Pdf these notes give a fully selfcontained introduction to the modular representation theory of the iwahorihecke algebras and the qschur. Hecke algebras are structures which are used to study the representation theory of padic groups. The ground eld fis assumed to contain a quantity qwhich might be an indeterminate or for some purposes an integer prime power or for other purposes a root of unity. Iwahorihecke algebras of weyl groups, and the drinfeldjimbo quantized enveloping algebras. Pdf iwahorihecke algebras and schur algebras of the. Affinelike hecke algebras and p adic representation theory. Iwahori hecke algebras, qschur algebras, cellular algebras. We explore the representation theory of renner monoids associated to classical groups and their hecke algebras. For example, they appear in the geometry of schubert varieties, where they are used in the definition of the kazhdanlusztig polynomials. Iwahorihecke algebras and schur algebras of the symmetric group. Algebras and representation theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including lie algebras and superalgebras, rings of differential operators, group rings and algebras, c algebras and hopf algebras, with particular emphasis on quantum groups. In particular by results of barbaschmoy and subsequently barbaschciubotaru, there is a precise relation between the unitary dual of a block of representations of a padic group and a particular iwahorihecke algebra.
There is a perfect pairing between the cocenter of the group algebra and the. Let f denote the fock space representation of the quantum group u v sl. These notes give a fully selfcontained introduction to the modular representation theory of the iwahori hecke algebras and the qschur algebras of the symmetric groups. We apply lusztigs theory of cells and asymptotic algebras to the iwahori hecke algebra of a nite weyl group extended by a group of graph automorphisms. Even more generally, the iwahorihecke algebra gives a quantum version of the weyl group situation. Abstract the iwahorihecke algebras of nite coxeter groups play an important role in many areas of mathematics. Their construction relies crucially on certain deformed versions of the underlying algebras, namely, iwahorihecke algebras of weyl groups, and the. In particular, this includes a combinatorial analysis of their structure and representation theory. This paper is concerned with the representation theory of the braid group bn.
Iwahorihecke algebras are fundamental in many areas of mathematics, ranging from the representation theory of lie groups and quantum groups, to knot theory and statistical mechanics. Iwahorihecke algebras and kazhdanlusztig polynomials. On schur algebras corresponding to hecke algebras of type b, joint work with chunju lai and dan nakano. However, it appears naturally in the context of the mod p representation theory of g, or rather, the prop iwahorihecke algebra h does. In this thesis we study the representation theory of the iwahorihe. Wenzlcrystal bases of quantum affine algebras and affine kazhdan. This result was recently applied to the representation theory of reductive padic. In this paper we introduce the notion of the stability of a sequence of modules over hecke algebras. These notes give a fully selfcontained introduction to the modular representation theory of the iwahorihecke algebras and the qschur algebras of the symmetric groups. Sothetheory of padic groups and their hecke algebras is over 1dimensional local. There are no new results here, and the same is essentially true of the proofs. The current article is a short survey on the theory of hecke algebras, and in.
Hence, the iwahorihecke algebra h0over a eld k with characteristic p is much less understood than in the complex case. A few more words to explain how iwahorihecke algebras arise as spaces of functions. Finite hecke algebras and their characters math user home pages. We show how the operation of conjugation in the coxeter group translates to the iwahorihecke algebra.
Algebras of iwahorihecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of lie type. Buy iwahorihecke algebras and schur algebras of the symmetric group university lecture series on free shipping on qualified orders. We also clarify some misunderstandings on vertex operator algebras, modular functors and intertwining operator algebras. In this thesis we study the representation theory of the iwahori he. Prop iwahori hecke algebras are gorenstein 3 algebras.
The answer is that even if one is only concerned with spherical representations, their theory naturally leads to the iwahori subgroup and the iwahori hecke algebra. Dirac 1928, in his work on the relativistic wave equation of the electron, introduced matrices that provide a representation of the cli. Algebras of iwahorihecke type are one of the tools and were, probably, first considered in the context of representation theory. Representation theories of some towers of algebras related. Iwahorihecke algebras and their representation theory. Representation theory, automorphic forms, and complex geometry. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. The associated iwahorihecke algebras and their trace functions are the subject of section 4. These results draw heavily on the work of howlett and lehrer 31 who successfully followed a similar approach for the representation theory of. The algebra h hr,q is called the affine hecke algebra or iwahori. Iwahori hecke algebras and their representation theory. In section 5, we introduce markov traces for iwahorihecke algebras of classical type, and study some of their basic properties.
The modular representation theory of iwahori hecke algebras and this theory s connection to groups of lie type is an area of rapidly expanding interest. In classifying the irreducible representations of iwahori. Homological representations of the iwahorihecke algebra 1. Some reducible specht modules for iwahorihecke algebras. This paper is a continuation of 3 in which the rst two authors have introduced the spherical hecke algebra and the satake isomorphism for an untwisted a ne kacmoody group over a nonarchimedian local eld. In this generality, iwahori hecke algebras have significance far beyond their origin in the representation theory of \p\adic groups. It offers a substantially simplified treatment of the original proofs. This article is a fairly selfcontained treatment of some of the basic facts about iwahori hecke algebras attached to split connected reductive padic groups. It will also serve as a good introduction to students and beginning researchers since each chapter contains exercises and there is an appendix containing a quick development of the representation theory of algebras. Starkeys rule, a computation of a type a hecke algebra character table, using the. Recently, the representation theory of algebraic groups over 2dimensional lo. This volume consists of notes of the courses on iwahorihecke algebras and their representation theory, given during the cime summer school which took place in 1999 in martina franca, italy.
Other readers will always be interested in your opinion of the books youve read. Representations of hecke algebras at roots of unity. Representations of iwahorihecke algebras induced from parabolic. The best explanation of why the rank of jif is independent of q is that the iwahorihecke algebras appear naturally in the representation theory of the general linear groups. This thesis lays the foundationsfor a theory of unipotent hecke algebras, a family of hecke algebras that includes both the classical gelfandgraev hecke algebra and a generalization of the iwahori hecke algebra the yokonuma hecke algebra. Representation theory an electronic journal of the american mathematical society volume 4, pages 370397 september 11, 2000 s 1088416500000935 on the representation theory o. You can find the proof for example in bourbaki, lie groups and algebras, chap. In mathematics, an affine hecke algebra is the algebra associated to an affine weyl group. Given a coxeter group was above, there is an algebra called the iwahori hecke algebra which we now describe. In this talk, i will give a purely algebraic realization for these quantum algebras and their canonical bases. In mathematics, the iwahori hecke algebra, or hecke algebra, named for erich hecke and nagayoshi iwahori, is a deformation of the group algebra of a coxeter group. Abstract this volume presents a fully selfcontained introduction to the modular representation theory of the iwahori hecke algebras of the symmetric groups and of the qschur algebras.
The vdecomposition numbers are the coefficients when the canonical basis for this representation is expanded in terms of the basis of partitions, and the evaluations at v 1 of these polynomials give the decomposition numbers for iwahorihecke algebras and qschur algebras over c. Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. There is a definition of iwahorihecke algebras for coxeter groups in terms of generators and relations and there is a definition of hecke algebras involving functions on locally compact groups. This volume consists of notes of the courses on iwahori hecke algebras and their representation theory, given during the cime summer school which took place in 1999 in martina franca, italy.
A geometric construction of the iwahorihecke algebra for. Support varieties for hecke algebras, joint work with dan nakano. Representation theory of vertex operator algebras and. Pdf iwahorihecke algebras and schur algebras of the symmetric. Representations of hecke algebras at roots of unity algebra. In particular by results of barbaschmoy and subsequently barbaschciubotaru, there is a precise relation between the unitary dual of a block of representations of a padic group and a particular iwahori hecke algebra.
Cocenters and representations of prop hecke algebras american. Utilizing new research and results, this book can be studied from a combinatorial, algorithmic or algebraic viewpoint. This will be the key step in describing our general plan for trace functions, along the lines of gp. These iwahorihecke algebras arise in the study of groups with bnpairs, and their representation theory bears a close relationship to the representation theory of the corresponding coxeter groups. Representation theory of lie superalgebras and related topics. On the classification of irreducible representations of affine hecke. Chapter 4 of introduction to soergel bimodules, to appear in the rsme springer series. In cartan type an, the hecke algebra is a natural deformation of the rook monoid. We study the representation theory of three towers of algebras which are related to the symmetric groups and their hecke algebras. Iwahorihecke algebras department of mathematics university of. Representations of iwahorihecke algebras induced from. Their construction relies crucially on certain deformed versions of the underlying algebras, namely, iwahorihecke algebras of weyl groups and the drinfeldjimboquantized enveloping algebras. Kazhdanlusztig theory is a key to understanding the representation theory of the iwahorihecke algebra hqw, l. Prop iwahorihecke algebras are gorenstein 3 algebras.
The key point in identifying the hecke algebra with an algebra of functions on a group is the presence of a tits system, or bnpair. Matrix coefficients and iwahorihecke algebra modules. The modular representation theory of iwahorihecke algebras and this theory s connection to groups of lie type is an area of rapidly expanding interest. The fock space has the structure of a umodule, and has important connections to the representation theory of iwahorihecke algebras. Hecke algebras and harmonic analysis 1229 the form hw,q for a certain af. Iwahori hecke algebras and their representation theory, in particular the theory of socalled imprimitive representations or imprimitive modules.
Summer school held in martina franca, italy, june 28 july 6, 1999. Their role in the representation theory of reductive padic groups lus6. Abstract the iwahori hecke algebras of nite coxeter groups play an important role in many areas of mathematics. Iwahorihecke algebras, qschur algebras, cellular algebras. Especially, a propiwahori hecke algebra which is attached to a propiwahori subgroup i1 has an important role in the study. Buy representations of hecke algebras at roots of unity. This geometric construction is rather complex but it enables us to apply powerful sheaftheoretic tools such as the bbd decomposition theorem to study klr algebras and their representation theory. Sep 09, 2003 this article gives a fairly selfcontained treatment of the basic facts about the iwahori hecke algebra of a split padic group, including bernsteins presentation, macdonalds formula, the casselmanshalika formula, and the lusztigkato formula. Abstract this volume presents a fully selfcontained introduction to the modular representation theory of the iwahorihecke algebras of the symmetric groups and of the qschur algebras. Selected titles in this series american mathematical society. 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.
Kazhdanlusztig polynomials to representation theory of the lie algebra g of g. If there is a repetition in the number of beads, we will look for the smaller 1 where it is. The study of these algebras was pioneered by dipper and james in a series of landmark papers. These crossingless matchings representations have found applications in both knot theory not geometric representation theory. Representation theories of some towers of algebras related to the symmetric groups and their hecke algebras florent hivert and nicolas m. Their construction relies crucially on certain deformed versions of the underlying algebras, namely, iwahorihecke algebras of weyl groups and the.
Characters of finite coxeter groups and iwahorihecke algebras. Iwahorihecke algebras and schur algebras of the symmetric. The modular representation theory of iwahorihecke algebras and this theorys connection to groups of lie type is an area of rapidly expanding interest. Murphyon the representation theory of the symmetric groups and associated hecke algebras. Kazhdanlusztig theory is a key to understanding the representation theory of the iwahorihecke algebra h. Abstract references similar articles additional information.
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. Some of these ideas have been generalized to weyl groups, a larger class of groups whose structure is essential in the study of lie groups and lie algebras. Hecke algebras, generalisations and representation theory hal. This volume presents a fully selfcontained introduction to the modular representation theory of the iwahori hecke algebras of the symmetric groups and of the \q\schur algebras. Iwahorihecke algebras of sl over dimensional local fields. Hecke algebras are quotients of the group rings of artin braid groups. Two basic problems of representation theory are to classify irreducible. Keywords iwahori, hecke algebra, representation, braid group, configu ration space. We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras.
Algebras of iwahori hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of lie type. These bases are the kazhdanlusztig bases from 7, and lusztigs canonical bases originating in 10 which are kashiwaras global crystal bases, respectively. Their construction relies crucially on certain deformed versions of. Algebras of iwahorihecke type are one of the tools and were probably. References ams representation theory of the american. Iwahorihecke algebras and their representation theory, in particular the theory of socalled imprimitive representations or imprimitive modules. This article gives a fairly selfcontained treatment of the basic facts about the iwahorihecke algebra of a split padic group, including bernsteins presentation, macdonalds formula, the casselmanshalika formula, and the lusztigkato formula. Throughout this paper we will use the terms module and representation. If v is a representation, let v be the contragredient representation. Another runner removal theorem for vdecomposition numbers of. Iwahorihecke algebras of typeaat roots of unity sciencedirect. This connection found a spectacular application in vaughan jones construction of new invariants of knots.
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