Representations of finite groups.

*(Russian)*Zbl 0606.20008This comprehensive survey article is devoted to ordinary and modular representation theory of finite groups by materials from R. Zh. Mat. for 1965-1984 [an earlier survey is due to the late S. D. Berman and was published ibid. 1964, 83-122 (1966; Zbl 0209.058)].

Contents. Ch. 1. Ordinary representations and characters. § 1.1. Representations over algebraically closed fields of characteristic 0. § 1.2. Characters of solvable groups and their generalizations. § 1.3. Monomial representations and M-groups. § 1.4. Representations over non-closed fields. Schur index. § 1.5. Abstract properties of groups defined by properties of characters. § 1.6. Projective representations. § 1.7. Representations and characters of certain types of groups. Ch. 2. Modular representations. § 2.1. General theory. § 2.2. Blocks with special types of defect groups. § 2.3. Brauer and Alperin-McKay hypotheses. § 2.4. Other results. Ch. 3. Representations of finite groups of Lie type. A. Complex representations and characters. § 3.1. General problems. Certain character series. § 3.2. Hecke algebras and decomposition of induced characters. § 3.3. Representations of reductive algebraic groups. Deligne-Lusztig theory. § 3.4. Unipotent characters. § 3.5. Representations of classical groups. § 3.6. Other results. B. Modular representations. § 3.7. Irreducible representations over algebraically closed fields of prime characteristic. § 3.8. Indecomposable and projective modules. § 3.9. Connection between complex and modular characters of Chevalley groups. § 3.10. Representations over fields of characteristic \(r\neq p\). Chap. 4. Further questions. § 4.1. Integral representations and lattices. § 4.2. Representations of permutations groups. Concluding remarks. Literature. The bibliography contains 620 items.

Contents. Ch. 1. Ordinary representations and characters. § 1.1. Representations over algebraically closed fields of characteristic 0. § 1.2. Characters of solvable groups and their generalizations. § 1.3. Monomial representations and M-groups. § 1.4. Representations over non-closed fields. Schur index. § 1.5. Abstract properties of groups defined by properties of characters. § 1.6. Projective representations. § 1.7. Representations and characters of certain types of groups. Ch. 2. Modular representations. § 2.1. General theory. § 2.2. Blocks with special types of defect groups. § 2.3. Brauer and Alperin-McKay hypotheses. § 2.4. Other results. Ch. 3. Representations of finite groups of Lie type. A. Complex representations and characters. § 3.1. General problems. Certain character series. § 3.2. Hecke algebras and decomposition of induced characters. § 3.3. Representations of reductive algebraic groups. Deligne-Lusztig theory. § 3.4. Unipotent characters. § 3.5. Representations of classical groups. § 3.6. Other results. B. Modular representations. § 3.7. Irreducible representations over algebraically closed fields of prime characteristic. § 3.8. Indecomposable and projective modules. § 3.9. Connection between complex and modular characters of Chevalley groups. § 3.10. Representations over fields of characteristic \(r\neq p\). Chap. 4. Further questions. § 4.1. Integral representations and lattices. § 4.2. Representations of permutations groups. Concluding remarks. Literature. The bibliography contains 620 items.

Reviewer: Ya.G.Berkovich

##### MSC:

20Cxx | Representation theory of groups |

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

20C15 | Ordinary representations and characters |

20C20 | Modular representations and characters |

20G05 | Representation theory for linear algebraic groups |