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Representations and invariants of the classical groups. Paperback ed. (English) Zbl 0948.22001

Encyclopedia of Mathematics and Its Applications. 68. Cambridge: Cambridge University Press. xvi, 685 p. (1999).
The central topic of the book under review is the theory of the classical complex groups and their invariants. The classical groups are approached from two different perspectives: as special algebraic complex reductive groups and as complex Lie groups which are complexifications of compact Lie groups. From the algebraic perspective, the book gives an introduction to the theory of complex linear algebraic groups with an emphasis on reductive groups (Chapters 1 and 2). To make the material more directly accessible to students, many general results are proved directly for the different types of classical groups. From the Lie theoretic perspective, it gives an introduction to the structure theory of reductive complex Lie algebras and groups (Chapter 2) as well as their representation theory (Chapters 5 and 7). This material sets the stage for the core of the book which is the invariant theory of the classical groups. Chapter 3 is devoted to the representation theory of finite groups and semisimple finite-dimensional algebras. Polynomial and tensor invariants are discussed in Chapter 4 not using the highest weight theory which is explained later in Chapter 5, followed by a concrete description of the irreducible representations of the classical groups. Chapter 6 introduces Clifford algebras, the spin groups, and the spin representations, hence completing the concretization of the irreducible representations of the orthogonal groups. Weyl’s character formula which is derived in Chapter 7 is used in Chapter 8 to derive the branching laws for the three series of classical groups. Chapters 9 and 10 discuss tensor representations, including Weyl duality of \(GL_n\) and \(S_k\), spherical harmonics and generalizations thereof. A nice link to topology is the connection between orthogonal invariants, braid groups and knot polynomials explained in Section 10.4. The two final Chapters 11 and 12 first introduce homogeneous spaces of algebraic groups, flag manifolds and the corresponding group decompositions. This prepares the study of representations in the algebra of regular functions on affine varieties with an emphasis on multiplicity free spaces and symmetric spaces. The appendices contain background material on algebraic geometry, multilinear algebra, Lie algebras and Lie groups. It should be emphasized that there are several appealing ideas underlying the general philosophy of the book. One is to put the representations of the classical groups into the center and to derive the delicate combinatorial results on the representations of the symmetric group via Weyl duality from the representation theory of \(GL_n\). Likewise representation theory is used in the last section to study the geometry of certain homogeneous spaces and not vice versa. The Weyl duality is only one incarnation of the duality between linear group representations and certain commuting algebras which is a unifying theme of the whole book, showing up in many different settings. It rests on the Double Commutant Theorem, is fundamental for Howe’s theory of dual pairs, the theory of spherical harmonics and its generalizations and many other settings discussed in the book. In the setting of group actions on smooth affine varieties this idea has been developed further in the dissertation of I. Agricola, “Die Frobenius-Zerlegung auf algebraischen \(G\)-Mannigfaltigkeiten”, Humboldt-Universität Berlin, 2000. The book grew out of various graduate courses taught by the authors and hence can be used as a source for various kinds of courses. This is further supported by the rich collections of exercises accompanying each section and by the notes on the chapters, containing information on the historical development of the main results. Local reading is well supported by the structure of the book. Since the material on the classical groups is exposed in an elementary fashion and the book is essentially self contained, it should be very well accessible for advanced students. The book under review is the paperback edition of the first edition of the book which has been reviewed in Zbl 0901.22001. Note communicated by the authors: Corrections and new material in the present version of the book can be found at http://math.rutgers.edu/~goodman/repbook.html.

MSC:

22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
20G15 Linear algebraic groups over arbitrary fields
20G45 Applications of linear algebraic groups to the sciences
20G05 Representation theory for linear algebraic groups
22E10 General properties and structure of complex Lie groups

Citations:

Zbl 0901.22001
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