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Group representation theory for physicists. 2nd revised and augmented ed. (English) Zbl 1015.20012

Singapore: World Scientific. xxvi, 574 p. (2002).
This book, when following the advertising comments in the Preface, intends to carry out a systematic reform of the standard, or better traditional representation theory of groups. The motivation of the authors emerges from their rather frustrating experience when applying standard methods of group theory to applications in physics. Their crucial arguing is that they want to establish a new approach to group representation theory where the concepts and methods are in accordance with those usually employed in quantum mechanics in order to alleviate the use of group-theoretical methods and to make them simply attractive to physicists and chemists. The basic idea of their approach can be summarized under the title ‘eigenfunction method’. This method is nothing but the search for a complete set of commuting, eventually self-adjoint operators whose simultaneous diagonalization is utilized to tackle and to solve the group-theoretical problems in question.
The merits of this book consist basically of the comprehensive treatment of their specific group-theoretical method applied to a broad range of physical problems. In fact, this book might be considered as a serious attempt to cover many areas, like finite groups with applications to point and space groups describing the symmetries of crystal properties, the permutation groups in relation to the representation theory of the unitary groups with applications in elementary particle physics by including systematically the Lie algebras and their representation theory with special emphasis on Dynkin diagrams. Likewise, applications to spectroscopy are thoroughly treated with the aim to give sufficiently many tables of isoscalar factors and coefficients of fractional parentage. Even the construction of irreducible vector bases emerging from non-orthogonal reducible bases are tackled systematically and successfully.
This book might be qualified as a valuable supplement to the existing literature dealing with group theory, representation theory, and last but not least with the application of group-theoretical methods to physical problems.
Reviewer: Rainer Dirl (Wien)

MSC:

20C35 Applications of group representations to physics and other areas of science
22E70 Applications of Lie groups to the sciences; explicit representations
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20-02 Research exposition (monographs, survey articles) pertaining to group theory
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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