Representation of Quantum Algebras and Combinatorics of Young Tableaux

University Lecture Series - volume 26

Susumu Ariki

158 pages, parution le 09/04/2003

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Résumé

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 ⁽¹⁾_{r-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.

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

L'auteur - Susumu Ariki

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

Caractéristiques techniques

	PAPIER
Éditeur(s)	American Mathematical Society (AMS)
Auteur(s)	Susumu Ariki
Parution	09/04/2003
Nb. de pages	158
Format	17,5 x 25
Couverture	Broché
Poids	295g
Intérieur	Noir et Blanc
EAN13	9780821832325
ISBN13	978-0-8218-3232-5