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

"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

• Introduction
• The Serre relations
• Kac-Moody Lie algebras
• Crystal bases of \(U_v\)-modules
• The tensor product of crystals
• Crystal bases of \(U_v^-\)
• The canonical basis
• Existence and uniqueness (part I)
• Existence and uniqueness (part II)
• The Hayashi realization
• Description of the crystal graph of \(V(\Lambda)\)
• An overview of the application to Hecke algebras
• The Hecke algebra of type \(G(m,1,n)\)
• The proof of Theorem 12.5
• Reference guide
• Bibliography
• Index