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Crystal Bases: Representations And Combinatorics

2017-01-17 Mathematics
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Release : 2017-01-17
Genre : Mathematics
Kind : eBook
Book Rating : 466/5 ( reviews)

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Book Synopsis Crystal Bases: Representations And Combinatorics by : Daniel Bump

Download or read book Crystal Bases: Representations And Combinatorics written by Daniel Bump. This book was released on 2017-01-17. Available in PDF, EPUB and Kindle. Book excerpt: This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.

Crystal Bases

2017 Combinatorial analysis
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Release : 2017
Genre : Combinatorial analysis
Kind : eBook
Book Rating : 441/5 ( reviews)

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Book Synopsis Crystal Bases by : Daniel Bump

Download or read book Crystal Bases written by Daniel Bump. This book was released on 2017. Available in PDF, EPUB and Kindle. Book excerpt: This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on the ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear group, and phenomena in combinatorics. The authors are both contributors to Sage, an open-source mathematical software system, which has strong support for crystal bases and combinatorics and the book takes advantage of this.

Introduction to Quantum Groups and Crystal Bases

2002 Mathematics
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Release : 2002
Genre : Mathematics
Kind : eBook
Book Rating : 746/5 ( reviews)

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Book Synopsis Introduction to Quantum Groups and Crystal Bases by : Jin Hong

Download or read book Introduction to Quantum Groups and Crystal Bases written by Jin Hong. This book was released on 2002. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.

Representations of Quantum Algebras and Combinatorics of Young Tableaux

2002 Mathematics
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Release : 2002
Genre : Mathematics
Kind : eBook
Book Rating : 335/5 ( reviews)

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Book Synopsis Representations of Quantum Algebras and Combinatorics of Young Tableaux by : Susumu Ariki

Download or read book Representations of Quantum Algebras and Combinatorics of Young Tableaux written by Susumu Ariki. This book was released on 2002. Available in PDF, EPUB and Kindle. Book excerpt: 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) Leclerc-Lascoux-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 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. The material 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.

Tensor Categories

2016-08-05 Mathematics
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Release : 2016-08-05
Genre : Mathematics
Kind : eBook
Book Rating : 415/5 ( reviews)

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Book Synopsis Tensor Categories by : Pavel Etingof

Download or read book Tensor Categories written by Pavel Etingof. This book was released on 2016-08-05. Available in PDF, EPUB and Kindle. Book excerpt: Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

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