##
**Methods of representation theory. With applications to finite groups and orders. Vol. II.**
*(English)*
Zbl 0616.20001

Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XV, 951 p.; £87.15 (1987).

This volume completes Curtis and Reiner’s monumental text-reference book on representation theory [Volume I was published in 1981; see Zbl 0469.20001 for a review]. Professor Reiner died in October, 1986 and never saw this book in its completed form.

The first volume contained a compendious survey of the established foundations of the representation theory of finite groups and of orders. The present volume surveys a series of topics of more recent origin.

The first chapter covers algebraic K-theory, with an eye toward its applications to integral representation theory (but with enough generality to get one started toward other applications as well). First, the Grothendieck groups \(G_ 0\) and \(K_ 0\) are treated, with Lam’s theory of Frobenius functors included. The techniques developed are then applied to the case of integral group rings, culminating in Lenstra’s formula for the Grothendieck group of the integral group ring of a finite abelian group. Next, \(K_ 1\) and the associated topics of stable range and basic elements are discussed, allowing the introduction of Mayer- Vietoris sequences in low-dimensional K-theory. An interlude on K-theory of polynomial rings includes Seshadri’s theorem and a statement of the Quillen-Suslin theorem. Relative K-theory is defined, and the exact sequence associated with an ideal is derived. The functor \(SK_ 1\) is discussed, featuring the remarkable computations of Keating and Oliver, as well as Bass’ generalization of the Dirichlet Unit Theorem and Wall’s extension of Higman’s work on units in group rings. The functor \(K_ 2\) is discussed; its power is illustrated by showing how Tate’s computation of \(K_ 2\) of the rational field yields classical reciprocity theorems. The chapter concludes with a survey of results on \(SK_ 1\) for integral group rings. Quillen’s higher K-functors are not covered.

Chapter 6 (the numbering of chapters continues that of Volume I) covers locally free class groups of orders. First, basic formulae and properties are developed and applied to a variety of specific computations. The Eichler condition and Jacobinski’s powerful cancellation law are then demonstrated. FrĂ¶hlich’s Hom-formula for the class group is presented, and the Swan subgroup is discussed together with its application to the Galois module theory of tame extensions. Martin Taylor’s p-adic logarithm method and its striking applications are discussed. Finally, the Picard group of invertible bimodules is discussed, in its connections with Morita theory, class groups and automorphism groups.

At this point, the discussion changes a bit abruptly to the theory of blocks, which is mainly part of modular representation theory, although it is intertwined with the theory of p-adic integral representations. After basic definitions are given, defect groups are discussed using Green’s G-algebras. The realizations of defect groups as vertices and as Sylow intersections are exhibited. Then, the Brauer correspondence is introduced, and Brauer’s Main Theorems are proved. Applications to character theory and to the structure of finite groups are given, as well as the basic facts about blocks with cyclic defect groups.

Chapter 8 gives an extensive introduction to the representations of the finite groups of Lie type. Topics include finite groups with BN-pairs and their homology representations, Hecke algebras, generic algebras, Levi decomposition, cuspidal characters, duality in the character ring and (briefly) the modular representations.

A chapter on rationality questions discusses the Frobenius-Schur classification of real representations, Serre’s induction theorem for orthogonal characters, and the Schur index up through the Benard- Schacher, Brauer-Speiser, Witt-Berman and Brauer-Witt theorems. The representations of the symmetric groups are discussed from the viewpoint of Specht modules and Solomon’s results on permutation characters. A section on Lam’s notion of Artin exponent rounds out the chapter.

Chapter 10, on indecomposable modules, is the shortest and most tentative in the book. The topics are the Bernstein-Gelfand-Ponomarev proof of Gabriel’s classification of quivers of finite representation type, and introduction to Auslander-Reiten sequences, and Yamagata’s version of Rojter’s proof of the first Brauer-Thrall conjecture. Since their emphasis is on group representations, the authors can be excused for omitting mention of the enormous literature on covering spaces, functor categories, tubes, tilting modules and so on that has appeared in the past twenty years or so. One does hope, however, that a definitive source on these matters will appear, since the activity in the area seems to be settling down.

The final chapter discusses two interesting characters (if the reader will excuse an egregious pun). Both are rings, and both revived after long periods of dormancy. The Burnside ring of a finite group G was defined implicitly long ago in Burnside’s famous group theory text. It is the Grothendieck ring of finite G-sets. Solomon revived interest in it in 1967, with applications to rational characters in mind. Dress studied it intensively, and found many remarkable results, some of which are given here. Since then, a lively interest in the Burnside ring has continued. The representation ring or Green ring was introduced by J. A. Green in 1964; he used it as the setting for his powerful transfer theorem. It is a Grothendieck ring with relations from direct sums only. Conlon showed how certain canonical idempotents in the Burnside algebra (the complexification of the Burnside ring) gave rise to useful decompositions of the representation algebra. Around 1970, Zemanek gave a few examples of nilpotent elements in modular representation rings, and then the subject lay essentially dormant for nearly fifteen years. Quite recently, it has been revived by D. Benson, in collaborations with Parker and Carlson. Their interesting results on duality, nilpotent elements and related structural matters conclude the book.

We should now summarize our impression of this enormous compendium (more than 1700 pages in the two volumes combined). With these two books, one has at hand all the basic information needed to start research in any area of the representation theory of finite groups or of orders. In principle, it would be possible to learn representation theory ab initio by reading this work. Of course, one would need a great deal of time to absorb a significant proportion of what is here, and occasional guidance from an expert would be nearly mandatory. The whole work is written clearly and carefully, with few errors and many interesting exercises. The community of algebraists must be deeply indebted to the authors for performing the magnificent service of writing everything down so beautifully.

The first volume contained a compendious survey of the established foundations of the representation theory of finite groups and of orders. The present volume surveys a series of topics of more recent origin.

The first chapter covers algebraic K-theory, with an eye toward its applications to integral representation theory (but with enough generality to get one started toward other applications as well). First, the Grothendieck groups \(G_ 0\) and \(K_ 0\) are treated, with Lam’s theory of Frobenius functors included. The techniques developed are then applied to the case of integral group rings, culminating in Lenstra’s formula for the Grothendieck group of the integral group ring of a finite abelian group. Next, \(K_ 1\) and the associated topics of stable range and basic elements are discussed, allowing the introduction of Mayer- Vietoris sequences in low-dimensional K-theory. An interlude on K-theory of polynomial rings includes Seshadri’s theorem and a statement of the Quillen-Suslin theorem. Relative K-theory is defined, and the exact sequence associated with an ideal is derived. The functor \(SK_ 1\) is discussed, featuring the remarkable computations of Keating and Oliver, as well as Bass’ generalization of the Dirichlet Unit Theorem and Wall’s extension of Higman’s work on units in group rings. The functor \(K_ 2\) is discussed; its power is illustrated by showing how Tate’s computation of \(K_ 2\) of the rational field yields classical reciprocity theorems. The chapter concludes with a survey of results on \(SK_ 1\) for integral group rings. Quillen’s higher K-functors are not covered.

Chapter 6 (the numbering of chapters continues that of Volume I) covers locally free class groups of orders. First, basic formulae and properties are developed and applied to a variety of specific computations. The Eichler condition and Jacobinski’s powerful cancellation law are then demonstrated. FrĂ¶hlich’s Hom-formula for the class group is presented, and the Swan subgroup is discussed together with its application to the Galois module theory of tame extensions. Martin Taylor’s p-adic logarithm method and its striking applications are discussed. Finally, the Picard group of invertible bimodules is discussed, in its connections with Morita theory, class groups and automorphism groups.

At this point, the discussion changes a bit abruptly to the theory of blocks, which is mainly part of modular representation theory, although it is intertwined with the theory of p-adic integral representations. After basic definitions are given, defect groups are discussed using Green’s G-algebras. The realizations of defect groups as vertices and as Sylow intersections are exhibited. Then, the Brauer correspondence is introduced, and Brauer’s Main Theorems are proved. Applications to character theory and to the structure of finite groups are given, as well as the basic facts about blocks with cyclic defect groups.

Chapter 8 gives an extensive introduction to the representations of the finite groups of Lie type. Topics include finite groups with BN-pairs and their homology representations, Hecke algebras, generic algebras, Levi decomposition, cuspidal characters, duality in the character ring and (briefly) the modular representations.

A chapter on rationality questions discusses the Frobenius-Schur classification of real representations, Serre’s induction theorem for orthogonal characters, and the Schur index up through the Benard- Schacher, Brauer-Speiser, Witt-Berman and Brauer-Witt theorems. The representations of the symmetric groups are discussed from the viewpoint of Specht modules and Solomon’s results on permutation characters. A section on Lam’s notion of Artin exponent rounds out the chapter.

Chapter 10, on indecomposable modules, is the shortest and most tentative in the book. The topics are the Bernstein-Gelfand-Ponomarev proof of Gabriel’s classification of quivers of finite representation type, and introduction to Auslander-Reiten sequences, and Yamagata’s version of Rojter’s proof of the first Brauer-Thrall conjecture. Since their emphasis is on group representations, the authors can be excused for omitting mention of the enormous literature on covering spaces, functor categories, tubes, tilting modules and so on that has appeared in the past twenty years or so. One does hope, however, that a definitive source on these matters will appear, since the activity in the area seems to be settling down.

The final chapter discusses two interesting characters (if the reader will excuse an egregious pun). Both are rings, and both revived after long periods of dormancy. The Burnside ring of a finite group G was defined implicitly long ago in Burnside’s famous group theory text. It is the Grothendieck ring of finite G-sets. Solomon revived interest in it in 1967, with applications to rational characters in mind. Dress studied it intensively, and found many remarkable results, some of which are given here. Since then, a lively interest in the Burnside ring has continued. The representation ring or Green ring was introduced by J. A. Green in 1964; he used it as the setting for his powerful transfer theorem. It is a Grothendieck ring with relations from direct sums only. Conlon showed how certain canonical idempotents in the Burnside algebra (the complexification of the Burnside ring) gave rise to useful decompositions of the representation algebra. Around 1970, Zemanek gave a few examples of nilpotent elements in modular representation rings, and then the subject lay essentially dormant for nearly fifteen years. Quite recently, it has been revived by D. Benson, in collaborations with Parker and Carlson. Their interesting results on duality, nilpotent elements and related structural matters conclude the book.

We should now summarize our impression of this enormous compendium (more than 1700 pages in the two volumes combined). With these two books, one has at hand all the basic information needed to start research in any area of the representation theory of finite groups or of orders. In principle, it would be possible to learn representation theory ab initio by reading this work. Of course, one would need a great deal of time to absorb a significant proportion of what is here, and occasional guidance from an expert would be nearly mandatory. The whole work is written clearly and carefully, with few errors and many interesting exercises. The community of algebraists must be deeply indebted to the authors for performing the magnificent service of writing everything down so beautifully.

Reviewer: W.H.Gustafson

### MSC:

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

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

20Cxx | Representation theory of groups |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

16Gxx | Representation theory of associative rings and algebras |

16E20 | Grothendieck groups, \(K\)-theory, etc. |

16P10 | Finite rings and finite-dimensional associative algebras |

16S34 | Group rings |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

20C10 | Integral representations of finite groups |

20C11 | \(p\)-adic representations of finite groups |

20C15 | Ordinary representations and characters |

20C20 | Modular representations and characters |

20C25 | Projective representations and multipliers |