Softcover ISBN:  9780821832325 
Product Code:  ULECT/26 
List Price:  $42.00 
MAA Member Price:  $37.80 
AMS Member Price:  $33.60 
Electronic ISBN:  9781470418335 
Product Code:  ULECT/26.E 
List Price:  $39.00 
MAA Member Price:  $35.10 
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Book DetailsUniversity Lecture SeriesVolume: 26; 2002; 158 ppMSC: Primary 05; 17; 20; Secondary 14; 16;
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 to apply 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 \(A_{r1}^{(1)}\) as a main example. The corresponding combinatorics, developed by Misra and Miwa, turns out to be the combinatorics of Young tableaux.
The second goal of this book is to explain the proof of the (generalized) LascouxLeclercThibon 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, and related fields.ReadershipGraduate students and research mathematicians interested in representation theory of algebraic groups and quantum groups, the theory of Hecke algebras, algebraic combinatorics, and related fields.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. The Serre relations

Chapter 3. KacMoody Lie algebras

Chapter 4. Crystal bases of $U_v$modules

Chapter 5. The tensor product of crystals

Chapter 6. Crystal bases of $U_v^$

Chapter 7. The canonical basis

Chapter 8. Existence and uniqueness (part I)

Chapter 9. Existence and uniqueness (part II)

Chapter 10. The Hayashi realization

Chapter 11. Description of the crystal graph of $V(\Lambda )$

Chapter 12. An overview of the application to Hecke algebras

Chapter 13. The Hecke algebra of type $G(m,1,n)$

Chapter 14. The proof of Theorem 12.5

Chapter 15. Reference guide


Reviews

The author gives a good introduction to the algebraic aspects of this fastdeveloping field … Overall, this is a wellwritten and clear exposition of the theory needed to understand the latest advances in the theory of the canonical/global crystal basis and the links with the representation theory of symmetric groups and Hecke algebras. The book finishes with an extensive bibliography of papers, which is well organised into different areas of the theory for easy reference.
Zentralblatt MATH 
Well written and covers ground quickly to get to the heart of the theory … should serve as a solid introduction … abundant references to the literature are given.
Mathematical Reviews


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 Book Details
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 Get Permissions
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 to apply 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 \(A_{r1}^{(1)}\) as a main example. The corresponding combinatorics, developed by Misra and Miwa, turns out to be the combinatorics of Young tableaux.
The second goal of this book is to explain the proof of the (generalized) LascouxLeclercThibon 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, and related fields.
Graduate students and research mathematicians interested in representation theory of algebraic groups and quantum groups, the theory of Hecke algebras, algebraic combinatorics, and related fields.

Chapters

Chapter 1. Introduction

Chapter 2. The Serre relations

Chapter 3. KacMoody Lie algebras

Chapter 4. Crystal bases of $U_v$modules

Chapter 5. The tensor product of crystals

Chapter 6. Crystal bases of $U_v^$

Chapter 7. The canonical basis

Chapter 8. Existence and uniqueness (part I)

Chapter 9. Existence and uniqueness (part II)

Chapter 10. The Hayashi realization

Chapter 11. Description of the crystal graph of $V(\Lambda )$

Chapter 12. An overview of the application to Hecke algebras

Chapter 13. The Hecke algebra of type $G(m,1,n)$

Chapter 14. The proof of Theorem 12.5

Chapter 15. Reference guide

The author gives a good introduction to the algebraic aspects of this fastdeveloping field … Overall, this is a wellwritten and clear exposition of the theory needed to understand the latest advances in the theory of the canonical/global crystal basis and the links with the representation theory of symmetric groups and Hecke algebras. The book finishes with an extensive bibliography of papers, which is well organised into different areas of the theory for easy reference.
Zentralblatt MATH 
Well written and covers ground quickly to get to the heart of the theory … should serve as a solid introduction … abundant references to the literature are given.
Mathematical Reviews