Kazhdan-Lusztig Cells with Unequal Parameters
Kazhdan-Lusztig Cells with Unequal Parameters

Author: Cédric Bonnafé

Publisher: Springer

Published: 2018-05-07

Total Pages: 350

ISBN-13: 3319707361

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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.

Kazhdan-Lusztig Cells in Type Bn with Unequal Parameters
Kazhdan-Lusztig Cells in Type Bn with Unequal Parameters

Author: Edmund Howse

Publisher:

Published: 2016

Total Pages: 0

ISBN-13:

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This mathematics thesis deals with combinatorial representation theory. Cells were introduced in a 1979 paper written by D. Kazhdan and G. Lusztig, and have intricate links with many areas of mathematics, including the representation theory of Coxeter groups, Iwahori-Hecke algebras, semisimple complex Lie algebras, reductive algebraic groups and Lie groups. One of the main problems in the theory of cells is their classification for all finite Coxeter groups. This thesis is a detailed study of cells in type Bn with respect to certain choices of parameters, and contributes to the classification by giving the first characterisation of left cells when b/a = n − 1. Other results include the introduction of a generalised version of the enhanced right descent set and exhibiting the asymptotic left cells of type Bn as left Vogan classes. Combinatorial results give rise to efficient algorithms so that cells can be determined with a computer; the methods involved in this work transfer to a new, faster way of calculating the cells with respect to the studied parameters. The appendix is a Python file containing code to make such calculations.

Kazhdan-Lusztig Cells in Affine Weyl Groups with Unequal Parameters
Kazhdan-Lusztig Cells in Affine Weyl Groups with Unequal Parameters

Author: Jérémie Guilhot

Publisher:

Published: 2008

Total Pages: 120

ISBN-13:

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Hecke algebras arise naturally in the representation theory of reductive groups over finite or p-adic fields. These algebras are specializations of Iwahori-Hecke algebras which can be defined in terms of a Coxeter group and a weight function without reference to reductive groups and this is the setting we are working in. Kazhdan-Lusztig cells play a crucial role in the study of Iwahori-Hecke algebras. The aim of this work is to study the Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters. More precisely, we show that the Kazhdan-Lusztig polynomials of an affine Weyl group are invariant under ``long enough'' translations, we decompose the lowest two-sided cell into left cells and we determine the decomposition of the affine Weyl group of type G into cells for a whole class of weight functions.

Hecke Algebras with Unequal Parameters
Hecke Algebras with Unequal Parameters

Author: George Lusztig

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 145

ISBN-13: 0821833561

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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.

Reflection Groups and Coxeter Groups
Reflection Groups and Coxeter Groups

Author: James E. Humphreys

Publisher: Cambridge University Press

Published: 1992-10

Total Pages: 222

ISBN-13: 9780521436137

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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.

Representation Theory of Algebraic Groups and Quantum Groups
Representation Theory of Algebraic Groups and Quantum Groups

Author: Akihiko Gyoja

Publisher: Springer Science & Business Media

Published: 2010-11-25

Total Pages: 356

ISBN-13: 0817646973

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Invited articles by top notch experts Focus is on topics in representation theory of algebraic groups and quantum groups Of interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics

Introduction to Soergel Bimodules
Introduction to Soergel Bimodules

Author: Ben Elias

Publisher: Springer Nature

Published: 2020-09-26

Total Pages: 588

ISBN-13: 3030488268

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This book provides a comprehensive introduction to Soergel bimodules. First introduced by Wolfgang Soergel in the early 1990s, they have since become a powerful tool in geometric representation theory. On the one hand, these bimodules are fairly elementary objects and explicit calculations are possible. On the other, they have deep connections to Lie theory and geometry. Taking these two aspects together, they offer a wonderful primer on geometric representation theory. In this book the reader is introduced to the theory through a series of lectures, which range from the basics, all the way to the latest frontiers of research. This book serves both as an introduction and as a reference guide to the theory of Soergel bimodules. Thus it is intended for anyone who wants to learn about this exciting field, from graduate students to experienced researchers.

Representations of Reductive Groups
Representations of Reductive Groups

Author: Monica Nevins

Publisher: Birkhäuser

Published: 2015-12-18

Total Pages: 545

ISBN-13: 3319234439

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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