Combinatorics and representations of finite groups.

*(English)*Zbl 0796.05095
Vorlesungen aus dem Fachbereich Mathematik der Universität Essen. 20. Essen: Univ. GH Essen, FB Math. 94 p. (1993).

The main purpose is to give a description of the interplay between combinatorics (especially the study of partitions and related objects) and irreducible representations of some classes of finite groups (especially symmetric groups and their coverings and some linear groups). The origin was the study of (numerical) properties of the representations of symmetric groups including the degrees and the distribution into blocks. Of course the symmetric groups are very special in several respects, but as it has turned out some of the combinatorial analysis applied there may be modified to give results for other groups.

In the first chapter we present the combinatorial concepts and results needed for our study of irreducible representations and blocks. Thereby an attempt has been made to make analogies very clear. Thus for instance cores and quotients, which are well known for partitions, may also be defined for bar partitions and symbols. Their definitions are inspired and even helped by the properties we want them to have to be able to apply them to study degrees of irreducible characters and to enumerate characters in blocks. Chapter II is devoted to the analysis of character degrees, especially the description of power of a given prime dividing a given character degree. This is important in chapter III, where we consider characters in blocks. It should be stressed that the combinatorial method may be used to obtain much more information about representations than described here.

In the first chapter we present the combinatorial concepts and results needed for our study of irreducible representations and blocks. Thereby an attempt has been made to make analogies very clear. Thus for instance cores and quotients, which are well known for partitions, may also be defined for bar partitions and symbols. Their definitions are inspired and even helped by the properties we want them to have to be able to apply them to study degrees of irreducible characters and to enumerate characters in blocks. Chapter II is devoted to the analysis of character degrees, especially the description of power of a given prime dividing a given character degree. This is important in chapter III, where we consider characters in blocks. It should be stressed that the combinatorial method may be used to obtain much more information about representations than described here.

##### MSC:

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

20C30 | Representations of finite symmetric groups |

05A17 | Combinatorial aspects of partitions of integers |

11P81 | Elementary theory of partitions |

05A15 | Exact enumeration problems, generating functions |

05A19 | Combinatorial identities, bijective combinatorics |

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |