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