**University Lecture Series**

Volume: 26;
2002;
158 pp;
Softcover

MSC: Primary 05; 17; 20;
Secondary 14; 16

Print ISBN: 978-0-8218-3232-5

Product Code: ULECT/26

List Price: $39.00

AMS Member Price: $31.20

MAA member Price: $35.10

**Electronic ISBN: 978-1-4704-1833-5
Product Code: ULECT/26.E**

List Price: $39.00

AMS Member Price: $31.20

MAA member Price: $35.10

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# Representations of Quantum Algebras and Combinatorics of Young Tableaux

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*Susumu Ariki*

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 \(A_{r-1}^{(1)}\) 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)
Lascoux-Leclerc-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, and related fields.

#### Readership

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.

#### Reviews & Endorsements

The author gives a good introduction to the algebraic aspects of this fast-developing field … Overall, this is a well-written 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

#### Table of Contents

# Table of Contents

## Representations of Quantum Algebras and Combinatorics of Young Tableaux

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Introduction 110 free
- The Serre relations 918 free
- Kac-Moody Lie algebras 1524
- Crystal bases of 𝑈ᵥ-modules 2332
- The tensor product of crystals 2938
- Crystal bases of 𝑈ᵥ⁻ 3746
- The canonical basis 4756
- Existence and uniqueness (part I) 5564
- Existence and uniqueness (part II) 6170
- The Hayashi realization 7584
- Description of the crystal graph of 𝑉(Λ) 8796
- An overview of the application to Hecke algebras 97106
- The Hecke algebra of type 𝐺(𝑚,1,𝑛) 105114
- The proof of Theorem 12.5 123132
- Reference guide 147156
- Bibliography 149158
- Index 157166 free
- Back Cover Back Cover1169