherical functions and basic hypergeometric series. Rational cherednik algebras, diagonal coinvariants, and other animals. Double a ne hecke algebras, also called cherednik algebras, were introduced by cherednik in 1993 as a tool in his proof of macdonalds conjectures about orthogonal polynomials for root systems. They were introduced by cherednik, who used them to prove macdonald s constant term conjecture for macdonald polynomials. Haiman, cherednik algebras, macdonald polynomials and combinatorics, proc.