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.

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.

Combinatorics of Coxeter Groups
Combinatorics of Coxeter Groups

Author: Anders Bjorner

Publisher: Springer Science & Business Media

Published: 2006-02-25

Total Pages: 371

ISBN-13: 3540275967

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Includes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

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

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.

Forty Years Of Algebraic Groups, Algebraic Geometry, And Representation Theory In China: In Memory Of The Centenary Year Of Xihua Cao's Birth
Forty Years Of Algebraic Groups, Algebraic Geometry, And Representation Theory In China: In Memory Of The Centenary Year Of Xihua Cao's Birth

Author: Jie Du

Publisher: World Scientific

Published: 2022-10-21

Total Pages: 490

ISBN-13: 9811263507

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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.

Finite Dimensional Algebras and Quantum Groups
Finite Dimensional Algebras and Quantum Groups

Author: Bangming Deng

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 790

ISBN-13: 0821841866

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"The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak{gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature."--Publisher's website.