##
**Symmetry, representations, and invariants.
Based on the book ‘Representations and invariants of the classical groups’ originally published by Cambridge University Press, 1998.**
*(English)*
Zbl 1173.22001

Graduate Texts in Mathematics 255. New York, NY: Springer (ISBN 978-0-387-79851-6/hbk; 978-0-387-79852-3/ebook). xx, 716 p. (2009).

The book under review is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups. The book is based on the authors’ earlier book [Representations and Invariants of the Classical Groups. Encyclopaedia of Mathematical Sciences. 68. (Cambridge): Cambridge University Press. (1998; Zbl 0901.22001) or (1999; Zbl 0948.22001)]. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. A wealth of examples and discussion prepares the reader for the general results now stated and proved in their natural generality. The parts of the previous book that have withstood the authors’ many revisions as they lectured from its material have been retained; these parts appear after substantial rewriting and reorganization. The first four chapters are, in large part, newly written and offer a more direct and elementary approach to the subject. Several of the later parts of the book are also new.

The book is divided into twelve chapters and four appendices. Chapter 1 gives an elementary approach to the classical groups, viewed either as Lie groups or algebraic groups, without using any deep results from differential manifold theory or algebraic geometry. Chapter 2 develops the basic structure of the classical Lie groups and their Lie algebras. Chapter 3 is devoted to Cartan’s highest-weight theory and the Weyl group. A general treatment of associative algebras and their representations occurs in Chapter 4, where the key result is the general duality theorem for locally regular representations of a reductive algebraic group. The machinery of Chapters 1–4 is then applied in Chapter 5 to obtain the principal results in classical representation and invariant theory: the first fundamental theorems for the classical groups and the application of invariant theory to representation theory via the duality theorem. Chapter 6 introduces Clifford algebra, the spin groups, and spin representations. Weyl’s character formula is derived in Chapter 7 and used in Chapter 8 to derive the branching laws for the classical groups. Chapters 9–10 apply all the machinery developed in previous chapters to analyze the tensor representations of the classical groups. The general study of algebraic groups over \(\mathbb C\) and homogeneous spaces begins in Chapter 11 as a preparation for the geometric approach to representations and invariant theory in Chapter 12. Chapter 12 is devoted to representations of reductive algebraic groups on spaces of regular functions on affine varieties. For example, in Chapter 12 a proof of the celebrated Kostant-Rallis theorem for symmetric spaces is given and every implication for the invariant theory of classical groups is explained. The appendices contain background material on algebraic geometry, multilinear algebra, Lie algebras and Lie groups.

The book can be used as a source for various kinds of courses. This is further supported by the rich collections of exercises (mostly with detailed hints for solutions) accompanying each section. Local reading is well supported by the structure of the book. The book can be recommended for a rather wide audience of readers: for graduate and postgraduate students as well as for researchers as a reference work.

The book is divided into twelve chapters and four appendices. Chapter 1 gives an elementary approach to the classical groups, viewed either as Lie groups or algebraic groups, without using any deep results from differential manifold theory or algebraic geometry. Chapter 2 develops the basic structure of the classical Lie groups and their Lie algebras. Chapter 3 is devoted to Cartan’s highest-weight theory and the Weyl group. A general treatment of associative algebras and their representations occurs in Chapter 4, where the key result is the general duality theorem for locally regular representations of a reductive algebraic group. The machinery of Chapters 1–4 is then applied in Chapter 5 to obtain the principal results in classical representation and invariant theory: the first fundamental theorems for the classical groups and the application of invariant theory to representation theory via the duality theorem. Chapter 6 introduces Clifford algebra, the spin groups, and spin representations. Weyl’s character formula is derived in Chapter 7 and used in Chapter 8 to derive the branching laws for the classical groups. Chapters 9–10 apply all the machinery developed in previous chapters to analyze the tensor representations of the classical groups. The general study of algebraic groups over \(\mathbb C\) and homogeneous spaces begins in Chapter 11 as a preparation for the geometric approach to representations and invariant theory in Chapter 12. Chapter 12 is devoted to representations of reductive algebraic groups on spaces of regular functions on affine varieties. For example, in Chapter 12 a proof of the celebrated Kostant-Rallis theorem for symmetric spaces is given and every implication for the invariant theory of classical groups is explained. The appendices contain background material on algebraic geometry, multilinear algebra, Lie algebras and Lie groups.

The book can be used as a source for various kinds of courses. This is further supported by the rich collections of exercises (mostly with detailed hints for solutions) accompanying each section. Local reading is well supported by the structure of the book. The book can be recommended for a rather wide audience of readers: for graduate and postgraduate students as well as for researchers as a reference work.

Reviewer: Sergei Platonov (Petrozavodsk)

### MSC:

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

20G05 | Representation theory for linear algebraic groups |

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

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

22E46 | Semisimple Lie groups and their representations |

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

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |