Representation theory : a first course / William Fulton, Joe Harris

The text is divided into four main parts and six appendices. Part I provides, mainly for didactical reasons, an account on the representation theory of finite groups, with emphasis on what is useful for Lie groups later on.

Part II gives an introduction to Lie groups, Lie algebras, and their finite-dimensional representations. A great deal of effort is spent on explaining things in the case of low-dimensional concrete examples. This leads to some initial classification theory for Lie algebras of low dimension and rank.

Part III forms the heart of the book: the finite-dimensional representations of the classical groups. For each series of classical Lie algebras an explicit construction for representations of highest given weight is performed, and the geometric meaning of the decompositions of the naturally occuring representations is comprehensively discussed as well. A special emphasis is laid on exhibiting the various relations among the representations of the classical Lie groups, which are caused by algebraic relations between the Lie algebras.

Part IV, entitled “Lie Theory”, is less example-oriented than the previous parts. It is designed to throw a bridge between the foregoing, rather concrete examples and the general theory. The approach is to interpret the phenomena encountered before in the framework of the abstract theory and modern terminology. This includes the classification of complex simple Lie algebras by Dynkin diagrams, exceptional Lie algebras, homogeneous spaces, Bruhat decompositions, the general Weyl character formula, and multiplicity formulae.

The six appendices at the end of the book give complete proofs of some facts from the general theory of Lie algebras and invariants, as they were used in the course of the text. Among the topics explained here are: symmetric functions and determinantal identities, tensor and exterior algebra, semisimple Lie algebras, Cartan subalgebras, the Weyl group, Ado’s and Levi’s theorems, and invariant theory for the classical groups.

Altogether, the present textbook is an excellent introduction to the representation theory of Lie groups and Lie algebras. The prerequisites are minimal, and the amount of information, motivation, inspiration, and guidance is enormous. The reader is skillfully led to a level from which he can study the general theory, together with the many topics which had to be omitted here, with profit and appreciation. With regard to its main features mentioned in the beginning, this textbook is an outstanding example of didactic mastery, and it serves the purpose of the series “Readings in Mathematics” in a perfect manner. (Zentralblatt)

Bibliogr. p.536-541. Index