Algebraic combinatorics via finite group actions. (English) Zbl 0726.05002

Kerber, Adalbert

Mannheim etc.: Wissenschaftsverlag. 436 p. (1991).

This book is an introduction to the theory of classification, enumeration, construction and generation of certain discrete structures in mathematics and sciences, which can be defined as equivalence classes on finite sets and in particular on finite sets of mappings. Such examples are graphs, switching functions, physical states, chemical isomers. The interest of such a theory has increased after the development of modern computer techniques.
The main methods consist in replacing the equivalence relations by finite group actions and in applying some algebraic tools like the Cauchy- Frobenius Lemma and its refinements. The author has given a survey of the present situation of this theory. Of course, many results could not be put inside the book; they are too numerous, and the book does not include asymptotic methods.
The first chapters give some theoretical aspects: actions of finite groups on sets, weights, marks, representations, while the rest of the book points out several algorithmic methods and constructions of such structures. Some beautiful examples, in particular in graph theory, appear in the book. Let us note the problem of chemical isomerism which led to its early development. The book ends with some historical remarks, comments and references, as well as with several suggestions for further reading.

Reviewer: M.Ştefănescu (Iaşi)

MSC:

05-02	Research exposition (monographs, survey articles) pertaining to combinatorics
05E20	Group actions on designs, etc. (MSC2000)
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures
20B30	Symmetric groups
05E15	Combinatorial aspects of groups and algebras (MSC2010)
05E30	Association schemes, strongly regular graphs
05E35	Orthogonal polynomials (combinatorics) (MSC2000)

Keywords:

discrete structures; equivalence classes on finite sets; graphs; switching functions; physical states; chemical isomers; Cauchy-Frobenius Lemma; theoretical aspects; algorithmic methods; beautiful examples