Hecke Algebras With Unequal Parameters

Author : George Lusztig
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 52,5 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.

Author : George Lusztig
Publisher : Unknown
Page : 136 pages
File Size : 48,5 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.

Author : Cédric Bonnafé
Publisher : Springer
Page : 348 pages
File Size : 44,5 Mb
Release : 2018-05-07
Category : Mathematics
ISBN : 9783319707365

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Kazhdan-Lusztig Cells with Unequal Parameters by Cédric Bonnafé Pdf

This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig’s seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group. Kazhdan-Lusztig Cells with Unequal Parameters will appeal to graduate students and researchers interested in related subject areas, such as Lie theory, representation theory, and combinatorics of Coxeter groups. Useful examples and various exercises make this book suitable for self-study and use alongside lecture courses. Information for readers: The character {\mathbb{Z}} has been corrupted in the print edition of this book and appears incorrectly with a diagonal line running through the symbol.

Author : Meinolf Geck,Götz Pfeiffer
Publisher : Oxford University Press
Page : 478 pages
File Size : 40,5 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.

Author : Meinolf Geck,Nicolas Jacon
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 49,5 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.

Author : Maria Chlouveraki
Publisher : Springer Science & Business Media
Page : 173 pages
File Size : 40,5 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.

Author : Nanhua Xi
Publisher : Springer
Page : 147 pages
File Size : 54,5 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

Author : Ara, Pere
Publisher : American Mathematical Soc.
Page : 326 pages
File Size : 49,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821852972

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New Trends in Noncommutative Algebra by Ara, Pere Pdf

This volume contains the proceedings of the conference `New Trends in Noncommutative Algebra', held at the University of Washington, Seattle, in August 2010. The articles will provide researchers and graduate students with an indispensable overview of topics of current interest. Specific fields covered include: noncommutative algebraic geometry, representation theory, Calabi-Yau algebras, quantum algebras and deformation quantization, Poisson algebras, group algebras, and noncommutative Iwasawa algebras.

Author : Monica Nevins,Peter E. Trapa
Publisher : Birkhäuser
Page : 532 pages
File Size : 42,6 Mb
Release : 2015-12-18
Category : Mathematics
ISBN : 9783319234434

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Representations of Reductive Groups by Monica Nevins,Peter E. Trapa Pdf

Over the last forty years, David Vogan has left an indelible imprint on the representation theory of reductive groups. His groundbreaking ideas have lead to deep advances in the theory of real and p-adic groups, and have forged lasting connections with other subjects, including number theory, automorphic forms, algebraic geometry, and combinatorics. Representations of Reductive Groups is an outgrowth of the conference of the same name, dedicated to David Vogan on his 60th birthday, which took place at MIT on May 19-23, 2014. This volume highlights the depth and breadth of Vogan's influence over the subjects mentioned above, and point to many exciting new directions that remain to be explored. Notably, the first article by McGovern and Trapa offers an overview of Vogan's body of work, placing his ideas in a historical context. Contributors: Pramod N. Achar, Jeffrey D. Adams, Dan Barbasch, Manjul Bhargava, Cédric Bonnafé, Dan Ciubotaru, Meinolf Geck, William Graham, Benedict H. Gross, Xuhua He, Jing-Song Huang, Toshiyuki Kobayashi, Bertram Kostant, Wenjing Li, George Lusztig, Eric Marberg, William M. McGovern, Wilfried Schmid, Kari Vilonen, Diana Shelstad, Peter E. Trapa, David A. Vogan, Jr., Nolan R. Wallach, Xiaoheng Wang, Geordie Williamson

Author : Anna Beliakova,Aaron D. Lauda
Publisher : American Mathematical Soc.
Page : 361 pages
File Size : 53,7 Mb
Release : 2017-02-21
Category : Algebra
ISBN : 9781470424602

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Categorification and Higher Representation Theory by Anna Beliakova,Aaron D. Lauda Pdf

The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory. This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.

Author : Werner Müller,Sug Woo Shin,Nicolas Templier
Publisher : Springer
Page : 453 pages
File Size : 44,7 Mb
Release : 2018-10-11
Category : Mathematics
ISBN : 9783319948331

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Geometric Aspects of the Trace Formula by Werner Müller,Sug Woo Shin,Nicolas Templier Pdf

The second of three volumes devoted to the study of the trace formula, these proceedings focus on automorphic representations of higher rank groups. Based on research presented at the 2016 Simons Symposium on Geometric Aspects of the Trace Formula that took place in Schloss Elmau, Germany, the volume contains both original research articles and articles that synthesize current knowledge and future directions in the field. The articles discuss topics such as the classification problem of representations of reductive groups, the structure of Langlands and Arthur packets, interactions with geometric representation theory, and conjectures on the global automorphic spectrum. Suitable for both graduate students and researchers, this volume presents the latest research in the field. Readers of the first volume Families of Automorphic Forms and the Trace Formula will find this a natural continuation of the study of the trace formula.

Author : David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,J. Toby Stafford,Michel Van den Bergh
Publisher : Cambridge University Press
Page : 463 pages
File Size : 42,5 Mb
Release : 2015-11-19
Category : Mathematics
ISBN : 9781107065628

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Commutative Algebra and Noncommutative Algebraic Geometry by David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,J. Toby Stafford,Michel Van den Bergh Pdf

This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 1 contains expository papers ideal for those entering the field.

Author : Jie Du,Jianpan Wang,Lei Lin
Publisher : World Scientific
Page : 490 pages
File Size : 48,9 Mb
Release : 2022-10-21
Category : Mathematics
ISBN : 9789811263507

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Forty Years Of Algebraic Groups, Algebraic Geometry, And Representation Theory In China: In Memory Of The Centenary Year Of Xihua Cao's Birth by Jie Du,Jianpan Wang,Lei Lin Pdf

Professor Xihua Cao (1920-2005) was a leading scholar at East China Normal University (ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is particularly the formation of a world-renowned 'ECNU School' in algebra, covering research areas include algebraic groups, quantum groups, algebraic geometry, Lie algebra, algebraic number theory, representation theory and other hot fields. In January 2020, in order to commemorate Professor Xihua Cao's centenary birthday, East China Normal University held a three-day academic conference. Scholars at home and abroad gave dedications or delivered lectures in the conference. This volume originates from the memorial conference, collecting the dedications of scholars, reminiscences of family members, and 16 academic articles written based on the lectures in the conference, covering a wide range of research hot topics in algebra. The book shows not only scholars' respect and memory for Professor Xihua Cao, but also the research achievements of Chinese scholars at home and abroad.

Author : Robert S. Doran, Paul J. Sally, Jr., and Loren Spice
Publisher : American Mathematical Soc.
Page : 294 pages
File Size : 44,7 Mb
Release : 2011
Category : Harmonic analysis
ISBN : 9780821874035

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Harmonic Analysis on Reductive, $p$-adic Groups by Robert S. Doran, Paul J. Sally, Jr., and Loren Spice Pdf

Author : Caterina Consani,Matilde Marcolli
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 41,8 Mb
Release : 2007-12-18
Category : Mathematics
ISBN : 9783834803528

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Noncommutative Geometry and Number Theory by Caterina Consani,Matilde Marcolli Pdf

In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.