Hecke Algebras

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Representations of Hecke Algebras at Roots of Unity

Meinolf Geck,Nicolas Jacon
Representations of Hecke Algebras at Roots of Unity Book PDF

Author : Meinolf Geck,Nicolas Jacon
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 49,7 Mb
Release : 2011-05-18
Category : Mathematics
ISBN : 9780857297167

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Representations of Hecke Algebras at Roots of Unity by Meinolf Geck,Nicolas Jacon Pdf

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; it is one that has also seen a number of breakthroughs in recent years. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of unity, this book is a particularly valuable addition to current research in this field. Using the framework provided by the Kazhdan-Lusztig theory of cells, the authors develop an analogue of James' (1970) "characteristic-free'' approach to the representation theory of Iwahori-Hecke algebras in general. Presenting a systematic and unified treatment of representations of Hecke algebras at roots of unity, this book is unique in its approach and includes new results that have not yet been published in book form. It also serves as background reading to further active areas of current research such as the theory of affine Hecke algebras and Cherednik algebras. The main results of this book are obtained by an interaction of several branches of mathematics, namely the theory of Fock spaces for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the theory of Kazhdan-Lusztig bases and cells, and computational methods. This book will be of use to researchers and graduate students in representation theory as well as any researchers outside of the field with an interest in Hecke algebras.

Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group

Andrew Mathas
Iwahori Hecke Algebras and Schur Algebras of the Symmetric Group Book PDF

Author : Andrew Mathas
Publisher : American Mathematical Soc.
Page : 204 pages
File Size : 41,6 Mb
Release : 1999
Category : Representations of algebras
ISBN : 9780821819265

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Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group by Andrew Mathas Pdf

This volume presents a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the $q$-Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems. The main results are proved by showing that the Iwahori-Hecke algebras and $q$-Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the $q$-Schur algebras. This book is the first of its kind covering the topic. It offers a substantially simplified treatment of the original proofs. The book is a solid reference source for experts. 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. A second appendix gives tables of decomposition numbers.

Hecke Algebras

Aloys Krieg
Hecke Algebras Book PDF

Author : Aloys Krieg
Publisher : American Mathematical Soc.
Page : 158 pages
File Size : 52,6 Mb
Release : 1990
Category : Mathematics
ISBN : 9780821824979

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Hecke Algebras by Aloys Krieg Pdf

This volume gives an introduction to the algebraic theory of Hecke algebras, which can be viewed as generalizations of group algebras. At first a careful look at the product leads to liftings of the basic isomorphism theorems and of anti-homomorphisms from the group level to the attached Hecke algebras.

Hecke Algebras with Unequal Parameters

George Lusztig
Hecke Algebras with Unequal Parameters Book PDF

Author : George Lusztig
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 55,5 Mb
Release : 2003
Category : Hecke algebras
ISBN : 9780821833568

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Hecke Algebras with Unequal Parameters by George Lusztig Pdf

Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over $p$-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives researchers and graduate students working in the theory of algebraic groups and their representations an invaluable insight and a wealth of new and useful information.

Double Affine Hecke Algebras

Ivan Cherednik
Double Affine Hecke Algebras Book PDF

Author : Ivan Cherednik
Publisher : Cambridge University Press
Page : 449 pages
File Size : 48,8 Mb
Release : 2005-03-21
Category : Mathematics
ISBN : 9780521609180

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Double Affine Hecke Algebras by Ivan Cherednik Pdf

This is an essentially self-contained monograph centered on the new double Hecke algebra technique.

Representations of Affine Hecke Algebras

Nanhua Xi
Representations of Affine Hecke Algebras Book PDF

Author : Nanhua Xi
Publisher : Springer
Page : 147 pages
File Size : 46,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540486824

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Representations of Affine Hecke Algebras by Nanhua Xi Pdf

Kazhdan and Lusztig classified the simple modules of an affine Hecke algebra Hq (q E C*) provided that q is not a root of 1 (Invent. Math. 1987). Ginzburg had some very interesting work on affine Hecke algebras. Combining these results simple Hq-modules can be classified provided that the order of q is not too small. These Lecture Notes of N. Xi show that the classification of simple Hq-modules is essentially different from general cases when q is a root of 1 of certain orders. In addition the based rings of affine Weyl groups are shown to be of interest in understanding irreducible representations of affine Hecke algebras. Basic knowledge of abstract algebra is enough to read one third of the book. Some knowledge of K-theory, algebraic group, and Kazhdan-Lusztig cell of Cexeter group is useful for the rest

Reflection Groups and Coxeter Groups

James E. Humphreys
Reflection Groups and Coxeter Groups Book PDF

Author : James E. Humphreys
Publisher : Cambridge University Press
Page : 222 pages
File Size : 48,6 Mb
Release : 1992-10
Category : Mathematics
ISBN : 0521436133

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Reflection Groups and Coxeter Groups by James E. Humphreys Pdf

This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.

Affine Hecke Algebras and Orthogonal Polynomials

I. G. Macdonald
Affine Hecke Algebras and Orthogonal Polynomials Book PDF

Author : I. G. Macdonald
Publisher : Cambridge University Press
Page : 200 pages
File Size : 43,7 Mb
Release : 2003-03-20
Category : Mathematics
ISBN : 0521824729

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Affine Hecke Algebras and Orthogonal Polynomials by I. G. Macdonald Pdf

First account of a theory, created by Macdonald, of a class of orthogonal polynomial, which is related to mathematical physics.

Blocks and Families for Cyclotomic Hecke Algebras

Maria Chlouveraki
Blocks and Families for Cyclotomic Hecke Algebras Book PDF

Author : Maria Chlouveraki
Publisher : Springer Science & Business Media
Page : 173 pages
File Size : 54,7 Mb
Release : 2009-09-14
Category : Mathematics
ISBN : 9783642030635

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Blocks and Families for Cyclotomic Hecke Algebras by Maria Chlouveraki Pdf

The definition of Rouquier for the families of characters introduced by Lusztig for Weyl groups in terms of blocks of the Hecke algebras has made possible the generalization of this notion to the case of complex reflection groups. The aim of this book is to study the blocks and to determine the families of characters for all cyclotomic Hecke algebras associated to complex reflection groups. This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory, and can also serve as an introduction to the Hecke algebras of complex reflection groups.

Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras

Meinolf Geck,Götz Pfeiffer
Characters of Finite Coxeter Groups and Iwahori Hecke Algebras Book PDF

Author : Meinolf Geck,Götz Pfeiffer
Publisher : Oxford University Press
Page : 478 pages
File Size : 44,7 Mb
Release : 2000
Category : Mathematics
ISBN : 0198502508

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Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras by Meinolf Geck,Götz Pfeiffer Pdf

Finite Coxeter groups and related structures arise naturally in several branches of mathematics such as the theory of Lie algebras and algebraic groups. The corresponding Iwahori-Hecke algebras are then obtained by a certain deformation process which have applications in the representation theory of groups of Lie type and the theory of knots and links. This book develops the theory of conjugacy classes and irreducible character, both for finite Coxeter groups and the associated Iwahori-Hecke algebras. Topics covered range from classical results to more recent developments and are clear and concise. This is the first book to develop these subjects both from a theoretical and an algorithmic point of view in a systematic way, covering all types of finite Coxeter groups.

Blocks and Families for Cyclotomic Hecke Algebras

Maria Chlouveraki
Blocks and Families for Cyclotomic Hecke Algebras Book PDF

Author : Maria Chlouveraki
Publisher : Springer
Page : 166 pages
File Size : 53,8 Mb
Release : 2009-08-29
Category : Mathematics
ISBN : 9783642030642

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Blocks and Families for Cyclotomic Hecke Algebras by Maria Chlouveraki Pdf

This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory. It can also serve as an introduction to the Hecke algebras of complex reflection groups.

Gelfand Triples and Their Hecke Algebras

Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli
Gelfand Triples and Their Hecke Algebras Book PDF

Author : Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli
Publisher : Springer Nature
Page : 153 pages
File Size : 48,7 Mb
Release : 2020-09-25
Category : Mathematics
ISBN : 9783030516079

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Gelfand Triples and Their Hecke Algebras by Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli Pdf

This monograph is the first comprehensive treatment of multiplicity-free induced representations of finite groups as a generalization of finite Gelfand pairs. Up to now, researchers have been somehow reluctant to face such a problem in a general situation, and only partial results were obtained in the one-dimensional case. Here, for the first time, new interesting and important results are proved. In particular, after developing a general theory (including the study of the associated Hecke algebras and the harmonic analysis of the corresponding spherical functions), two completely new highly nontrivial and significant examples (in the setting of linear groups over finite fields) are examined in full detail. The readership ranges from graduate students to experienced researchers in Representation Theory and Harmonic Analysis.

Iwahori-Hecke Algebras and Their Representation Theory

Ivan Cherednik
Iwahori Hecke Algebras and Their Representation Theory Book PDF

Author : Ivan Cherednik
Publisher : Springer Science & Business Media
Page : 132 pages
File Size : 55,9 Mb
Release : 2002-12-19
Category : Mathematics
ISBN : 3540002243

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Iwahori-Hecke Algebras and Their Representation Theory by Ivan Cherednik Pdf

Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. 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. 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.

Hecke Algebras with Unequal Parameters

George Lusztig
Hecke Algebras with Unequal Parameters Book PDF

Author : George Lusztig
Publisher : Unknown
Page : 136 pages
File Size : 52,5 Mb
Release : 2003
Category : Hecke algebras
ISBN : 1470438631

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Hecke Algebras with Unequal Parameters by George Lusztig Pdf

Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over p-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interestin.

Representations of Quantum Algebras and Combinatorics of Young Tableaux

Susumu Ariki
Representations of Quantum Algebras and Combinatorics of Young Tableaux Book PDF

Author : Susumu Ariki
Publisher : American Mathematical Soc.
Page : 169 pages
File Size : 49,6 Mb
Release : 2002
Category : Bases (Linear topological spaces).
ISBN : 9780821832325

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Representations of Quantum Algebras and Combinatorics of Young Tableaux by Susumu Ariki Pdf

This book contains most of the nonstandard material necessary to get acquainted with this new rapidly developing area. It can be used as a good entry point into the study of representations of quantum groups. Among several tools used in studying representations of quantum groups (or quantum algebras) are the notions of Kashiwara's crystal bases and Lusztig's canonical bases. Mixing both approaches allows us to use a combinatorial approach to representations of quantum groups and toapply the theory to representations of Hecke algebras. The primary goal of this book is to introduce the representation theory of quantum groups using quantum groups of type $A {r-1 {(1) $ as a main example. The corresponding combinatorics, developed by Misra and Miwa, turns out to be thecombinatorics of Young tableaux. The second goal of this book is to explain the proof of the (generalized) Leclerc-Lascoux-Thibon conjecture. This conjecture, which is now a theorem, is an important breakthrough in the modular representation theory of the Hecke algebras of classical type. The book is suitable for graduate students and research mathematicians interested in representation theory of algebraic groups and quantum groups, the theory of Hecke algebras, algebraic combinatorics, andrelated fields.