##
**\(G\)-algebras and modular representation theory.**
*(English)*
Zbl 0837.20015

Oxford Mathematical Monographs. Oxford: Clarendon Press. xxvii, 570 p. (1995).

There are two – by now classical – approaches to the theory of modular representations of a finite group: the first, due to R. Brauer, is mainly concerned with blocks of characters, defect groups and (generalized) decomposition numbers; the second, due to J. A. Green, concentrates on indecomposable modules, vertices and sources. It was also shown by Green that central parts of both approaches can be unified by introducing the concept of a \(G\)-algebra, for a finite group \(G\). This unified approach was largely extended by L. Puig who introduced a number of other important concepts like interior \(G\)-algebras, pointed groups and source algebras.

The book by Thévenaz presents an introduction into modular representation theory, starting with this unified approach and treating the classical approaches as special cases. Chapter 1 develops the theory of algebras over a complete local ring, together with their points and exomorphisms. In chapter 2, \(G\)-algebras and interior \(G\)-algebras are introduced, and (local) pointed groups and multiplicity modules are defined. In chapter 3, the induction and defect theory is presented, together with the Puig and the Green correspondence. These topics form the core of the approach. Chapter 4 deals with some additional subjects, like the \(G\)-algebra version of Green’s indecomposability theorem and the concept of covering exomorphisms.

In chapter 5, the theory is specialized to the theory of (indecomposable) modules. It shows that isomorphism classes of indecomposable modules are determined uniquely by three invariants: their vertices, their sources and their multiplicity modules. This result is then applied to the special case of \(p\)-permutation (trivial source) modules. For later applications, the author develops the basic theory of endo-permutation modules. Their classification, due to E. C. Dade, in the case of abelian \(p\)-groups is not treated, however. The author stresses that the theory of \(G\)-algebras applies equally well to diagrams of modules, not only single modules. In particular, he presents Auslander-Reiten sequences from the point of view of defect theory.

In chapter 6, the other classical approach to representation theory is dealt with: the theory of group algebras and blocks. Brauer’s three main theorems are presented. A major theme is the relationship between block algebras and their source algebras, in particular in the special cases of blocks with central or, more generally, normal defect groups. Chapter 7 shows that the theory not only gives a unified approach to classical results but can also be used to prove results which were not accessible by the traditional methods. It does so by proving Puig’s main results on the structure of nilpotent blocks and their source algebras. The last chapter of the book deviates a little from the theme of \(G\)-algebras by showing that the theory can be generalized further by considering (Mackey and) Green functors, introducing pointed groups on these functors and developing a defect theory for them.

The book is not completely self-contained. For basic results on rings and modules, the reader is referred to other textbooks. Also occasionally more specialized results are cited without proof: on Brauer characters, Cartan and decomposition matrices, for example. Moreover, the book concentrates on topics which can be dealt with nicely within the theory; other topics are left out completely. For example, the book does not even compute the number of simple modules of a finite group algebra. Thus its main purpose is to complement other textbooks, not to replace them.

Although the presentation is more than 500 pages long, it is by no means complete. Topics which had to be left out include Puig’s generalization of Brauer’s second main theorem to modules, Linckelmann’s approach to cyclic defect groups and the Külshammer-Puig result on extensions of nilpotent blocks. Nevertheless, the author has done an excellent job in trying to make the theory accessible to a wider audience.

The book by Thévenaz presents an introduction into modular representation theory, starting with this unified approach and treating the classical approaches as special cases. Chapter 1 develops the theory of algebras over a complete local ring, together with their points and exomorphisms. In chapter 2, \(G\)-algebras and interior \(G\)-algebras are introduced, and (local) pointed groups and multiplicity modules are defined. In chapter 3, the induction and defect theory is presented, together with the Puig and the Green correspondence. These topics form the core of the approach. Chapter 4 deals with some additional subjects, like the \(G\)-algebra version of Green’s indecomposability theorem and the concept of covering exomorphisms.

In chapter 5, the theory is specialized to the theory of (indecomposable) modules. It shows that isomorphism classes of indecomposable modules are determined uniquely by three invariants: their vertices, their sources and their multiplicity modules. This result is then applied to the special case of \(p\)-permutation (trivial source) modules. For later applications, the author develops the basic theory of endo-permutation modules. Their classification, due to E. C. Dade, in the case of abelian \(p\)-groups is not treated, however. The author stresses that the theory of \(G\)-algebras applies equally well to diagrams of modules, not only single modules. In particular, he presents Auslander-Reiten sequences from the point of view of defect theory.

In chapter 6, the other classical approach to representation theory is dealt with: the theory of group algebras and blocks. Brauer’s three main theorems are presented. A major theme is the relationship between block algebras and their source algebras, in particular in the special cases of blocks with central or, more generally, normal defect groups. Chapter 7 shows that the theory not only gives a unified approach to classical results but can also be used to prove results which were not accessible by the traditional methods. It does so by proving Puig’s main results on the structure of nilpotent blocks and their source algebras. The last chapter of the book deviates a little from the theme of \(G\)-algebras by showing that the theory can be generalized further by considering (Mackey and) Green functors, introducing pointed groups on these functors and developing a defect theory for them.

The book is not completely self-contained. For basic results on rings and modules, the reader is referred to other textbooks. Also occasionally more specialized results are cited without proof: on Brauer characters, Cartan and decomposition matrices, for example. Moreover, the book concentrates on topics which can be dealt with nicely within the theory; other topics are left out completely. For example, the book does not even compute the number of simple modules of a finite group algebra. Thus its main purpose is to complement other textbooks, not to replace them.

Although the presentation is more than 500 pages long, it is by no means complete. Topics which had to be left out include Puig’s generalization of Brauer’s second main theorem to modules, Linckelmann’s approach to cyclic defect groups and the Külshammer-Puig result on extensions of nilpotent blocks. Nevertheless, the author has done an excellent job in trying to make the theory accessible to a wider audience.

Reviewer: B.Külshammer (Jena)

### MSC:

20C20 | Modular representations and characters |

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

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

16S34 | Group rings |

19A22 | Frobenius induction, Burnside and representation rings |