218 - Österreichische Mathematische Gesellschaft

J. Berstel, D. Perrin, C. Reutenauer: Codes and Automata. (Encyclopedia
of Mathematics and Its Applications 129.) Cambridge University Press, 2010,
xiii+619 S. ISBN 978-0-521-88831-8 H/b £ 75,–.
This is a strongly rewritten and considerably expanded version of the monograph
“Theory of codes” written by the first two authors in the mid 1980’s. It gives an
extensive presentation of the theory of codes. The term code here is meant to be
simply a subset X of a free monoid that freely generates a free submonoid; in
other words, X is a set of words such that each finite product of members of X
has a unique factorization into members of X. The theory of codes essentially
goes back to and is based on fundamental results by M.-P. Schützenberger and his
collaborators, and is nowadays an important part of theoretical computer science,
with strong connections to the fields of combinatorics of words, automata theory,
formal language theory and the theory of semigroups.
The principal problem in connection with codes treated in the book is to give
constructive descriptions of all of them. However, this problem can be solved only
for special classes; for example, it is easy for prefix codes and still possible, but
already quite difficult for bifix codes. In contrast, no general method is known up
to now to construct all finite codes. The types of methods treated in the book are
essentially twofold: direct ones (which are based on combinatorics of words) and
indirect ones (which are based on semigroups and automata). The novelty in the
present book compared to its predecessor is that much more material is included
concerning (i) connections to other fields of mathematics (symbolic dynamics,
probability theory, algebra) and (ii) algorithmic problems on graphs and words
(for example, a proof of the recently solved road coloring problem is included).
The book is very carefully written and requires as prerequisites only “basic mathematical
culture”. Every chapter contains carefully selected exercises (altogether
arond 200, including solutions), therefore the book is well suited for self-study;
various chapters of it my well serve as a basis for various courses, such as discrete
mathematics, algorithms, etc.
K. Auinger (Wien)
T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli: Representation Theory of the
Symmetric Groups. The Okounkov-Vershik Approach, Character Formulas, and
Partition Algebras. (Cambridge Studies in Advanced Mathematics 121.) Cambridge
University Press, 2010, xv+412 S. ISBN 978-0-521-11817-0 H/b £ 45,–.
This well written book provides an almost self-contained presentation of the representation
theory over C of the symmetric groups S n and the related combinatorics
– from the classical theory to the forefront of recent research. It is suitable
for (mature) readers, but requires as prerequisite only basic knowledge in group
theory and linear algebra. The general concepts of representation theory of finite
groups and finite dimensional associative algebras are developed in Chapter
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