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Brauer had already introduced the defect of a block and opened the way towards a classification by solving all the problems in defects zero and one, and by providing some evidence for the finiteness of the set of blocks with a given defect. In 1959 he discovered the defect group, and in 1964 Dade determined the blocks with cyclic defect groups. In 1978 Alperin and Broué discovered the Brauer category, and Broué and the author determined the blocks having a nilpotent Brauer category. In 1979, the author discovered the source algebra which determines all the other current invariants, representing faithfully the block – and found its structure in the nilpotent blocks. Recently, the discovery by Rickard that all blocks with the same cyclic defect group and the same Brauer category have the same homotopic category focussed great interest on the new, loose relationship between blocks called Rickard equivalence. This book describes the source algebra of a block from the source algebra of a Rickard equivalent block and the source of the Rickard equivalence.
"Although the book reaches the most advanced level, a special effort has been made to make it accessible to graduate students interested in finite groups or noncommutative algebras. A full chapter is devoted to reviewing the terminology, and the particular case of Morita equivalences is discussed separately, as an introduction. Two appendices, one on Weiss's criterion for permutation modules and the other on tensor induction of graded differential algebras, are of interest on their own."--Jacket.
Thisseries is devoted to the publication of monographs, lecture resp. seminar notes, and other materials arising from programs of the OSU Mathemaical Research Institute. This includes proceedings of conferences or workshops held at the Institute, and other mathematical writings.
A handbook of key articles providing both an introduction and reference for newcomers and experts alike.
A self-contained introduction is given to J. Rickard's Morita theory for derived module categories and its recent applications in representation theory of finite groups. In particular, Broué's conjecture is discussed, giving a structural explanation for relations between the p-modular character table of a finite group and that of its "p-local structure". The book is addressed to researchers or graduate students and can serve as material for a seminar. It surveys the current state of the field, and it also provides a "user's guide" to derived equivalences and tilting complexes. Results and proofs are presented in the generality needed for group theoretic applications.
This book is an outgrowth of a Research Symposium on the Modular Representation Theory of Finite Groups, held at the University of Virginia in May 1998. The main themes of this symposium were representations of groups of Lie type in nondefining (or cross) characteristic, and recent developments in block theory. Series of lectures were given by M. Geck, A. Kleshchev and R. Rouquier, and their brief was to present material at the leading edge of research but accessible to graduate students working in the field. The first three articles are substantial expansions of their lectures, and each provides a complete account of a significant area of the subject together with an extensive bibliography. The remaining articles are based on some of the other lectures given at the symposium; some again are full surveys of the topic covered while others are short, but complete, research articles. The opportunity has been taken to produce a book of enduring value so that this is not a conference proceedings in the conventional sense. Material has been updated so that this book, through its own content and in its extensive bibliographies, will serve as an invaluable resource for all those working in the area, whether established researchers or graduate students who wish to gain a general knowledge of the subject starting from a single source.
Over the last three decades representation theory of groups, Lie algebras and associative algebras has undergone a rapid development through the powerful tool of almost split sequences and the Auslander-Reiten quiver. Further insight into the homology of finite groups has illuminated their representation theory. The study of Hopf algebras and non-commutative geometry is another new branch of representation theory which pushes the classical theory further. All this can only be seen in connection with an understanding of the structure of special classes of rings. The aim of this book is to introduce the reader to some modern developments in: Lie algebras, quantum groups, Hopf algebras and algebraic groups; non-commutative algebraic geometry; representation theory of finite groups and cohomology; the structure of special classes of rings.
Surveys developments in the representation theory of finite dimensional algebras and related topics in seven papers illustrating different techniques developed over the recent years. For graduate students and researchers with a background in commutative algebra, including rings, modules, and homological algebra. Suitable as a text for an advanced graduate course. No index. Member prices are $31 for institutions and $23 for individuals, and are available to members of the Canadian Mathematical Society. Annotation copyrighted by Book News, Inc., Portland, OR
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.