# representations and cohomology volume 2 cohomology of groups and modules cambridge studies in advanced mathematics

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## Representations And Cohomology Volume 2 Cohomology Of Groups And Modules

**Author :**D. J. Benson

**ISBN :**0521636523

**Genre :**Mathematics

**File Size :**27. 81 MB

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This is the second of two volumes which will provide an introduction to modern developments in the representation theory of finite groups and associative algebras. The subject is viewed from the perspective of homological algebra and the theory of representations of finite dimensional algebras; the author emphasises modular representations and the homological algebra associated with their categories. This volume concentrates on the cohomology of groups, always with representations in view, however. It begins with a background reference chapter, then proceeds to an overview of the algebraic topology and K-theory associated with cohomology of groups, especially the work of Quillen. Later chapters look at algebraic and topological proofs of the finite generation of the cohomology ring of a finite group, and an algebraic approach to the Steenrod operations in group cohomology. The book cumulates in a chapter dealing with the theory of varieties for modules. Much of the material presented here has never appeared before in book form. Consequently students and research workers studying group theory, and indeed algebra in general, will be grateful to Dr Benson for supplying an exposition of a good deal of the essential results of modern representation theory.

## Representations And Cohomology Volume 1 Basic Representation Theory Of Finite Groups And Associative Algebras

**Author :**D. J. Benson

**ISBN :**0521636531

**Genre :**Mathematics

**File Size :**77. 37 MB

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This is the first of two volumes providing an introduction to modern developments in the representation theory of finite groups and associative algebras, which have transformed the subject into a study of categories of modules. Thus, Dr. Benson's unique perspective in this book incorporates homological algebra and the theory of representations of finite-dimensional algebras. This volume is primarily concerned with the exposition of the necessary background material, and the heart of the discussion is a lengthy introduction to the (Auslander-Reiten) representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost-split sequences are discussed in some detail.

## Modular Forms And Galois Cohomology

**Author :**Haruzo Hida

**ISBN :**052177036X

**Genre :**Mathematics

**File Size :**90. 66 MB

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Comprehensive account of recent developments in arithmetic theory of modular forms, for graduates and researchers.

## Elements Of The Representation Theory Of Associative Algebras Volume 2 Tubes And Concealed Algebras Of Euclidean Type

**Author :**Ibrahim Assem

**ISBN :**9780521836104

**Genre :**Mathematics

**File Size :**21. 34 MB

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Volume two of this modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field

## Local Representation Theory

**Author :**J. L. Alperin

**ISBN :**052144926X

**Genre :**Mathematics

**File Size :**82. 15 MB

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The aim of this text is to present some of the key results in the representation theory of finite groups. In order to keep the account reasonably elementary, so that it can be used for graduate-level courses, Professor Alperin has concentrated on local representation theory, emphasising module theory throughout. In this way many deep results can be obtained rather quickly. After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a discussion of Feit's work connecting maps and the Green correspondence. The work concludes with a treatment, new in part, of the Brauer-Dade theory. As a text, this book contains ample material for a one semester course. Exercises are provided at the end of most sections; the results of some are used later in the text. Representation theory is applied in number theory, combinatorics and in many areas of algebra. This book will serve as an excellent introduction to those interested in the subject itself or its applications.

## The Block Theory Of Finite Group Algebras

**Author :**Markus Linckelmann

**ISBN :**9781108575317

**Genre :**Mathematics

**File Size :**87. 83 MB

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This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.

## Elements Of The Representation Theory Of Associative Algebras Volume 3 Representation Infinite Tilted Algebras

**Author :**Ibrahim Assem

**ISBN :**9780521882187

**Genre :**Mathematics

**File Size :**34. 74 MB

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Volume three of this modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field.

## Flag Varieties

**Author :**V Lakshmibai

**ISBN :**9789811313936

**Genre :**Mathematics

**File Size :**67. 37 MB

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This book discusses the importance of flag varieties in geometric objects and elucidates its richness as interplay of geometry, combinatorics and representation theory. The book presents a discussion on the representation theory of complex semisimple Lie algebras, as well as the representation theory of semisimple algebraic groups. In addition, the book also discusses the representation theory of symmetric groups. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Many of the geometric results admit elegant combinatorial description because of the root system connections, a typical example being the description of the singular locus of a Schubert variety. This discussion is carried out as a consequence of standard monomial theory. Consequently, this book includes standard monomial theory and some important applications—singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. The two recent results on Schubert varieties in the Grassmannian have also been included in this book. The first result gives a free resolution of certain Schubert singularities. The second result is about certain Levi subgroup actions on Schubert varieties in the Grassmannian and derives some interesting geometric and representation-theoretic consequences.

## An Introduction To Homological Algebra

**Author :**Charles A. Weibel

**ISBN :**9781139643078

**Genre :**Mathematics

**File Size :**44. 23 MB

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The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.

## Cohomology Of Drinfeld Modular Varieties Part 2 Automorphic Forms Trace Formulas And Langlands Correspondence

**Author :**Gérard Laumon

**ISBN :**0521109906

**Genre :**Mathematics

**File Size :**37. 36 MB

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Cohomology of Drinfeld Modular Varieties aims to provide an introduction to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the ArthurSHSelberg trace formula, and to the proof in some cases of the Ramanujan-Petersson conjecture and the global Langlands conjecture for function fields. The author uses techniques that are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. This book will be of much interest to all researchers in algebraic number theory and representation theory.