Group theoretical methods and their applications. Transl. from the German by Baoswan Dzung Wong.Rev. transl. Rev. transl. (English) Zbl 0769.20002

Fässler, Albert; Stiefel, Eduard

Boston, MA etc.: Birkhäuser. xii, 296 p. (1992).

[For a review of the German original (1979) see Zbl 0413.20001.]
The contents of the English edition of the book under review covers the following topics of such a highly developed branch of mathematics as the theory of groups and their representations is.
Chapters 1 and 2 serve as the prerequisite where the authors present the basic definitions of a group and its representations and where they give some examples of groups, starting with the problem of constructing an adequate price index in economics and ending with the point groups, the groups of symmetry of molecules.
Chapter 3 demonstrates how to build a basis of functions which will be symmetry adapted to either group by going thoroughly through that procedure for some familiar point groups like, for example, the dihedral groups. This chapter is also dedicated to the group theoretical treatment of certain partial differential equations used in engineering sciences.
Chapter 4 is just the routine introduction to continuous groups made via treating with the group of 3D rotations, Lorentz group, and to the traditional technique of the spherical functions and Clebsch-Gordan coefficients.
In Chapter 5 the authors construct the general algorithm for symmetry adapted basis sets and show its application to the classical problem of molecular vibrations. In the remaining part, they define the character of a representation and prove some fundamental theorems concerning the properties of characters.
Chapter 6 is a brief outline of diverse applications of group theory, from the bifurcation problems with symmetries, with the explicit application to Brusselator, to the model of diffusion in theory of probability.
Chapter 7 is an introduction to the theory of Lie algebras with some textbook examples of its application to quantum mechanics like the central-force field and the spin-orbit coupling.
Chapter 8 links the concept of groups with symmetries of crystals and treats some solid state physics applications.
The theory of representations of compact Lie groups \(U(n)\), \(SU(n)\), and \(SO(n)\) is presented in Chapter 9. Particularly, an algorithm to determine the invariant irreducible subspaces for \(SU(3)\) is treated in detail there.

Reviewer: E.Kryachko (München)

MSC:

20-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20C35	Applications of group representations to physics and other areas of science
20H15	Other geometric groups, including crystallographic groups
22E70	Applications of Lie groups to the sciences; explicit representations
82-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics
20G45	Applications of linear algebraic groups to the sciences
00A69	General applied mathematics
70-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems
74-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids
81R05	Finite-dimensional groups and algebras motivated by physics and their representations
81V45	Atomic physics
81V55	Molecular physics

Keywords:

theory of groups; representations; point groups; groups of symmetry; Lorentz group; Clebsch-Gordan coefficients; Lie algebras

Software:

Mathematica