Representations of groups. A computational approach.

*(English)*Zbl 1208.20011
Cambridge Studies in Advanced Mathematics 124. Cambridge: Cambridge University Press (ISBN 978-0-521-76807-8/hbk). x, 460 p. (2010).

Let \(G\) be a finite group. A homomorphism \(T\colon G\to\mathrm{GL}(V,F)\), where \(V\) is a finite dimensional vector space over the field \(F\), is called a matrix representation of \(G\). If the characteristic of \(F\) does not divide the order of \(G\), \(T\) is called an ordinary representation of \(G\), otherwise it is called a modular representation. These two theories of representations of a group \(G\) lead to essentially different theorems about \(G\). The theory of ordinary representations of finite groups was developed around 1900 by Frobenius, Schur and Burnside. But the theory of modular representations of \(G\) was developed around 1935 mainly by R. Brauer. In both theories there are long standing conjectures as well as challenging problems and these make the subject more attractive and interesting. Existence of algorithms and computational methods have made some impact on the subject.

The classical references [C. W. Curtis and I. Reiner, Methods of representation theory, with applications to finite groups and orders. Vol. I. New York etc.: John Wiley & Sons (1981; Zbl 0469.20001) and Vol. II. New York etc.: John Wiley & Sons (1987; Zbl 0616.20001) and W. Feit, The representation theory of finite groups. Amsterdam - New York - Oxford: North-Holland Publishing Company (1982; Zbl 0493.20007)] served as sources for the ordinary and modular representation theory of finite groups for many years. But development of computational methods and use of computers in calculations make it necessary to have a different source with illustration of results.

The book under review grew out of courses given to graduate students at Aachen University by the authors and has emphasis on computational methods of representation theory. Both ordinary and modular representation theory are treated together because there are many common interactions. Each concept and theorem in the book is followed by a couple of examples and programming with Groups, Algorithms, Programming, GAP, is introduced to show how it can be used to solve different problems, and the worked examples are carried to the end to find the complete answer.

Apart from representation theory of finite groups, ordinary characters of finite groups and their applications are also treated in the book. Particularly, a few algorithms for computing character tables of groups, such as the Dixon-Schneider algorithm, are included in the book.

The chapter on modular representations and characters of finite groups covers most of important research topics such as \(p\)-modular systems, Brauer characters, decomposition numbers, defect groups, etc. \(p\)-Brauer character tables of some simple groups are computed, where \(p\) divides the order of the group in question.

The book covers areas in ordinary and modular representation theory of finite groups with emphasis on the computational problems concerning the subject. The computer algebra system GAP is well-introduced in the book and several illustrations using GAP are demonstrated in the book. The book is well written and many graduate students can benefit from the book to enhance their research work.

The classical references [C. W. Curtis and I. Reiner, Methods of representation theory, with applications to finite groups and orders. Vol. I. New York etc.: John Wiley & Sons (1981; Zbl 0469.20001) and Vol. II. New York etc.: John Wiley & Sons (1987; Zbl 0616.20001) and W. Feit, The representation theory of finite groups. Amsterdam - New York - Oxford: North-Holland Publishing Company (1982; Zbl 0493.20007)] served as sources for the ordinary and modular representation theory of finite groups for many years. But development of computational methods and use of computers in calculations make it necessary to have a different source with illustration of results.

The book under review grew out of courses given to graduate students at Aachen University by the authors and has emphasis on computational methods of representation theory. Both ordinary and modular representation theory are treated together because there are many common interactions. Each concept and theorem in the book is followed by a couple of examples and programming with Groups, Algorithms, Programming, GAP, is introduced to show how it can be used to solve different problems, and the worked examples are carried to the end to find the complete answer.

Apart from representation theory of finite groups, ordinary characters of finite groups and their applications are also treated in the book. Particularly, a few algorithms for computing character tables of groups, such as the Dixon-Schneider algorithm, are included in the book.

The chapter on modular representations and characters of finite groups covers most of important research topics such as \(p\)-modular systems, Brauer characters, decomposition numbers, defect groups, etc. \(p\)-Brauer character tables of some simple groups are computed, where \(p\) divides the order of the group in question.

The book covers areas in ordinary and modular representation theory of finite groups with emphasis on the computational problems concerning the subject. The computer algebra system GAP is well-introduced in the book and several illustrations using GAP are demonstrated in the book. The book is well written and many graduate students can benefit from the book to enhance their research work.

Reviewer: Mohammad-Reza Darafsheh (Tehran)

##### MSC:

20C40 | Computational methods (representations of groups) (MSC2010) |

20C15 | Ordinary representations and characters |

20C20 | Modular representations and characters |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |