By Steven H. Weintraub

'We discover extensively within the valley of standard representations, and we take the reader over the mountain cross 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 may be sufficiently serious 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. specifically, it's been one of many nice instruments within the examine and class of finite teams. the idea comprises a few really appealing effects: Frobenius' theorem, Burnside's theorem, Artin's theorem, Brauer's theorem - all of that are coated during this textbook. a few appear uninspiring first and foremost yet turn out to be rather worthwhile. Others are truly deep from the outset.And whilst a bunch (finite or in a different way) acts on anything else (as a collection of symmetries, for example), one finally ends up with a average illustration of the crowd. This publication is an creation to the illustration thought of finite teams from an algebraic perspective, concerning representations as modules over the crowd algebra. The procedure is to boost the needful algebra in average generality after which to specialize it to the case of crew representations. tools and effects specific to crew representations, equivalent to characters and brought on representations, are built intensive. mathematics comes into play while contemplating the sector of definition of a illustration, particularly for subfields of the advanced numbers.The publication 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 sector divides the order of the crowd. The e-book assumes merely the cloth of a typical graduate path in algebra. it's compatible as a textual content for a year-long graduate path. the topic is of curiosity to scholars of algebra, quantity conception and algebraic geometry. The systematic remedy awarded right here makes the e-book additionally helpful as a reference.