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Representations of Quantum Algebras and Combinatorics of Young Tableaux

Representations of Quantum Algebras and Combinatorics of Young Tableaux

Buch | Softcover
2002
American Mathematical Society (Verlag)
978-0-8218-3232-5 (ISBN)
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Introduces the representation theory of quantum groups using quantum groups of type $A_{r-1}^{(1)}$ as a main example. This book also explains the proof of the (generalized) Lascoux-Leclerc-Thibon conjecture. It is suitable for those interested in representation theory of algebraic groups and quantum groups, and the theory of Hecke algebras.
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.

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.

Erscheint lt. Verlag 1.8.2002
Reihe/Serie University Lecture Series
Zusatzinfo bibliography, index
Verlagsort Providence
Sprache englisch
Gewicht 312 g
Themenwelt Mathematik / Informatik Mathematik Algebra
Mathematik / Informatik Mathematik Geometrie / Topologie
Mathematik / Informatik Mathematik Graphentheorie
ISBN-10 0-8218-3232-8 / 0821832328
ISBN-13 978-0-8218-3232-5 / 9780821832325
Zustand Neuware
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