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Schur Algebras and Representation Theory

1993 Mathematics
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Release : 1993
Genre : Mathematics
Kind : eBook
Book Rating : 918/5 ( reviews)

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Book Synopsis Schur Algebras and Representation Theory by : Stuart Martin

Download or read book Schur Algebras and Representation Theory written by Stuart Martin. This book was released on 1993. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Schur Algebras and Representation Theory

2009-01-18 Mathematics
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Release : 2009-01-18
Genre : Mathematics
Kind : eBook
Book Rating : 465/5 ( reviews)

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Book Synopsis Schur Algebras and Representation Theory by : Stuart Martin

Download or read book Schur Algebras and Representation Theory written by Stuart Martin. This book was released on 2009-01-18. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group

1999 Mathematics
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Release : 1999
Genre : Mathematics
Kind : eBook
Book Rating : 267/5 ( reviews)

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Book Synopsis Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group by : Andrew Mathas

Download or read book Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group written by Andrew Mathas. This book was released on 1999. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Q-Schur Algebra

1998-12-10 Mathematics
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Release : 1998-12-10
Genre : Mathematics
Kind : eBook
Book Rating : 581/5 ( reviews)

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Book Synopsis The Q-Schur Algebra by : Stephen Donkin

Download or read book The Q-Schur Algebra written by Stephen Donkin. This book was released on 1998-12-10. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Algebras and Representation Theory

2018-09-07 Mathematics
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Release : 2018-09-07
Genre : Mathematics
Kind : eBook
Book Rating : 989/5 ( reviews)

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Book Synopsis Algebras and Representation Theory by : Karin Erdmann

Download or read book Algebras and Representation Theory written by Karin Erdmann. This book was released on 2018-09-07. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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