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**Representations and invariants of the classical groups.**
*(English)*
Zbl 0901.22001

Encyclopedia of Mathematics and its Applications 68. Cambridge: Cambridge University Press (ISBN 0-521-58273-3/hbk). xvi, 685 p. (1998).

The present book is about classical groups, their representations and their invariants. Classical groups are studied in this book from two different perspectives. One is that of viewing classical groups over \({\mathcal C}\) as complexification of compact Lie groups whereas the other perspective is to consider them as a special case of reductive algebraic groups over \({\mathcal C}\).

The book’s layout is the following. It is divided into twelve chapters some of which are supplemented by appendices. Chapter 1 treats classical groups as linear algebraic groups. It includes necessary definitions of linear algebraic groups, their Lie algebras, Jordan decomposition, and real forms of classical groups. The basic structure of classical groups is the focus of Chapter 2 that covers such terms as semisimple and unipotent elements, irreducible representations of SL(2, \({\mathcal C}\)), adjoint representations, reductivity of classical groups, and Weyl groups and weight lattice. Algebras and their representations are studied in the next Chapter 3. Chapter 4 deals with polynomial and tensor invariants. Irreducible representations of classical groups with some examples are the theme of Chapter 5. Chapter 6 focuses on spinors while, Chapter 7, on cohomology and characters. Branching patterns of classical groups are considered in Chapter 8. Chapter 9 is dedicated to tensor representations of \(GL(V)\) and includes Schur duality and Young tableaux. Tensor representations of the groups \(O(V)\) and \(Sp(V)\) are presented in Chapter 10. Chapter 11 returns to the study of the classical groups and their homogeneous spaces in order to prepare the reader to the geometric approach to representations and invariant theory in Chapter 12.

I consider this book as a valuable book and recommend it for a rather wide audience of readers. It can particularly be used for different courses.

The book’s layout is the following. It is divided into twelve chapters some of which are supplemented by appendices. Chapter 1 treats classical groups as linear algebraic groups. It includes necessary definitions of linear algebraic groups, their Lie algebras, Jordan decomposition, and real forms of classical groups. The basic structure of classical groups is the focus of Chapter 2 that covers such terms as semisimple and unipotent elements, irreducible representations of SL(2, \({\mathcal C}\)), adjoint representations, reductivity of classical groups, and Weyl groups and weight lattice. Algebras and their representations are studied in the next Chapter 3. Chapter 4 deals with polynomial and tensor invariants. Irreducible representations of classical groups with some examples are the theme of Chapter 5. Chapter 6 focuses on spinors while, Chapter 7, on cohomology and characters. Branching patterns of classical groups are considered in Chapter 8. Chapter 9 is dedicated to tensor representations of \(GL(V)\) and includes Schur duality and Young tableaux. Tensor representations of the groups \(O(V)\) and \(Sp(V)\) are presented in Chapter 10. Chapter 11 returns to the study of the classical groups and their homogeneous spaces in order to prepare the reader to the geometric approach to representations and invariant theory in Chapter 12.

I consider this book as a valuable book and recommend it for a rather wide audience of readers. It can particularly be used for different courses.

Reviewer: E.Kryachko (Baltimore)

### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

20G05 | Representation theory for linear algebraic groups |

20G15 | Linear algebraic groups over arbitrary fields |

20G45 | Applications of linear algebraic groups to the sciences |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

22E10 | General properties and structure of complex Lie groups |

### Keywords:

third order; classical groups; representations; invariants; compact Lie groups; linear algebraic groups; Jordan decomposition; Weyl groups; weight lattice; polynomial and tensor invariants; cohomology; characters; Schur duality; Young tableau; tensor representations; homogeneous spaces
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\textit{R. Goodman} and \textit{N. R. Wallach}, Representations and invariants of the classical groups. Cambridge: Cambridge University Press (1998; Zbl 0901.22001)