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Author: Stuart Martin
Publisher: Cambridge University Press
ISBN: 0521415918
Category : Mathematics
Languages : en
Pages : 256
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Book Description
The Schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups (both finite and infinite). In the text Dr Martin gives a full, self-contained account of this algebra and these links, covering both the basic theory of Schur algebras and related areas. He discusses the usual representation-theoretic topics such as constructions of irreducible modules, the blocks containing them, their modular characters and the problem of computing decomposition numbers; moreover deeper properties such as the quasi-hereditariness of the Schur algebra are discussed. The opportunity is taken to give an account of quantum versions of Schur algebras and their relations with certain q-deformations of the coordinate rings of the general linear group. The approach is combinatorial where possible, making the presentation accessible to graduate students. This is the first comprehensive text in this important and active area of research; it will be of interest to all research workers in representation theory.
Author: Stuart Martin
Publisher: Cambridge University Press
ISBN: 0521415918
Category : Mathematics
Languages : en
Pages : 256
Get Book Here
Book Description
The Schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups (both finite and infinite). In the text Dr Martin gives a full, self-contained account of this algebra and these links, covering both the basic theory of Schur algebras and related areas. He discusses the usual representation-theoretic topics such as constructions of irreducible modules, the blocks containing them, their modular characters and the problem of computing decomposition numbers; moreover deeper properties such as the quasi-hereditariness of the Schur algebra are discussed. The opportunity is taken to give an account of quantum versions of Schur algebras and their relations with certain q-deformations of the coordinate rings of the general linear group. The approach is combinatorial where possible, making the presentation accessible to graduate students. This is the first comprehensive text in this important and active area of research; it will be of interest to all research workers in representation theory.
Author: Stuart Martin
Publisher: Cambridge University Press
ISBN: 9780521100465
Category : Mathematics
Languages : en
Pages : 0
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Book Description
Schur algebras are an algebraic system that provide a link between the representation theory of the symmetric and general linear groups. Dr. Martin gives a self-contained account of this algebra and those links, covering the basic ideas and their quantum analogues. He discusses not only the usual representation-theoretic topics (such as constructions of irreducible modules, the structure of blocks containing them, decomposition numbers and so on) but also the intrinsic properties of Schur algebras, leading to a discussion of their cohomology theory. He also investigates the relationship between Schur algebras and other algebraic structures. Throughout, the approach uses combinatorial language where possible, thereby making the presentation accessible to graduate students. Some topics require results from algebraic group theory, which are contained in an appendix.
Author: Stephen Donkin
Publisher: Cambridge University Press
ISBN: 0521645581
Category : Mathematics
Languages : en
Pages : 193
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Book Description
This book focuses on the representation theory of q-Schur algebras and connections with the representation theory of Hecke algebras and quantum general linear groups. The aim is to present, from a unified point of view, quantum analogs of certain results known already in the classical case. The approach is largely homological, based on Kempf's vanishing theorem for quantum groups and the quasi-hereditary structure of the q-Schur algebras. Beginning with an introductory chapter dealing with the relationship between the ordinary general linear groups and their quantum analogies, the text goes on to discuss the Schur Functor and the 0-Schur algebra. The next chapter considers Steinberg's tensor product and infinitesimal theory. Later sections of the book discuss tilting modules, the Ringel dual of the q-Schur algebra, Specht modules for Hecke algebras, and the global dimension of the q-Schur algebras. An appendix gives a self-contained account of the theory of quasi-hereditary algebras and their associated tilting modules. This volume will be primarily of interest to researchers in algebra and related topics in pure mathematics.
Author: Andrew Mathas
Publisher: American Mathematical Soc.
ISBN: 0821819267
Category : Mathematics
Languages : en
Pages : 204
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Book Description
This volume presents a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the $q$-Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems. The main results are proved by showing that the Iwahori-Hecke algebras and $q$-Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the $q$-Schur algebras. This book is the first of its kind covering the topic. It offers a substantially simplified treatment of the original proofs. The book is a solid reference source for experts. It will also serve as a good introduction to students and beginning researchers since each chapter contains exercises and there is an appendix containing a quick development of the representation theory of algebras. A second appendix gives tables of decomposition numbers.
Author: Karin Erdmann
Publisher: Springer
ISBN: 3319919989
Category : Mathematics
Languages : en
Pages : 304
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Book Description
This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers. The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel's celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams. Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses.
Author: Pavel I. Etingof
Publisher: American Mathematical Soc.
ISBN: 0821853511
Category : Mathematics
Languages : en
Pages : 240
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Book Description
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.
Author: S. Donkin
Publisher:
ISBN: 9781107367647
Category : Electronic books
Languages : en
Pages : 192
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Book Description
This book focuses on the representation theory of q-Schur algebras.
Author: Bangming Deng
Publisher: Cambridge University Press
ISBN: 1107608600
Category : Mathematics
Languages : en
Pages : 217
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Book Description
The first book of its kind to present an algebraic approach to affine q-Schur algebras and affine quantum Schur-Weyl theory.
Author: Amritanshu Prasad
Publisher: Cambridge University Press
ISBN: 1316222705
Category : Mathematics
Languages : en
Pages : 206
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Book Description
This book discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. The first chapter provides a detailed account of necessary representation-theoretic background. An important highlight of this book is an innovative treatment of the Robinson–Schensted–Knuth correspondence and its dual by extending Viennot's geometric ideas. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. The treatment of Schur–Weyl duality reveals the directness and simplicity of Schur's original treatment of the subject. In addition, each exercise is assigned a difficulty level to test readers' learning. Solutions and hints to most of the exercises are provided at the end.
Author: Brian Hall
Publisher: Springer
ISBN: 3319134671
Category : Mathematics
Languages : en
Pages : 452
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Book Description
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula. Review of the first edition: This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended. — The Mathematical Gazette
