By Steven H. Weintraub
'We discover extensively within the valley of standard representations, and we take the reader over the mountain move resulting in the valley of modular representations, to some degree from which (s)he can survey this valley, yet we don't try and greatly discover it. we are hoping the reader could be sufficiently fascinated about the surroundings to additional discover either valleys on his/her personal' - from the Preface. illustration idea performs vital roles in geometry, algebra, research, and mathematical physics. particularly, it's been one of many nice instruments within the examine and category of finite teams. the speculation comprises a few fairly appealing effects: Frobenius' theorem, Burnside's theorem, Artin's theorem, Brauer's theorem - all of that are lined during this textbook. a few appear uninspiring first and foremost yet turn out to be fairly worthy. Others are in actual fact deep from the outset.And while a bunch (finite or in a different way) acts on whatever else (as a collection of symmetries, for example), one finally ends up with a common illustration of the crowd. This booklet is an creation to the illustration idea of finite teams from an algebraic standpoint, concerning representations as modules over the crowd algebra. The technique is to strengthen the needful algebra in moderate generality after which to specialize it to the case of team representations. tools and effects specific to workforce representations, reminiscent of characters and brought on representations, are constructed extensive. mathematics comes into play while contemplating the sphere of definition of a illustration, particularly for subfields of the complicated numbers.The e-book has an intensive improvement of the semisimple case, the place the attribute of the sector is 0 or is fundamental to the order of the crowd, and builds the principles of the modular case, the place the attribute of the sphere divides the order of the crowd. The e-book assumes basically the cloth of a typical graduate direction in algebra. it really is appropriate as a textual content for a year-long graduate path. the topic is of curiosity to scholars of algebra, quantity idea and algebraic geometry. The systematic remedy awarded right here makes the publication additionally worthwhile as a reference.