To install the package simply extract the archive into the pkgfolder of your gapinstallation. Two actions of the hecke algebra of type a on the corresponding polynomial ring are studied. What we actually compute is the commuting ring of the condensed algebra eae. A result relating the dirac cohomology with central characters is established for a subclass of algebras, which we call barbaschsahi algebras. Jan, 2017 affine hecke algebras for langlands parameters 3 the main goal of this p aper is the construction of an a. Kottwitz, and amritanshu prasad our aim here is to give a fairly selfcontained exposition of some basic facts about the iwahorihecke algebra hof a split padic group g, including bernsteins presentation and description of the center, macdonalds formula, the casselman. This result simultaneously generalizes known results on such a relation for real reductive lie groups and for various kinds of hecke algebras, which all go back to a conjecture of david vogan 1997. At the end of the talk i will say some words on an analog construction given.
Download affine hecke algebras and orthogonal polynomials i. Jul 17, 20 double affine hecke algebras download here. Jones abstract by studying representations of the braid group satisfying a certain quadratic relation we obtain a polynomial invariant in two variables for oriented links. The hook fusion procedure for hecke algebras james grime department of mathematics, university of york, york, yo10 5dd, uk received 26 may 2006 available online july 2006 communicated by jeanyves thibon abstract we derive a new expression for the qanalogue of the young symmetrizer which generate irreducible representations of the hecke. Hecke algebras are quotients of the group rings of artin braid groups. The hecke algebra of the pair g,kis the convolution algebra of zvalued functions with. The algebra is a commutative ring in the classical elliptic modular form theory, the hecke operators t n with n coprime to the level acting on the space of cusp forms of a given weight are selfadjoint with respect to the petersson inner product. Hecke algebra representations of braid groups and link polynomials by v. The iwahori hecke algebra is a deformation of the group algebra of a weyl group or, more generally, a coxeter group. Double affine hecke algebras and algebraic geometry. A reason for formulating the bernsteinzelevinsky derivatives for graded hecke algebras is that one can apply representation theory of symmetric groups, in particular the littlewoodrichardson rule, to compute the bernsteinzelevinsky derivatives of generalized speh representations, see section 7, for details.
Hecke algebras daniel bump may 11, 2010 by a hecke algebra we will usually mean an iwahori hecke algebra. Elliptic hecke algebras and modified cherednik algebras takebayashi, tadayoshi, proceedings of the japan academy, series a, mathematical sciences, 2003 the quantum knizhnikzamolodchikov equation and nonsymmetric macdonald polynomials kasatani, masahiro and takeyama, yoshihiro, 2009. A hecke algebra describes the most reasonable way to convolve functions or measures on a homogeneous space. W has an alternative description as the quotient of the complex group algebra cb w by the hecke relations.
Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. We introduce the term protonormal to refer to a subgroup h of a group g such that for every x in g the subgroups x1hx and h commute as sets. This result is used to translate the theory of bernsteinzelevinsky derivatives in the language of representations of hecke algebras, where we develop a comprehensive theory. Lectures on modular forms and hecke operators kennetha. These algebras replace the hecke algebra and the symmetric group in the schurweyl duality theorems for the symmetrised powers of the fundamental representation of gln and their quantum version. May 25, 2005 we introduce the term protonormal to refer to a subgroup h of a group g such that for every x in g the subgroups x1hx and h commute as sets. Cambridge core algebra affine hecke algebras and orthogonal polynomials by i. Download the ebook affine hecke algebras and orthogonal polynomials i macdonald cambridge 2003 in pdf or epub format and read it directly on your mobile phone, computer or any device. But this can be used to get the action of the full double affine hecke algebra. Macdonald in pdf or epub format and read it directly on your mobile phone, computer or any device. Pdf theory of hecke algebras to association schemes. Representations of coxeter groups and hecke algebras. This tag is for questions regarding the hecke algebra or, iwahori hecke algebra, which is the algebra generated by hecke operators. Group characters symmetric functions and the hecke algebra.
As a consequence we show that there exists a maximum cnorm on hg,h, generalizing. In mathematics, the iwahorihecke algebra, or hecke algebra, named for erich hecke and nagayoshi iwahori, is a deformation of the group algebra of a coxeter group. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic hecke algebra r n, which had originally been described by solomon. Hecke algebras for symmetric groups university of georgia. Affine hecke algebras and orthogonal polynomials i macdonald. Second, run these programs with various inputs and analyze the outputs. Joseph, a w module structure in the primitive spectrum of the enveloping algebra of a semisimple lie algebra, preprint, telaviv, 1978. In w1, one of the authors initiated a program of constructing the socalled spin hecke algebras associated to. In w1, one of the authors initiated a program of constructing the socalled spin hecke algebras associated to weyl groups with nontrivial 2. Hecke algebras and finite groups of lie type springerlink. A construction of drinfeld produces an analogue of a graded hecke algebra for any finite subgroup of glv. The book is made up of lecture notes from a course taught by goldschmidt at the university of california at berkeley in 1989.
That is, it is the algebra of all matrices that commute with the algebra a. Suppose that you have seen the definition of convolution of functions on a vector space, or on a discrete group the latter is just the group algebra of the group or some completion. Hecke algebra representations of braid groups and link. The main ingredient will be a new graded extalgebra associated to a quiver with involution and morita equivalent to the affine hecke algebra of type b. A corollary of this result is that the correspondence of hecke algebra modules preserves formal degree. Hecke algebra representations sage reference manual v9. Representations of the symmetric group and its hecke algebra n. In mathematics, the hecke algebra is the algebra generated by hecke operators properties. Wthe specialisation depends on the choice of parameters for the rational cherednik algebraevery specialised hecke. Double affine hecke algebras by minniemcfarland issuu. In particular, cur79, mar are great for motivation of the hecke algebra, hum92 goes deeper into the structure of the hecke algebra and the.
A characteristic variety for the primitive spectrum of a semisimple lie algebra, preprint, bonn, 1976 8. Affine hecke algebras and orthogonal polynomials by i. Jacon abstract this paper is an expository paper on the representation theory of the symmetric group and its hecke algebra in arbitrary characteristic. Computing and experimenting with hecke algebras the goals of this project are twofold. We study both the semisimple and the non semisimple case and give an introduction to. Cherednik algebra, the category o, and an exact functor, the kzfunctor, from o to the category of representations of a certain specialised cyclotomic hecke algebra h. Quiver hecke algebras and 2lie algebras 3 this article is based on a series of lectures at the national taiwan university, taipei, in december 2008 and a series of lectures at bicmr, peking university, in marchapril 2010. Hecke algebra in representation theory, centers play special roles and we shall be looking at the case of hecke algebras. Download the ebook affine hecke algebras and orthogonal polynomials i. The graded hecke algebra for a finite weyl group is intimately related to the geometry of the springer correspondence. H is, in fact, an isomorphism of hilbert algebras with involution, thus giving a coincidence of induced plancherel measures under suitable normalization. Click download or read online button to get group characters symmetric functions and the hecke algebra book now. It is expressed using a trace, discovered by ocneanu, on the hecke algebras of type a. Newest heckealgebras questions mathematics stack exchange.
Group characters, symmetric functions, and the hecke algebra free epub, mobi, pdf ebooks download, ebook torrents download. We use the notation t w for w2w to denote both elements of the braid group and their images in h. Hecke algebra, plancherel measure, tempered dual, ktheory. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic hecke algebra r n. Let g be a finite group, x a transitive gset and h z the corresponding hecke algebra, namely, the endomorphism ring of the zgmodule zx. Both are deformations of the natural action of the symmetric group on. Ivan cherednik submitted on 17 apr 2004, last revised 26 sep 2004 this version, v3 abstract. Group characters, symmetric functions, and the hecke algebra.
A new appendix is added for the projectivity of the gelfandgraev representation, and references are updated. This connection found a spectacular application in vaughan jones construction of new invariants of knots. The iwahorihecke algebra is a deformation of the group algebra of a weyl group or, more generally, a coxeter group. For the curious audience member, i suggest reading any of the well written expositions given in the references. These algebras are defined by generators and relations and they depend on a deformation parameter \q\. A corollary of this result is that the correspondence of. The hecke algebra is the kgendomorphism ring of the module m. Even without the hecke algebra conjecture, theorem 1. We study the modular representation theory of g and show how our previous results on cell data and canonical basic sets leads to a natural parametrization of the modular irreducible representations of g which admit nonzero vectors. Directed at graduate students and mathematicians, this book covers an unusual set of interrelated topics, presenting a selfcontained exposition of the algebra behind the jones polynomial along with various excursions into related areas.
As a consequence we show that there exists a maximum cnorm on hg,h. Here we explain the fundamental connections between the theory of iwahorihecke algebras and representations of a finite group of lie type g. The iwahorihecke algebra of the symmetric group is the convolution algebra of gl ninvariant functions on the variety of pairs of complete flags over a finite field. In fact it shows the relationship is much more basic than the satake isomorphism. First, to write computer programs, using the compute algebra software sage, to compute localizations of hecke algebras. Download fulltext pdf theory of hecke algebras to association schemes article pdf available in sut journal of mathematics 381 january 2002 with 51 reads. The algebra eae is morita equivalent to the algebra a. Affine hecke algebras and orthogonal polynomials i macdonald cambridge 2003. If moreover g,h is a hecke pair we show that the hecke algebra hg,h is generated by the range of a canonical partial representation of g vanishing on h. Representations of the symmetric group and its hecke algebra. Hecke algebra known as the rational cherednik algebra or cherednikdunkl algebra has been studied extensively eg, do.
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