An atlas of Brauer characters.

*(English)*Zbl 0831.20001
London Mathematical Society Monographs. New Series. 11. Oxford: Clarendon Press. xvii, 327 p. (1995).

The classification of finite simple groups is one of the most important mathematical results and has countless applications. Often in applications, questions about finite groups can be reduced to simple, nonabelian groups. However usually it is difficult to answer such a specific question, since the structure of simple groups is in general very complex. With respect to this problem the ATLAS of finite groups [J. H. Conway et al., Clarendon Press, Oxford, 1985; Zbl 0568.20001] has proved to be of invaluable help and is probably the most quoted reference for finite groups. The ATLAS of finite groups provides the ordinary character tables and the subgroup structure of nonabelian simple groups of low order, in particular of the sporadic groups and groups of order \(<10^9\).

In more intricate applications however the knowledge of modular characters is required. The new ATLAS of Brauer characters under review gives these most important informations and displays the Brauer character tables for the nonabelian simple groups of order \(<10^9\).

The book starts with a detailed explanation of the notions which are used. A short introduction into the basic facts of modular character theory is given too. The introduction closes with an explanation of how to read the character tables. The notations follow closely the conventions of the ATLAS of finite groups. This has the advantage that a lot of material is covered on a small space, but has the disadvantage that the tables are sometimes not easy to read. The main bulk of the book consists of the Brauer character tables for all primes dividing the order of the specific simple group. For each Brauer character the indicator is provided, which shows if the corresponding modular representation is equivalent to an orthogonal representation, a symplectic but not orthogonal representation, or if neither of the two former cases occurs. A very extensive bibliography closes the main part of the book.

Two appendices are given: The first appendix tabulates irrationalities and the so-called Conway polynomials. This information is essential to read the character tables properly. The second appendix lists improvements and corrections of errors and misprints of the ATLAS of finite groups.

We summarize: This ATLAS of Brauer characters is another most important step to improve the applicability of the classification of finite simple groups. If any property for finite simple groups has to be checked, the first source to turn to should be the two ATLASES.

In more intricate applications however the knowledge of modular characters is required. The new ATLAS of Brauer characters under review gives these most important informations and displays the Brauer character tables for the nonabelian simple groups of order \(<10^9\).

The book starts with a detailed explanation of the notions which are used. A short introduction into the basic facts of modular character theory is given too. The introduction closes with an explanation of how to read the character tables. The notations follow closely the conventions of the ATLAS of finite groups. This has the advantage that a lot of material is covered on a small space, but has the disadvantage that the tables are sometimes not easy to read. The main bulk of the book consists of the Brauer character tables for all primes dividing the order of the specific simple group. For each Brauer character the indicator is provided, which shows if the corresponding modular representation is equivalent to an orthogonal representation, a symplectic but not orthogonal representation, or if neither of the two former cases occurs. A very extensive bibliography closes the main part of the book.

Two appendices are given: The first appendix tabulates irrationalities and the so-called Conway polynomials. This information is essential to read the character tables properly. The second appendix lists improvements and corrections of errors and misprints of the ATLAS of finite groups.

We summarize: This ATLAS of Brauer characters is another most important step to improve the applicability of the classification of finite simple groups. If any property for finite simple groups has to be checked, the first source to turn to should be the two ATLASES.

Reviewer: U.Dempwolff (Kaiserslautern)

##### MSC:

20-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to group theory |

20C20 | Modular representations and characters |

20C33 | Representations of finite groups of Lie type |

20C34 | Representations of sporadic groups |

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

20D05 | Finite simple groups and their classification |

20D06 | Simple groups: alternating groups and groups of Lie type |

20D08 | Simple groups: sporadic groups |